Sample records for order bellman-isaacs equations

1. On third order integrable vector Hamiltonian equations

Meshkov, A. G.; Sokolov, V. V.

2017-03-01

A complete list of third order vector Hamiltonian equations with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.

2. Next-order structure-function equations

Hill, Reginald J.; Boratav, Olus N.

2001-01-01

Kolmogorov's equation [Dokl. Akad. Nauk SSSR 32, 16 (1941)] relates the two-point second- and third-order velocity structure functions and the energy dissipation rate. The analogous next higher-order two-point equation relates the third- and fourth-order velocity structure functions and the structure function of the product of pressure-gradient difference and two factors of velocity difference, denoted Tijk. The equation is simplified on the basis of local isotropy. Laboratory and numerical simulation data are used to evaluate and compare terms in the equation, examine the balance of the equation, and evaluate components of Tijk. Atmospheric surface-layer data are used to evaluate Tijk in the inertial range. Combined with the random sweeping hypothesis, the equation relates components of the fourth-order velocity structure function. Data show the resultant error of this application of random sweeping. The next-order equation constrains the relationships that have been suggested among components of the fourth-order velocity structure function. The pressure structure function, pressure-gradient correlation, and mean-squared pressure gradient are related to Tijk. Inertial range formulas are discussed.

3. Third-order integrable difference equations generated by a pair of second-order equations

Matsukidaira, Junta; Takahashi, Daisuke

2006-02-01

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson (QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.

4. Tachyons and Higher Order Wave Equations

Barci, D. G.; Bollini, C. G.; Rocca, M. C.

We consider a fourth order wave equation having normal as well as tachyonic solutions. The propagators are, respectively, the Feynman causal function and the Wheeler-Green function (half advanced and half retarded). The latter is consistent with the elimination of tachyons from free asymptotic states. We verify the absence of absorptive parts from convolutions involving the tachyon propagator.

5. Higher Order Equations and Constituent Fields

Barci, D. G.; Bollini, C. G.; Oxman, L. E.; Rocca, M.

We consider a simple wave equation of fourth degree in the D'Alembertian operator. It contains the main ingredients of a general Lorentz-invariant higher order equation, namely, a normal bradyonic sector, a tachyonic state and a pair of complex conjugate modes. The propagators are respectively the Feynman causal function and three Wheeler-Green functions (half-advanced and half-retarded). The latter are Lorentz-invariant and consistent with the elimination of tachyons and complex modes from free asymptotic states. We also verify the absence of absorptive parts from convolutions involving Wheeler propagators.

6. Nonlocal diffusion second order partial differential equations

Benedetti, I.; Loi, N. V.; Malaguti, L.; Taddei, V.

2017-02-01

The paper deals with a second order integro-partial differential equation in Rn with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces.

7. Local dynamics for high-order semilinear hyperbolic equations

Volevich, L. R.; Shirikyan, A. R.

2000-06-01

This paper is devoted to studying high-order semilinear hyperbolic equations. It is assumed that the equation is a small perturbation of an equation with real constant coefficients and that the roots of the full symbol of the unperturbed equation with respect to the variable \\tau dual to time are either separated from the imaginary axis or lie outside the domain \

8. From differential to difference equations for first order ODEs

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Walker, Kevin P.

1991-01-01

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

9. High-Order CESE Methods for the Euler Equations

DTIC Science & Technology

2010-11-01

Technical Paper 3. DATES COVERED (From - To) 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER High-Order CESE Methods for the Euler Equations 5b. GRANT NUMBER...of high-order CESE methods for solving nonlinear hyperbolic partial differential equations. A series of high-order algorithms have been developed...based on a systematic, recursive formulation that achieves fourth-, sixth-, and eighth-order accuracy. The new high-order CESE method shares many

10. A New Factorisation of a General Second Order Differential Equation

ERIC Educational Resources Information Center

Clegg, Janet

2006-01-01

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

11. High-order rogue waves for the Hirota equation

SciTech Connect

Li, Linjing; Wu, Zhiwei; Wang, Lihong; He, Jingsong

2013-07-15

The Hirota equation is better than the nonlinear Schrödinger equation when approximating deep ocean waves. In this paper, high-order rational solutions for the Hirota equation are constructed based on the parameterized Darboux transformation. Several types of this kind of solutions are classified by their structures. -- Highlights: •The determinant representation of the N-fold Darboux transformation of the Hirota equation. •Properties of the fundamental pattern of the higher order rogue wave. •Ring structure and triangular structure of the higher order rogue waves.

12. Third order equations of motion and the Ostrogradsky instability

Motohashi, Hayato; Suyama, Teruaki

2015-04-01

It is known that any nondegenerate Lagrangian containing time derivative terms higher than first order suffers from the Ostrogradsky instability, pathological excitation of positive and negative energy degrees of freedom. We show that, within the framework of analytical mechanics of point particles, any Lagrangian for third order equations of motion, which evades the nondegeneracy condition, still leads to the Ostrogradsky instability. Extension to the case of higher odd order equations of motion is also considered.

13. Vector order parameter in general relativity: Covariant equations

Meierovich, Boris E.

2010-07-01

Phase transitions with spontaneous symmetry breaking and vector order parameter are considered in multidimensional theory of general relativity. Covariant equations, describing the gravitational properties of topological defects, are derived. The topological defects are classified in accordance with the symmetry of the covariant derivative of the vector order parameter. The abilities of the derived equations are demonstrated in application to the braneworld concept. New solutions of the Einstein equations with a transverse vector order parameter are presented. In the vicinity of phase transition, the solutions are found analytically.

14. Vector order parameter in general relativity: Covariant equations

SciTech Connect

Meierovich, Boris E.

2010-07-15

Phase transitions with spontaneous symmetry breaking and vector order parameter are considered in multidimensional theory of general relativity. Covariant equations, describing the gravitational properties of topological defects, are derived. The topological defects are classified in accordance with the symmetry of the covariant derivative of the vector order parameter. The abilities of the derived equations are demonstrated in application to the braneworld concept. New solutions of the Einstein equations with a transverse vector order parameter are presented. In the vicinity of phase transition, the solutions are found analytically.

15. Numerical integration of ordinary differential equations of various orders

NASA Technical Reports Server (NTRS)

Gear, C. W.

1969-01-01

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

16. Second-order variational equations for N-body simulations

Rein, Hanno; Tamayo, Daniel

2016-07-01

First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton's method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection. We provide an implementation of first- and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first- and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

17. On the solutions of fractional order of evolution equations

Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.

2017-01-01

In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.

18. Second-order envelope equation of graphene electrons

Luo, Ji

2014-10-01

A treatment of graphene's electronic states based on the tight-binding method is presented. Like Dirac equation, this treatment uses envelope functions to eliminate crystal potential. Besides, a density-functional-theory Kohn-Sham (KS) orbital of an isolated carbon atom is employed. By locally expanding envelope functions into second-order polynomials and by involving up to third-nearest atoms in calculating orbital integrals, the second-order envelope equation is obtained. This equation does not contain any experimental data except graphene's crystal structure, and its coefficients are determined through several kinds of integrals of the carbon KS orbital. As an improvement, it leads to more accurate energy dispersion than Dirac equation including the triangular warping effect and asymmetry for electrons and holes, and gives the Fermi velocity which is in good agreement with the experimental value.

19. Bounded solutions of a second order evolution equation and applications

Leiva, Hugo

2001-02-01

In this paper we study the following abstract second order differential equation with dissipation in a Hilbert space H: u″+cu'+dA u+kG(u)=P(t), u∈H, t∈R, where c, d and k are positive constants, G:H→H is a Lipschitzian function and P:R→H is a continuous and bounded function. A:D(A)⊂H→H is an unbounded linear operator which is self-adjoint, positive definite and has compact resolvent. Under these conditions we prove that for some values of d, c and k this system has a bounded solution which is exponentially asymptotically stable. Moreover; if P(t) is almost periodic, then this bounded solution is also almost periodic. These results are applied to a very well known second order system partial differential equations; such as the sine-Gordon equation, The suspension bridge equation proposed by Lazer and McKenna, etc.

20. A uniformly second order fast sweeping method for eikonal equations

Luo, Songting

2013-05-01

A uniformly second order method with a local solver based on the piecewise linear discontinuous Galerkin formulation is introduced to solve the eikonal equation with Dirichlet boundary conditions. The method utilizes an interesting phenomenon, referred as the superconvergence phenomenon, that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth. This phenomenon greatly simplifies the local solver based on the discontinuous Galerkin formulation by reducing its local degrees of freedom from two (1-D) (or three (2-D), or four (3-D)) to one with the information of the gradient frozen. When considering the eikonal equation with point-source conditions, we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts, either multiplicatively or additively. One part is known and captures the source singularities; the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations. We extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy, then recover the eikonal with second order accuracy. Numerical examples are presented to demonstrate the performance of the method.

1. Time regularity of the solutions to second order hyperbolic equations

Kinoshita, Tamotu; Taglialatela, Giovanni

2011-04-01

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class γ^{s0} and the Cauchy data belong to γ^{s1}, then the Cauchy problem has a solution in γ^{s0}([0,T^{*}];γ^{s1}({R})) for some T *>0, provided 1≤ s 1≤2-1/ s 0. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤ s 1≤ s 0.

2. Absorbing boundary conditions for second-order hyperbolic equations

NASA Technical Reports Server (NTRS)

Jiang, Hong; Wong, Yau Shu

1989-01-01

A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.

3. Optimization of High-order Wave Equations for Multicore CPUs

SciTech Connect

Williams, Samuel

2011-11-01

This is a simple benchmark to guage the performance of a high-order isotropic wave equation grid. The code is optimized for both SSE and AVX and is parallelized using OpenMP (see Optimization section). Structurally, the benchmark begins, reads a few command-line parameters, allocates and pads the four arrays (current, last, next wave fields, and the spatially varying but isotropic velocity), initializes these arrays, then runs the benchmark proper. The code then benchmarks the naive, SSE (if supported), and AVX (if supported implementations) by applying the wave equation stencil 100 times and taking the average performance. Boundary conditions are ignored and would noiminally be implemented by the user. THus, the benchmark measures only the performance of the wave equation stencil and not a full simulation. The naive implementation is a quadruply (z,y,x, radius) nested loop that can handle arbitrarily order wave equations. The optimized (SSE/AVX) implentations are somewhat more complex as they operate on slabs and include a case statement to select an optimized inner loop depending on wave equation order.

4. Second-order numerical solution of time-dependent, first-order hyperbolic equations

NASA Technical Reports Server (NTRS)

Shah, Patricia L.; Hardin, Jay

1995-01-01

A finite difference scheme is developed to find an approximate solution of two similar hyperbolic equations, namely a first-order plane wave and spherical wave problem. Finite difference approximations are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.

5. Higher order matrix differential equations with singular coefficient matrices

SciTech Connect

Fragkoulis, V. C.; Kougioumtzoglou, I. A.; Pantelous, A. A.; Pirrotta, A.

2015-03-10

In this article, the class of higher order linear matrix differential equations with constant coefficient matrices and stochastic process terms is studied. The coefficient of the highest order is considered to be singular; thus, rendering the response determination of such systems in a straightforward manner a difficult task. In this regard, the notion of the generalized inverse of a singular matrix is used for determining response statistics. Further, an application relevant to engineering dynamics problems is included.

6. Solving Second-Order Differential Equations with Variable Coefficients

ERIC Educational Resources Information Center

Wilmer, A., III; Costa, G. B.

2008-01-01

A method is developed in which an analytical solution is obtained for certain classes of second-order differential equations with variable coefficients. By the use of transformations and by repeated iterated integration, a desired solution is obtained. This alternative method represents a different way to acquire a solution from classic power…

7. Neumann problems for second order ordinary differential equations across resonance

Yong, Li; Huaizhong, Wang

1995-05-01

This paper deals with the existence-uniqueness problem to Neumann problems for second order ordinary differential equations probably across resonance. By the optimal control theory method, some global optimality results about the unique solvability for such boundary value problems are established.

8. Negative-order Korteweg-de Vries equations.

PubMed

Qiao, Zhijun; Fan, Engui

2012-07-01

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions.

9. Higher order parabolic approximations of the reduced wave equation

NASA Technical Reports Server (NTRS)

Mcaninch, G. L.

1986-01-01

Asymptotic solutions of order k to the nth are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory, and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.

10. Second-order neutral impulsive stochastic evolution equations with delay

Ren, Yong; Sun, Dandan

2009-10-01

In this paper, we study the second-order neutral stochastic evolution equations with impulsive effect and delay (SNSEEIDs). We establish the existence and uniqueness of mild solutions to SNSEEIDs under non-Lipschitz condition with Lipschitz condition being considered as a special case by the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the theory.

11. Oscillation properties of some functional fourth order hyperbolic differential equations

Petrova, Z.

2012-11-01

In this paper, we apply our recent results for fourth order functional ordinary differential equations and inequalities and obtain sufficient conditions for oscillation of all sufficiently smooth solutions of the following equation ∑ i+j = 2;4ai,j∂i+ju(x,y)/∂xi∂yj+ ∑ i = 1nbi(x,y)u(x-σi,y-τi)+c(x,y,u) = f(x,y), where x>0,y>0,ai,j∈R,σi≥0 and τi ≥ 0 are constants for all the indices. Also, we suppose that n∈N,bi(x,y)∈C(R+2;R+), ∀i = 1-n;c(x,y,u)∈C(R+2,R;R) and f(x,y)∈C(R+2;R). In particular, we establish sufficient conditions for the distribution of zeros this equation.

12. On fractional Langevin equation involving two fractional orders

Baghani, Omid

2017-01-01

In numerical analysis, it is frequently needed to examine how far a numerical solution is from the exact one. To investigate this issue quantitatively, we need a tool to measure the difference between them and obviously this task is accomplished by the aid of an appropriate norm on a certain space of functions. For example, Sobolev spaces are indispensable part of theoretical analysis of partial differential equations and boundary integral equations, as well as are necessary for the analysis of some numerical methods for the solving of such equations. But most of articles that appear in this field usually use ‖.‖∞ in the space of C[a, b] which is very restrictive. In this paper, we introduce a new norm that is convenient for the fractional and singular differential equations. Using this norm, the existence and uniqueness of initial value problems for nonlinear Langevin equation with two different fractional orders are studied. In fact, the obtained results could be used for the classical cases. Finally, by two examples we show that we cannot always speak about the existence and uniqueness of solutions just by using the previous methods.

13. Einstein-Weyl spaces and third-order differential equations

Tod, K. P.

2000-08-01

The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, "On the null surface formalism," Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., "Null surfaces formation in 3D," J. Math Phys. (submitted)] are extended to describe Einstein-Weyl spaces, following Cartan [E. Cartan, "Les espaces généralisées et l'integration de certaines classes d'equations différentielles," C. R. Acad. Sci. 206, 1425-1429 (1938); "La geometria de las ecuaciones diferenciales de tercer order," Rev. Mat. Hispano-Am. 4, 1-31 (1941)]. In the resulting formalism, Einstein-Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein-Weyl spaces are given.

14. Stabilisation of second-order nonlinear equations with variable delay

Berezansky, Leonid; Braverman, Elena; Idels, Lev

2015-08-01

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.

15. A Solution to the Fundamental Linear Fractional Order Differential Equation

NASA Technical Reports Server (NTRS)

Hartley, Tom T.; Lorenzo, Carl F.

1998-01-01

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

16. Order Reduction of the Chemical Master Equation via Balanced Realisation

PubMed Central

López-Caamal, Fernando; Marquez-Lago, Tatiana T.

2014-01-01

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator. PMID:25121581

17. Order reduction of the chemical master equation via balanced realisation.

PubMed

López-Caamal, Fernando; Marquez-Lago, Tatiana T

2014-01-01

We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.

18. Second order upwind Lagrangian particle method for Euler equations

SciTech Connect

Samulyak, Roman; Chen, Hsin -Chiang; Yu, Kwangmin

2016-06-01

A new second order upwind Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) an upwind second-order particle-based algorithm with limiter, providing accuracy and long term stability, and (c) accurate resolution of states at free interfaces. In conclusion, numerical verification tests demonstrating the convergence order for fixed domain and free surface problems are presented.

19. Second order upwind Lagrangian particle method for Euler equations

DOE PAGES

Samulyak, Roman; Chen, Hsin -Chiang; Yu, Kwangmin

2016-06-01

A new second order upwind Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) an upwind second-order particle-based algorithm with limiter, providing accuracy and longmore » term stability, and (c) accurate resolution of states at free interfaces. In conclusion, numerical verification tests demonstrating the convergence order for fixed domain and free surface problems are presented.« less

20. Point model equations for neutron correlation counting: Extension of Böhnel's equations to any order

Favalli, Andrea; Croft, Stephen; Santi, Peter

2015-09-01

Various methods of autocorrelation neutron analysis may be used to extract information about a measurement item containing spontaneously fissioning material. The two predominant approaches being the time correlation analysis (that make use of a coincidence gate) methods of multiplicity shift register logic and Feynman sampling. The common feature is that the correlated nature of the pulse train can be described by a vector of reduced factorial multiplet rates. We call these singlets, doublets, triplets etc. Within the point reactor model the multiplet rates may be related to the properties of the item, the parameters of the detector, and basic nuclear data constants by a series of coupled algebraic equations - the so called point model equations. Solving, or inverting, the point model equations using experimental calibration model parameters is how assays of unknown items is performed. Currently only the first three multiplets are routinely used. In this work we develop the point model equations to higher order multiplets using the probability generating functions approach combined with the general derivative chain rule, the so called Faà di Bruno Formula. Explicit expression up to 5th order are provided, as well the general iterative formula to calculate any order. This work represents the first necessary step towards determining if higher order multiplets can add value to nondestructive measurement practice for nuclear materials control and accountancy.

1. Point model equations for neutron correlation counting: Extension of Böhnel's equations to any order

SciTech Connect

Favalli, Andrea; Croft, Stephen; Santi, Peter

2015-06-15

Various methods of autocorrelation neutron analysis may be used to extract information about a measurement item containing spontaneously fissioning material. The two predominant approaches being the time correlation analysis (that make use of a coincidence gate) methods of multiplicity shift register logic and Feynman sampling. The common feature is that the correlated nature of the pulse train can be described by a vector of reduced factorial multiplet rates. We call these singlets, doublets, triplets etc. Within the point reactor model the multiplet rates may be related to the properties of the item, the parameters of the detector, and basic nuclear data constants by a series of coupled algebraic equations – the so called point model equations. Solving, or inverting, the point model equations using experimental calibration model parameters is how assays of unknown items is performed. Currently only the first three multiplets are routinely used. In this work we develop the point model equations to higher order multiplets using the probability generating functions approach combined with the general derivative chain rule, the so called Faà di Bruno Formula. Explicit expression up to 5th order are provided, as well the general iterative formula to calculate any order. This study represents the first necessary step towards determining if higher order multiplets can add value to nondestructive measurement practice for nuclear materials control and accountancy.

2. Point model equations for neutron correlation counting: Extension of Böhnel's equations to any order

DOE PAGES

Favalli, Andrea; Croft, Stephen; Santi, Peter

2015-06-15

Various methods of autocorrelation neutron analysis may be used to extract information about a measurement item containing spontaneously fissioning material. The two predominant approaches being the time correlation analysis (that make use of a coincidence gate) methods of multiplicity shift register logic and Feynman sampling. The common feature is that the correlated nature of the pulse train can be described by a vector of reduced factorial multiplet rates. We call these singlets, doublets, triplets etc. Within the point reactor model the multiplet rates may be related to the properties of the item, the parameters of the detector, and basic nuclearmore » data constants by a series of coupled algebraic equations – the so called point model equations. Solving, or inverting, the point model equations using experimental calibration model parameters is how assays of unknown items is performed. Currently only the first three multiplets are routinely used. In this work we develop the point model equations to higher order multiplets using the probability generating functions approach combined with the general derivative chain rule, the so called Faà di Bruno Formula. Explicit expression up to 5th order are provided, as well the general iterative formula to calculate any order. This study represents the first necessary step towards determining if higher order multiplets can add value to nondestructive measurement practice for nuclear materials control and accountancy.« less

3. The first-order Euler-Lagrange equations and some of their uses

2016-12-01

In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.

4. A high-order accurate embedded boundary method for first order hyperbolic equations

Mattsson, Ken; Almquist, Martin

2017-04-01

A stable and high-order accurate embedded boundary method for first order hyperbolic equations is derived. Where the grid-boundaries and the physical boundaries do not coincide, high order interpolation is used. The boundary stencils are based on a summation-by-parts framework, and the boundary conditions are imposed by the SAT penalty method, which guarantees linear stability for one-dimensional problems. Second-, fourth-, and sixth-order finite difference schemes are considered. The resulting schemes are fully explicit. Accuracy and numerical stability of the proposed schemes are demonstrated for both linear and nonlinear hyperbolic systems in one and two spatial dimensions.

5. Pseudospectral collocation methods for fourth order differential equations

NASA Technical Reports Server (NTRS)

Malek, Alaeddin; Phillips, Timothy N.

1994-01-01

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

6. Fourth order wave equations with nonlinear strain and source terms

Liu, Yacheng; Xu, Runzhang

2007-07-01

In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u0)[greater-or-equal, slanted]0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.

7. Octonic second-order equations of relativistic quantum mechanics

SciTech Connect

Mironov, Victor L.; Mironov, Sergey V.

2009-01-15

We demonstrate a generalization of relativistic quantum mechanics using eight-component value ''octons'' that generate an associative noncommutative spatial algebra. It is shown that the octonic second-order equation for the eight-component octonic wave function, obtained from the Einstein relation for energy and momentum, describes particles with spin 1/2. It is established that the octonic wave function of a particle in the state with defined spin projection has a specific spatial structure that takes the form of an octonic oscillator with two spatial polarizations: longitudinal linear and transverse circular.

8. Collinear limits beyond the leading order from the scattering equations

Nandan, Dhritiman; Plefka, Jan; Wormsbecher, Wadim

2017-02-01

The structure of tree-level scattering amplitudes for collinear massless bosons is studied beyond their leading splitting function behavior. These near-collinear limits at sub-leading order are best studied using the Cachazo-He-Yuan (CHY) formulation of the S-matrix based on the scattering equations. We compute the collinear limits for gluons, gravitons and scalars. It is shown that the CHY integrand for an n-particle gluon scattering amplitude in the collinear limit at sub-leading order is expressed as a convolution of an ( n - 1)-particle gluon integrand and a collinear kernel integrand, which is universal. Our representation is shown to obey recently proposed amplitude relations in which the collinear gluons of same helicity are replaced by a single graviton. Finally, we extend our analysis to effective field theories and study the collinear limit of the non-linear sigma model, Einstein-Maxwell-Scalar and Yang-Mills-Scalar theory.

9. Partially Ordered Sets of Quantum Measurements and the Dirac Equation

Knuth, Kevin H.

2012-02-01

Events can be ordered according to whether one event influences another. This results in a partially ordered set (poset) of events often referred to as a causal set. In this framework, an observer can be represented by a chain of events. Quantification of events and pairs of events, referred to as intervals, can be performed by projecting them onto an observer chain, or even a pair of observer chains, which in specific situations leads to a Minkowski metric replete with Lorentz transformations (Bahreyni & Knuth, 2011. APS B21.00007). In this work, we unify this picture with the Process Calculus, which coincides with the Feynman rules of quantum mechanics (Goyal, Knuth, Skilling, 2010, arXiv:0907.0909; Goyal & Knuth, Symmetry 2011, 3(2), 171), by considering quantum measurements to be events. This is performed by quantifying pairs of events, which represent transitions, with a pair of numbers, or a quantum amplitude. In the 1+1D case this results in the Feynman checkerboard model of the Dirac equation (Feynman & Hibbs, 1965). We further demonstrate that in the case of 3+1 dimensions, we recover Bialnycki-Birula's (1994, Phys. Rev. D, 49(12), 6920) body-centered cubic cellular automata model of the Dirac equation studied more recently by Earle (2011, arXiv:1102.1200v1).

10. Generation and application of the equations of condition for high order Runge-Kutta methods

NASA Technical Reports Server (NTRS)

Haley, D. C.

1972-01-01

This thesis develops the equations of condition necessary for determining the coefficients for Runge-Kutta methods used in the solution of ordinary differential equations. The equations of condition are developed for Runge-Kutta methods of order four through order nine. Once developed, these equations are used in a comparison of the local truncation errors for several sets of Runge-Kutta coefficients for methods of order three up through methods of order eight.

11. Higher-order Hamiltonian fluid reduction of Vlasov equation

SciTech Connect

Perin, M.; Chandre, C.; Morrison, P.J.; Tassi, E.

2014-09-15

From the Hamiltonian structure of the Vlasov equation, we build a Hamiltonian model for the first three moments of the Vlasov distribution function, namely, the density, the momentum density and the specific internal energy. We derive the Poisson bracket of this model from the Poisson bracket of the Vlasov equation, and we discuss the associated Casimir invariants.

12. Transverse Laser Patterns: Quantitative Validation of the Order Parameter Equation

DTIC Science & Technology

2006-04-28

Fundamentos Matemáticos E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid 28040 Madrid, SPAIN Contents 1 Introduction...derivation of the OP equations from the MB equations is based on the slow envelope assumption, i.e., on the assumption that the amplitudes of the

13. Unsplit complex frequency shifted perfectly matched layer for second-order wave equation using auxiliary differential equations.

PubMed

Gao, Yingjie; Zhang, Jinhai; Yao, Zhenxing

2015-12-01

The complex frequency shifted perfectly matched layer (CFS-PML) can improve the absorbing performance of PML for nearly grazing incident waves. However, traditional PML and CFS-PML are based on first-order wave equations; thus, they are not suitable for second-order wave equation. In this paper, an implementation of CFS-PML for second-order wave equation is presented using auxiliary differential equations. This method is free of both convolution calculations and third-order temporal derivatives. As an unsplit CFS-PML, it can reduce the nearly grazing incidence. Numerical experiments show that it has better absorption than typical PML implementations based on second-order wave equation.

14. A fourth-order box method for solving the boundary layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1977-01-01

A fourth order box method for calculating high accuracy numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations is presented. The method is the natural extension of the second order Keller Box scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary layer equations. Numerical results for high accuracy test cases show the method to be significantly faster than other higher order and second order methods.

15. Higher-order Schrödinger and Hartree–Fock equations

SciTech Connect

Carles, Rémi; Lucha, Wolfgang; Moulay, Emmanuel

2015-12-15

The domain of validity of the higher-order Schrödinger equations is analyzed for harmonic-oscillator and Coulomb potentials as typical examples. Then, the Cauchy theory for higher-order Hartree–Fock equations with bounded and Coulomb potentials is developed. Finally, the existence of associated ground states for the odd-order equations is proved. This renders these quantum equations relevant for physics.

16. APL and the numerical solution of high-order linear differential equations

Gershenfeld, Neil A.; Schadler, Edward H.; Bilaniuk, O. M.

1983-08-01

An Nth-order linear ordinary differential equation is rewritten as a first-order equation in an N×N matrix. Taking advantage of the matrix manipulation strength of the APL language this equation is then solved directly, yielding a great simplification over the standard procedure of solving N coupled first-order scalar equations. This eases programming and results in a more intuitive algorithm. Example applications of a program using the technique are given from quantum mechanics and control theory.

17. Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

ERIC Educational Resources Information Center

Seaman, Brian; Osler, Thomas J.

2004-01-01

A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

18. An efficient technique for higher order fractional differential equation.

PubMed

2016-01-01

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of [Formula: see text]-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

19. Exploration of POD-Galerkin Techniques for Developing Reduced Order Models of Reaction-Advection Equations

DTIC Science & Technology

2014-04-01

downstream boundary (when needed) is obtained by extrapolation, taking into account the hyperbolic character of the equation . By separating the...for Developing Reduced Order Models of Reaction-Advection Equations 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT...advection scalar equation is used as a representative equation to investigate the overall approach. Both linear and nonlinear model equations are

20. Discrete integration of continuous Kalman filtering equations for time invariant second-order structural systems

NASA Technical Reports Server (NTRS)

Park, K. C.; Belvin, W. Keith

1990-01-01

A general form for the first-order representation of the continuous second-order linear structural-dynamics equations is introduced to derive a corresponding form of first-order continuous Kalman filtering equations. Time integration of the resulting equations is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete Kalman filtering equations involving only symmetric sparse N x N solution matrices.

1. Solving cochlear mechanics problems with higher-order differential equations.

PubMed

de Boer, E; van Bienema, E

1982-11-01

Since most "exact" solution methods for cochlear models are rather unwieldy, they do not lend themselves to easy and multi-purpose application. In this paper a new solution method is described that is more flexible in this respect. A three-dimensional cochlear model is considered. It can be described by an integral equation in terms of the wavenumber k. The kernel Q (k) of that equation is approximated by a rational function of k and this makes it possible to reformulate the problem as a differential equation. The latter can be solved by a straightforward and well-known method. Results of computations with this technique are presented in two forms: an overview of the entire cochlear wave pattern and a detailed representation of the response peak. The method is also used to determine whether a discernible reflected wave is produced in the cochlea or not. For this purpose the wavenumber spectrum of the cochlear wave is studied: it is found to be a one-sided function of k. With surprisingly simple means it is thus shown that no appreciable reflection occurs from the inhomogeneity that is characteristic in cochlear wave propagation. This holds true for values of damping constant delta as low as 0.01, a factor of 5 smaller than is commonly used in cochlear modeling.

2. Soliton solutions of the KdV equation with higher-order corrections

Wazwaz, Abdul-Majid

2010-10-01

In this work, the Korteweg-de Vries (KdV) equation with higher-order corrections is examined. We studied the KdV equation with first-order correction and that with second-order correction that include the terms of the fifth-order Lax, Sawada-Kotera and Caudrey-Dodd-Gibbon equations. The simplified form of the bilinear method was used to show the integrability of the first-order models and therefore to obtain multiple soliton solutions for each one. The obstacles to integrability of some of the models with second-order corrections are examined as well.

3. Integrability Test and Travelling-Wave Solutions of Higher-Order Shallow- Water Type Equations

2010-04-01

We apply the Weiss-Tabor-Carnevale (WTC) Painlevé test to members of a sequence of higher-order shallow-water type equations. We obtain the result that the equations considered are non-integrable, although compatibility conditions at real resonances are satisfied. We also construct travelling-wave solutions for these and related equations.

4. Compact high-order schemes for the Euler equations

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Kumar, Ajay

1988-01-01

An implicit approximate factorization (AF) algorithm is constructed which has the following characteristics. In 2-D: the scheme is unconditionally stable, has a 3 x 3 stencil and at steady state has a fourth order spatial accuracy. The temporal evolution is time accurate either to first or second order through choice of parameter. In 3-D: the scheme has almost the same properties as in 2-D except that it is now only conditionally stable, with the stability condition (the CFL number) being dependent on the cell aspect ratios, delta y/delta x and delta z/delta x. The stencil is still compact and fourth order accuracy at steady state is maintained.

5. Compact high order schemes for the Euler equations

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Kumar, Ajay

1988-01-01

An implicit approximate factorization (AF) algorithm is constructed which has the following characteistics. In 2-D: The scheme is unconditionally stable, has a 3 x 3 stencil and at steady state has a fourth order spatial accuracy. The temporal evolution is time accurate either to first or second order through choice of parameter. In 3-D: The scheme has almost the same properties as in 2-D except that it is now only conditionally stable, with the stability condition (the CFL number) being dependent on the cell aspect ratios, delta y/delta x and delta z/delta x. The stencil is still compact and fourth order accuracy at steady state is maintained. Numerical experiments on a 2-D shock-reflection problem show the expected improvement over lower order schemes, not only in accuracy (measured by the L sub 2 error) but also in the dispersion. It is also shown how the same technique is immediately extendable to Runge-Kutta type schemes resulting in improved stability in addition to the enhanced accuracy.

6. Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

ERIC Educational Resources Information Center

Robin, W.

2007-01-01

The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

7. Second-order discrete Kalman filtering equations for control-structure interaction simulations

NASA Technical Reports Server (NTRS)

Park, K. C.; Belvin, W. Keith; Alvin, Kenneth F.

1991-01-01

A general form for the first-order representation of the continuous, second-order linear structural dynamics equations is introduced in order to derive a corresponding form of first-order Kalman filtering equations (KFE). Time integration of the resulting first-order KFE is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete KFE involving only symmetric, N x N solution matrix.

8. Consistency of Equations in the Second-Order Gauge-Invariant Cosmological Perturbation Theory

Nakamura, K.

2009-06-01

Along the general framework of the gauge-invariant perturbation theory developed in the papers [K.~Nakamura, Prog.~Theor.~Phys. 110 (2003), 723; Prog.~Theor.~Phys. 113 (2005), 481], we rederive the second-order Einstein equation on four-dimensional homogeneous isotropic background universe in a gauge-invariant manner without ignoring any mode of perturbations. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We also confirmed the consistency of all the equations of the second-order Einstein equation and the equations of motion for matter fields, which are derived in the paper [K.~Nakamura, arXiv:0804.3840]. This confirmation implies that all the derived equations of the second order are self-consistent and these equations are correct in this sense.

9. Localized modes of the Hirota equation: Nth order rogue wave and a separation of variable technique

Mu, Gui; Qin, Zhenyun; Chow, Kwok Wing; Ee, Bernard K.

2016-10-01

The Hirota equation is a special extension of the intensively studied nonlinear Schrödinger equation, by incorporating third order dispersion and one form of the self-steepening effect. Higher order rogue waves of the Hirota equation can be calculated theoretically through a Darboux-dressing transformation by a separation of variable approach. A Taylor expansion is used and no derivative calculation is invoked. Furthermore, stability of these rogue waves is studied computationally. By tracing the evolution of an exact solution perturbed by random noise, it is found that second order rogue waves are generally less stable than first order ones.

10. Fractional-order difference equations for physical lattices and some applications

SciTech Connect

Tarasov, Vasily E.

2015-10-15

Fractional-order operators for physical lattice models based on the Grünwald-Letnikov fractional differences are suggested. We use an approach based on the models of lattices with long-range particle interactions. The fractional-order operators of differentiation and integration on physical lattices are represented by kernels of lattice long-range interactions. In continuum limit, these discrete operators of non-integer orders give the fractional-order derivatives and integrals with respect to coordinates of the Grünwald-Letnikov types. As examples of the fractional-order difference equations for physical lattices, we give difference analogs of the fractional nonlocal Navier-Stokes equations and the fractional nonlocal Maxwell equations for lattices with long-range interactions. Continuum limits of these fractional-order difference equations are also suggested.

11. Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities.

PubMed

Khare, Avinash; Rasmussen, Kim Ø; Salerno, Mario; Samuelsen, Mogens R; Saxena, Avadh

2006-07-01

A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.

12. A novel unsplit perfectly matched layer for the second-order acoustic wave equation.

PubMed

Ma, Youneng; Yu, Jinhua; Wang, Yuanyuan

2014-08-01

When solving acoustic field equations by using numerical approximation technique, absorbing boundary conditions (ABCs) are widely used to truncate the simulation to a finite space. The perfectly matched layer (PML) technique has exhibited excellent absorbing efficiency as an ABC for the acoustic wave equation formulated as a first-order system. However, as the PML was originally designed for the first-order equation system, it cannot be applied to the second-order equation system directly. In this article, we aim to extend the unsplit PML to the second-order equation system. We developed an efficient unsplit implementation of PML for the second-order acoustic wave equation based on an auxiliary-differential-equation (ADE) scheme. The proposed method can benefit to the use of PML in simulations based on second-order equations. Compared with the existing PMLs, it has simpler implementation and requires less extra storage. Numerical results from finite-difference time-domain models are provided to illustrate the validity of the approach.

13. Approach to first-order exact solutions of the Ablowitz-Ladik equation.

PubMed

Ankiewicz, Adrian; Akhmediev, Nail; Lederer, Falk

2011-05-01

We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solutions of the nonlinear Schrödinger equation (NLSE).

14. Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrödinger Equations

Asad-uz-zaman, M.; Chachou Samet, H.; Khawaja, U. Al

2016-08-01

We derive the Lax pairs and integrability conditions of the nonlinear Schrödinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrödinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrödinger equation.

15. Initial-value problem for a linear ordinary differential equation of noninteger order

SciTech Connect

Pskhu, Arsen V

2011-04-30

An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the equation; these conditions are necessary for the equation under consideration. The problem is reduced to an integral equation; an explicit representation of the solution in terms of the Wright function is constructed. As a consequence of these results, necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. Bibliography: 7 titles.

16. High-Order Central WENO Schemes for 1D Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Bryan A. (Technical Monitor)

2002-01-01

In this paper we derive fully-discrete Central WENO (CWENO) schemes for approximating solutions of one dimensional Hamilton-Jacobi (HJ) equations, which combine our previous works. We introduce third and fifth-order accurate schemes, which are the first central schemes for the HJ equations of order higher than two. The core ingredient is the derivation of our schemes is a high-order CWENO reconstructions in space.

17. Higher-order nonlinear Schrodinger equations for simulations of surface wavetrains

Slunyaev, Alexey

2016-04-01

Numerous recent results of numerical and laboratory simulations of waves on the water surface claim that solutions of the weakly nonlinear theory for weakly modulated waves in many cases allow a smooth generalization to the conditions of strong nonlinearity and dispersion, even when the 'envelope' is difficult to determine. The conditionally 'strongly nonlinear' high-order asymptotic equations still imply the smallness of the parameter employed in the asymptotic series. Thus at some (unknown a priori) level of nonlinearity and / or dispersion the asymptotic theory breaks down; then the higher-order corrections become useless and may even make the description worse. In this paper we use the higher-order nonlinear Schrodinger (NLS) equation, derived in [1] (the fifth-order NLS equation, or next-order beyond the classic Dysthe equation [2]), for simulations of modulated deep-water wave trains, which attain very large steepness (below or beyond the breaking limit) due to the Benjamin - Feir instability. The results are compared with fully nonlinear simulations of the potential Euler equations as well as with the weakly nonlinear theories represented by the nonlinear Schrodinger equation and the classic Dysthe equation with full linear dispersion [2]. We show that the next-order Dysthe equation can significantly improve the description of strongly nonlinear wave dynamics compared with the lower-order asymptotic models. [1] A.V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water. JETP 101, 926-941 (2005). [2] K. Trulsen, K.B. Dysthe, A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281-289 (1996).

18. A posteriori error estimation for hp -adaptivity for fourth-order equations

Moore, Peter K.; Rangelova, Marina

2010-04-01

A posteriori error estimates developed to drive hp-adaptivity for second-order reaction-diffusion equations are extended to fourth-order equations. A C^1 hierarchical finite element basis is constructed from Hermite-Lobatto polynomials. A priori estimates of the error in several norms for both the interpolant and finite element solution are derived. In the latter case this requires a generalization of the well-known Aubin-Nitsche technique to time-dependent fourth-order equations. We show that the finite element solution and corresponding Hermite-Lobatto interpolant are asymptotically equivalent. A posteriori error estimators based on this equivalence for solutions at two orders are presented. Both are shown to be asymptotically exact on grids of uniform order. These estimators can be used to control various adaptive strategies. Computational results for linear steady-state and time-dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings.

19. Fourth-order master equation for a charged harmonic oscillator coupled to an electromagnetic field

Kurt, Arzu; Eryigit, Resul

Using Krylov averaging method, we have derived a fourth-order master equation for a charged harmonic oscillator weakly coupled to an electromagnetic field. Interaction is assumed to be of velocity coupling type which also takes into account the diagmagnetic term. Exact analytical expressions have been obtained for the second, the third and the fourth-order corrections to the diffusion and the drift terms of the master equation. We examined the validity range of the second order master equation in terms of the coupling constant and the bath cutoff frequency and found that for the most values of those parameters, the contribution from the third and the fourth order terms have opposite signs and cancel each other. Inclusion of the third and the fourth-order terms is found to not change the structure of the master equation. Bolu, Turkey.

20. Un-collided-flux preconditioning for the first order transport equation

SciTech Connect

Rigley, M.; Koebbe, J.; Drumm, C.

2013-07-01

Two codes were tested for the first order neutron transport equation using finite element methods. The un-collided-flux solution is used as a preconditioner for each of these methods. These codes include a least squares finite element method and a discontinuous finite element method. The performance of each code is shown on problems in one and two dimensions. The un-collided-flux preconditioner shows good speedup on each of the given methods. The un-collided-flux preconditioner has been used on the second-order equation, and here we extend those results to the first order equation. (authors)

1. Stationary axisymmetric solutions involving a third order equation irreducible to Painlevé transcendents

Gariel, J.; Marcilhacy, G.; Santos, N. O.

2008-02-01

We extend the method of separation of variables, studied by Léauté and Marcilhacy [Ann. Inst. Henri Poincare, Sect. A 331, 363 (1979)], to obtain transcendent solutions of the field equations for stationary axisymmetric systems. These solutions depend on transcendent functions satisfying a third order differential equation. For some solutions this equation satisfies the necessary conditions, but not sufficient, to have fixed critical points.

2. On the basic equations for the second-order modeling of compressible turbulence

NASA Technical Reports Server (NTRS)

Liou, W. W.; Shih, T.-H.

1991-01-01

Equations for the mean and turbulent quantities for compressible turbulent flows are derived. Both the conventional Reynolds average and the mass-weighted, Favre average were employed to decompose the flow variable into a mean and a turbulent quality. These equations are to be used later in developing second order Reynolds stress models for high speed compressible flows. A few recent advances in modeling some of the terms in the equations due to compressibility effects are also summarized.

3. Special polynomials associated with the fourth order analogue to the Painlevé equations

Kudryashov, Nikolai A.; Demina, Maria V.

2007-04-01

Rational solutions of the fourth order analogue to the Painlevé equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulae for their coefficients and degrees are derived. It is shown that special solutions of the Fordy Gibbons, the Caudrey Dodd Gibbon and the Kaup Kupershmidt equations can be expressed through solutions of the equation studied.

4. New solutions for two integrable cases of a generalized fifth-order nonlinear equation

Wazwaz, Abdul-Majid

2015-05-01

Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.

5. On Picard boundary value problem for second order asymptotically homogeneous equations

Dong, Y.

Using the Leray-Schauder continuation principle we give some existence results for the Picard boundary value problem of second order asymptotically homogeneous equations. Some previous results by Tippett, Gaines-Mawhin, Lazer-Leach will be extended.

6. The stability of numerical methods for second order ordinary differential equations

NASA Technical Reports Server (NTRS)

Gear, C. W.

1978-01-01

An important characterization of a numerical method for first order ODE's is the region of absolute stability. If all eigenvalues of the linear problem dy/dt = Ay are inside this region, the numerical method is stable. If the second order system d/dt(dy/dt) = 2Ady/dt - By is solved as a first order system, the same result applies to the eigenvalues of the generalized eigenvalue problem (lambda-squared)I 2(lambda)A + B. No such region exists for general methods for second order equations, but in some cases a region of absolute stability can be defined for methods for the single second order equation d/dt(dy/dt) = 2ady/dt - by. The absence of a region of absolute stability can occur when different members of a system of first order equations are solved by different methods.

7. Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions.

PubMed

Ankiewicz, Adrian; Wang, Yan; Wabnitz, Stefan; Akhmediev, Nail

2014-01-01

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota and the Lakshmanan-Porsezian-Daniel equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue wave solutions of the corresponding equations.

8. Spectral methods for some singularly perturbed third order ordinary differential equations

Temsah, R.

2008-01-01

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton?s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.

9. First-Order System Least-Squares for the Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Bochev, P.; Cai, Z.; Manteuffel, T. A.; McCormick, S. F.

1996-01-01

This paper develops a least-squares approach to the solution of the incompressible Navier-Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier-Stokes equations as a first-order system by introducing a velocity flux variable and associated curl and trace equations. We show that the resulting system is well-posed, and that an associated least-squares principle yields optimal discretization error estimates in the H(sup 1) norm in each variable (including the velocity flux) and optimal multigrid convergence estimates for the resulting algebraic system.

10. Novel asymmetric representation method for solving the higher-order Ginzburg-Landau equation

PubMed Central

Wong, Pring; Pang, Lihui; Wu, Ye; Lei, Ming; Liu, Wenjun

2016-01-01

In ultrafast optics, optical pulses are generated to be of shorter pulse duration, which has enormous significance to industrial applications and scientific research. The ultrashort pulse evolution in fiber lasers can be described by the higher-order Ginzburg-Landau (GL) equation. However, analytic soliton solutions for this equation have not been obtained by use of existing methods. In this paper, a novel method is proposed to deal with this equation. The analytic soliton solution is obtained for the first time, and is proved to be stable against amplitude perturbations. Through the split-step Fourier method, the bright soliton solution is studied numerically. The analytic results here may extend the integrable methods, and could be used to study soliton dynamics for some equations in other disciplines. It may also provide the other way to obtain two-soliton solutions for higher-order GL equations. PMID:27086841

11. A second-order differential equation for a point charged particle

Torromé, Ricardo Gallego

A model for the dynamics of a classical point charged particle interacting with higher order jet fields is introduced. In this model, the dynamics of the charged particle is described by an implicit ordinary second-order differential equation. Such equation is free of run-away and pre-accelerated solutions of Dirac’s type. The theory is Lorentz invariant, compatible with the first law of Newton and Larmor’s power radiation formula. Few implications of the new equation in the phenomenology of non-neutral plasmas is considered.

12. A fifth order implicit method for the numerical solution of differential-algebraic equations

Skvortsov, L. M.

2015-06-01

An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.

13. Asymptotic solutions of a fourth—order analogue for the Painlevé equations

Gaiur, I. Yu; Kudryashov, N. A.

2017-01-01

Asymptotic solutions of a fourth-order analogue for the Painlevé equations that is self-similar reduction of the modified Sawada-Kotera and Kaup-Kupershmidt equation is considered. The Boutroux variables of two types have been found which allows us to find asymptotic solutions of the equation in the neighbourhood of the infinity. It was shown that asymptotic of self-similar solution for the modified Sawada-Kotera and Kaup-Kupershmidt equations can be determined as solutions of autonomous differential equations. Asymptotic solutions expressed by elementary functions have been found too. Besides asymptotic solutions expressed by logarithmic derivative of two elliptic Weierstrass functions have been found. Connection between obtained asymptotic solutions and asymptotic solutions of the Sawada-Kotera and Kaup-Kupershmidt equations has been discussed.

14. A model of the nerve impulse using two first-order differential equations

Hindmarsh, J. L.; Rose, R. M.

1982-03-01

The Hodgkin-Huxley model1 of the nerve impulse consists of four coupled nonlinear differential equations, six functions and seven constants. Because of the complexity of these equations and the necessity for numerical solution, it is difficult to use them in simulations of interactions in small neural networks. Thus, it would be useful to have a second-order differential equation which predicted correctly properties such as the frequency-current relationship. Fitzhugh2 introduced a second-order model of the nerve impulse, but his equations predict an action potential duration which is similar to the inter-spike interval3 and they do not give a reasonable frequency-current relationship. To develop a second-order model having few parameters but which does not have these disadvantages, we have generalized the second-order Fitzhugh equations2, and based the form of the functions in the new equations on voltage-clamp data obtained from a snail neurone. We report here an unexpected property of the resulting equations-the x and y null clines in the phase plane lie close together when the phase point is on the recovery side of the phase plane. The resulting slow movement along the phase path gives a long inter-spike interval, a property not shown clearly by previous models2,4. The model also predicts the linearity of the frequency-current relationship, and may be useful for studying detailed interactions in networks containing small numbers of neurones.

15. Critical study of higher order numerical methods for solving the boundary-layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1978-01-01

A fourth order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method, which is the natural extension of the second order box scheme to fourth order, was demonstrated with application to the incompressible, laminar and turbulent, boundary layer equations. The efficiency of the present method is compared with two point and three point higher order methods, namely, the Keller box scheme with Richardson extrapolation, the method of deferred corrections, a three point spline method, and a modified finite element method. For equivalent accuracy, numerical results show the present method to be more efficient than higher order methods for both laminar and turbulent flows.

16. A critical study of higher-order numerical methods for solving the boundary-layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1977-01-01

A fourth-order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations. The efficiency of the present method is compared with other two-point and three-point higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, and the three-point spline methods. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

17. Application of higher-order numerical methods to the boundary-layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1978-01-01

A fourth-order method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method is the natural extension of the second-order Keller Box Scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary-layer equations for both attached and separated flows. The efficiency of the present method is compared with other higher-order methods; namely, the Keller Box Scheme with Richardson extrapolation, the method of deferred corrections, the three-point spline methods, and a modified finite-element method. For equivalent accuracy, numerical results show the present method to be more efficient than the other higher-order methods for both laminar and turbulent flows.

18. Using Kernel Equating to Assess Item Order Effects on Test Scores

ERIC Educational Resources Information Center

Moses, Tim; Yang, Wen-Ling; Wilson, Christine

2007-01-01

This study explored the use of kernel equating for integrating and extending two procedures proposed for assessing item order effects in test forms that have been administered to randomly equivalent groups. When these procedures are used together, they can provide complementary information about the extent to which item order effects impact test…

19. SR-52 program for the solution of two first order differential equations.

PubMed

Yakush, S A

1979-03-01

This paper presents a program written for the Texas Instruments SR-52 programmable calculator to numerically solve a pair of first order ordinary differential equations. The program uses a fourth order Runga-Kutta method and a typical sample run is presented.

20. Introducing graded meshes in the numerical approximation of distributed-order diffusion equations

2016-10-01

In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.

1. Approximated Lax pairs for the reduced order integration of nonlinear evolution equations

Gerbeau, Jean-Frédéric; Lombardi, Damiano

2014-05-01

A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations. It is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the basis on which the solution is searched for evolves in time according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive front or wave propagation. Another difference with other reduced-order methods is that it is not based on an off-line/on-line strategy. Numerical examples are shown for the linear advection, KdV and FKPP equations, in one and two dimensions.

2. First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

NASA Technical Reports Server (NTRS)

Cai, Z.; Manteuffel, T. A.; McCormick, S. F.

1996-01-01

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

3. Dynamics and Control of a Reduced Order System of the 2-d Navier-Stokes Equations

Smaoui, Nejib; Zribi, Mohamed

2014-11-01

The dynamics and control problem of a reduced order system of the 2-d Navier-Stokes (N-S) equations is analyzed. First, a seventh order system of nonlinear ordinary differential equations (ODE) which approximates the dynamical behavior of the 2-d N-S equations is obtained by using the Fourier Galerkin method. We show that the dynamics of this ODE system transforms from periodic solutions to chaotic attractors through a sequence of bifurcations including a period doubling scenarios. Then three Lyapunov based controllers are designed to either control the system of ODEs to a desired fixed point or to synchronize two ODE systems obtained from the truncation of the 2-d N-S equations under different conditions. Numerical simulations are presented to show the effectiveness of the proposed controllers. This research was supported and funded by the Research Sector, Kuwait University under Grant No. SM02/14.

4. Estimates of solutions of certain classes of second-order differential equations in a Hilbert space

SciTech Connect

Artamonov, N V

2003-08-31

Linear second-order differential equations of the form u''(t)+(B+iD)u'(t)+(T+iS)u(t)=0 in a Hilbert space are studied. Under certain conditions on the (generally speaking, unbounded) operators T, S, B and D the correct solubility of the equation in the 'energy' space is proved and best possible (in the general case) estimates of the solutions on the half-axis are obtained.

5. On integration of the first order differential equations in a finite terms

Malykh, M. D.

2017-01-01

There are several approaches to the description of the concept called briefly as integration of the first order differential equations in a finite terms or symbolical integration. In the report three of them are considered: 1.) finding of a rational integral (Beaune or Poincaré problem), 2.) integration by quadratures and 3.) integration when the general solution of given differential equation is an algebraical function of a constant (Painlevé problem). Their realizations in Sage are presented.

6. Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

7. Green's functional for a higher order ordinary integro-differential equation with nonlocal conditions

Özen, Kemal

2016-12-01

One of the little-known techniques for ordinary integro-differential equations in literature is Green's functional method, the origin of which dates back to Azerbaijani scientist Seyidali S. Akhiev. According to this method, Green's functional concepts for some simple forms of such equations have been introduced in the several studies. In this study, we extend Green's functional concept to a higher order ordinary integro-differential equation involving generally nonlocal conditions. A novel kind of adjoint problem and Green's functional are constructed for completely nonhomogeneous problem. By means of the obtained Green's functional, the solution to the problem is identified.

8. Second order scheme for Maxwell's equations with discontinuous dielectric permittivity on structured meshes

Ismagilov, Timur Z.

2013-10-01

A second order finite volume scheme for numerical solution of Maxwell's equations with discontinuous dielectric permittivity on structured meshes is suggested. The scheme is based on approaches of Godunov, Lax-Wendroff and Van Leer. The distinctive feature of the suggested scheme is calculation and limitation of derivatives that ensures second order of approximation even in the cells adjacent to dielectric permittivity discontinuity. Numerical tests for problems with linear and curvilinear dielectric permittivity discontinuities confirm second order of approximation.

9. Properties-preserving high order numerical methods for a kinetic eikonal equation

Luo, Songting; Payne, Nicholas

2017-02-01

For the BGK (Bhatnagar-Gross-Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf-Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton-Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge-Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.

10. A high-order element-based Galerkin Method for the global shallow water equations.

SciTech Connect

Nair, Ramachandran D.; Tufo, Henry M.; Levy, Michael Nathan

2010-08-01

The shallow water equations are used as a test for many atmospheric models because the solution mimics the horizontal aspects of atmospheric dynamics while the simplicity of the equations make them useful for numerical experiments. This study describes a high-order element-based Galerkin method for the global shallow water equations using absolute vorticity, divergence, and fluid depth (atmospheric thickness) as the prognostic variables, while the wind field is a diagnostic variable that can be calculated from the stream function and velocity potential (the Laplacians of which are the vorticity and divergence, respectively). The numerical method employed to solve the shallow water system is based on the discontinuous Galerkin and spectral element methods. The discontinuous Galerkin method, which is inherently conservative, is used to solve the equations governing two conservative variables - absolute vorticity and atmospheric thickness (mass). The spectral element method is used to solve the divergence equation and the Poisson equations for the velocity potential and the stream function. Time integration is done with an explicit strong stability-preserving second-order Runge-Kutta scheme and the wind field is updated directly from the vorticity and divergence at each stage, and the computational domain is the cubed sphere. A stable steady-state test is run and convergence results are provided, showing that the method is high-order accurate. Additionally, two tests without analytic solutions are run with comparable results to previous high-resolution runs found in the literature.

11. The Generation of a Series of Multiwing Chaotic Attractors Using Integer and Fractional Order Differential Equation Systems

Xu, Fei

In this article, we present a systematic approach to design chaos generators using integer order and fractional order differential equation systems. A series of multiwing chaotic attractors and grid multiwing chaotic attractors are obtained using linear integer order differential equation systems with switching controls. The existence of chaotic attractors in the corresponding fractional order differential equation systems is also investigated. We show that, using the nonlinear fractional order differential equation system, or linear fractional order differential equation systems with switching controls, a series of multiwing chaotic attractors can be obtained.

12. A space-time spectral collocation algorithm for the variable order fractional wave equation.

PubMed

Bhrawy, A H; Doha, E H; Alzaidy, J F; Abdelkawy, M A

2016-01-01

The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space-time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi-Gauss-Lobatto collocation scheme for the spatial discretization and the shifted Jacobi-Gauss-Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

13. On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations

Zhang, Xiangxiong

2017-01-01

We construct a local Lax-Friedrichs type positivity-preserving flux for compressible Navier-Stokes equations, which can be easily extended to multiple dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge-Kutta time discretizations satisfy a weak positivity property. With a simple and efficient positivity-preserving limiter, high order explicit Runge-Kutta DG schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier-Stokes equations is accurately resolved.

14. Toward order-by-order calculations of the nuclear and neutron matter equations of state in chiral effective field theory

Sammarruca, F.; Coraggio, L.; Holt, J. W.; Itaco, N.; Machleidt, R.; Marcucci, L. E.

2015-05-01

We calculate the nuclear and neutron matter equations of state from microscopic nuclear forces at different orders in chiral effective field theory and with varying momentum-space cutoff scales. We focus attention on how the order-by-order convergence depends on the choice of resolution scale and the implications for theoretical uncertainty estimates on the isospin asymmetry energy. Specifically we study the equations of state using consistent NLO and N2LO (next-to-next-to-leading order) chiral potentials where the low-energy constants cD and cE associated with contact vertices in the N2LO chiral three-nucleon force are fitted to reproduce the binding energies of H3 and He3 as well as the beta-decay lifetime of H3 . At these low orders in the chiral expansion there is little sign of convergence, while an exploratory study employing the N3LO two-nucleon force together with the N2LO three-nucleon force give first indications for (slow) convergence with low-cutoff potentials and poor convergence with higher-cutoff potentials. The consistent NLO and N2LO potentials described in the present work provide the basis for estimating theoretical uncertainties associated with the order-by-order convergence of nuclear many-body calculations in chiral effective field theory.

15. High-Order Central WENO Schemes for Multi-Dimensional Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)

2002-01-01

We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the third-order scheme: one scheme that is based on a genuinely two-dimensional Central WENO reconstruction, and another scheme that is based on a simpler dimension-by-dimension reconstruction. The simpler dimension-by-dimension variant is then extended to a multi-dimensional fifth-order scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes.

16. Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

DOE PAGES

Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

2015-01-23

In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

17. Multilevel solvers of first-order system least-squares for Stokes equations

SciTech Connect

Lai, Chen-Yao G.

1996-12-31

Recently, The use of first-order system least squares principle for the approximate solution of Stokes problems has been extensively studied by Cai, Manteuffel, and McCormick. In this paper, we study multilevel solvers of first-order system least-squares method for the generalized Stokes equations based on the velocity-vorticity-pressure formulation in three dimensions. The least-squares functionals is defined to be the sum of the L{sup 2}-norms of the residuals, which is weighted appropriately by the Reynolds number. We develop convergence analysis for additive and multiplicative multilevel methods applied to the resulting discrete equations.

18. Constructing conservation laws for fractional-order integro-differential equations

Lukashchuk, S. Yu.

2015-08-01

In a class of functions depending on linear integro-differential fractional-order variables, we prove an analogue of the fundamental operator identity relating the infinitesimal operator of a point transformation group, the Euler-Lagrange differential operator, and Noether operators. Using this identity, we prove fractional-differential analogues of the Noether theorem and its generalizations applicable to equations with fractional-order integrals and derivatives of various types that are Euler-Lagrange equations. In explicit form, we give fractional-differential generalizations of Noether operators that gives an efficient way to construct conservation laws, which we illustrate with three examples.

19. Exact Nonlinear Fourth-order Equation for Two Coupled Oscillators: Metamorphoses of Resonance Curves

Kyzioł, J.; Okniński, A.

We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov-Bogoliubov-Mitropolsky approach - we compute the amplitude profiles, which from mathematical point of view are algebraic curves. In the present paper we investigate metamorphoses of amplitude profiles induced by changes of control parameters near singular points of these curves. It follows that dynamics changes qualitatively in the neighbourhood of a singular point.

20. A note on the nonlocal boundary value problem for a third order partial differential equation

Belakroum, Kheireddine; Ashyralyev, Allaberen; Guezane-Lakoud, Assia

2016-08-01

The nonlocal boundary-value problem for a third order partial differential equation d/3u (t ) d t3 +A d/u (t ) d t =f (t ), 0 order partial differential equations are obtained.

1. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

NASA Technical Reports Server (NTRS)

Yan, Jue; Shu, Chi-Wang; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

2. A method for solving stochastic equations by reduced order models and local approximations

SciTech Connect

Grigoriu, M.

2012-08-01

A method is proposed for solving equations with random entries, referred to as stochastic equations (SEs). The method is based on two recent developments. The first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments. The second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector, referred to as stochastic reduced order model (SROM), and uses it to construct a SROM for the solution of this equation. The proposed method is a direct extension of these two methods. It uses SROMs to select expansion points, rather than selecting these points by geometrical considerations, and represents the solution by linear and/or higher order local approximations. The implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic/differential equations. The method is conceptually simple, non-intrusive, efficient relative to classical Monte Carlo simulation, accurate, and guaranteed to converge to the exact solution.

3. Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients

Pandit, Sapna; Kumar, Manoj; Tiwari, Surabhi

2015-02-01

In this article, the authors proposed a numerical scheme based on Crank-Nicolson finite difference scheme and Haar wavelets to find numerical solutions of different types of second order hyperbolic telegraph equations (i.e. telegraph equation with constant coefficients, with variable coefficients, and singular telegraph equation). This work is an extension of the scheme by Jiwari (2012) for hyperbolic equations. The use of Haar basis function is made with multiresolution analysis to get the fast and accurate results on collocation points. The convergence of the proposed scheme is proved by doing its error analysis. Four test examples are considered to demonstrate the accuracy and efficiency of the scheme. The scheme is easy and very suitable for computer implementation and provides numerical solutions close to the exact solutions and available in the literature.

4. Evolution of higher order nonlinear equation for the dust ion-acoustic waves in nonextensive plasma

SciTech Connect

Yasmin, S.; Asaduzzaman, M.; Mamun, A. A.

2012-10-15

There are three different types of nonlinear equations, namely, Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and mixed modified K-dV (mixed mK-dV) equations, for the nonlinear propagation of the dust ion-acoustic (DIA) waves. The effects of electron nonextensivity on DIA solitary waves propagating in a dusty plasma (containing negatively charged stationary dust, inertial ions, and nonextensive q distributed electrons) are examined by solving these nonlinear equations. The basic features of mixed mK-dV (higher order nonlinear equation) solitons are found to exist beyond the K-dV limit. The properties of mK-dV solitons are compared with those of mixed mK-dV solitons. It is found that both positive and negative solitons are obtained depending on the q (nonextensive parameter).

5. Second order finite volume scheme for Maxwell's equations with discontinuous electromagnetic properties on unstructured meshes

Ismagilov, Timur Z.

2015-02-01

This paper presents a second order finite volume scheme for numerical solution of Maxwell's equations with discontinuous dielectric permittivity and magnetic permeability on unstructured meshes. The scheme is based on Godunov scheme and employs approaches of Van Leer and Lax-Wendroff to increase the order of approximation. To keep the second order of approximation near dielectric permittivity and magnetic permeability discontinuities a novel technique for gradient calculation and limitation is applied near discontinuities. Results of test computations for problems with linear and curvilinear discontinuities confirm second order of approximation. The scheme was applied to modelling propagation of electromagnetic waves inside photonic crystal waveguides with a bend.

6. Fourth-order master equation for a charged harmonic oscillator interacting with the electromagnetic field

Kurt, Arzu; Eryigit, Resul

2015-12-01

The master equation for a charged harmonic oscillator coupled to an electromagnetic reservoir is investigated up to fourth order in the interaction strength by using Krylov averaging method. The interaction is in the velocity-coupling form and includes a diamagnetic term. Exact analytical expressions for the second-, the third-, and the fourth-order contributions to mass renormalization, decay constant, normal and anomalous diffusion coefficients are obtained for the blackbody type environment. It is found that, generally, the third- and the fourth-order contributions have opposite signs when their magnitudes are comparable to that of the second-order one.

7. Second-order curved boundary treatments of the lattice Boltzmann method for convection-diffusion equations

Huang, Juntao; Hu, Zexi; Yong, Wen-An

2016-04-01

In this paper, we present a kind of second-order curved boundary treatments for the lattice Boltzmann method solving two-dimensional convection-diffusion equations with general nonlinear Robin boundary conditions. The key idea is to derive approximate boundary values or normal derivatives on computational boundaries, with second-order accuracy, by using the prescribed boundary condition. Once the approximate information is known, the second-order bounce-back schemes can be perfectly adopted. Our boundary treatments are validated with a number of numerical examples. The results show the utility of our boundary treatments and very well support our theoretical predications on the second-order accuracy thereof. The idea is quite universal. It can be directly generalized to 3-dimensional problems, multiple-relaxation-time models, and the Navier-Stokes equations.

8. A Generalized 4th-Order Runge-Kutta Method for the Gross-Pitaevskii Equation

Kandes, Martin

2015-04-01

We present the implementation of a method-of-lines approach for numerically approximating solutions of the time-dependent Gross-Pitaevksii equation in non-uniformly rotating reference frames. Implemented in parallel using a hybrid MPI + OpenMP framework, which will allow for scalable, high-resolution numerical simulations, we utilize an explicit, generalized 4th-order Runge-Kutta time-integration scheme with 2nd- and 4th-order central differences to approximate the spatial derivatives in the equation. The principal objective of this project is to model the effect(s) of inertial forces on quantized vortices within weakly-interacting dilute atomic gas Bose-Einstein condensates in the mean-field limit of the Gross-Pitaevskii equation. Here, we discuss our work-to-date and preliminary results.

9. Existence and Stability Results for Second-Order Stochastic Equations Driven by Fractional Brownian Motion

Revathi, P.; Sakthivel, R.; Song, D.-Y.; Ren, Yong; Zhang, Pei

2013-09-01

Fractional Brownian motion has been widely used to model a number of phenomena in diverse fields of science and engineering. In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of second-order nonautonomous neutral stochastic evolution equations with infinite delay driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in Hilbert spaces. More precisely, using semigroup theory and successive approximation approach, we establish a set of sufficient conditions for obtaining the required result under the assumption that coefficients satisfy non-Lipschitz condition with Lipschitz condition being considered as a special case. Further, the result is deduced to study the second-order autonomous neutral stochastic equations with fBm. The results generalize and improve some known results. Finally, as an application, stochastic wave equation with infinite delay driven by fractional Brownian motion is provided to illustrate the obtained theory.

10. Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

Kim, Sunghan; Lee, Ki-Ahm

2016-03-01

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.

11. Efficiency of perfectly matched layers for seismic wave modeling in second-order viscoelastic equations

Ping, Ping; Zhang, Yu; Xu, Yixian; Chu, Risheng

2016-12-01

In order to improve the perfectly matched layer (PML) efficiency in viscoelastic media, we first propose a split multi-axial PML (M-PML) and an unsplit convolutional PML (C-PML) in the second-order viscoelastic wave equations with the displacement as the only unknown. The advantage of these formulations is that it is easy and efficient to revise the existing codes of the second-order spectral element method (SEM) or finite-element method (FEM) with absorbing boundaries in a uniform equation, as well as more economical than the auxiliary differential equations PML. Three models which are easily suffered from late time instabilities are considered to validate our approaches. Through comparison the M-PML with C-PML efficiency of absorption and stability for long time simulation, it can be concluded that: (1) for an isotropic viscoelastic medium with high Poisson's ratio, the C-PML will be a sufficient choice for long time simulation because of its weak reflections and superior stability; (2) unlike the M-PML with high-order damping profile, the M-PML with second-order damping profile loses its stability in long time simulation for an isotropic viscoelastic medium; (3) in an anisotropic viscoelastic medium, the C-PML suffers from instabilities, while the M-PML with second-order damping profile can be a better choice for its superior stability and more acceptable weak reflections than the M-PML with high-order damping profile. The comparative analysis of the developed methods offers meaningful significance for long time seismic wave modeling in second-order viscoelastic wave equations.

12. Using 4th order Runge-Kutta method for solving a twisted Skyrme string equation

Hadi, Miftachul; Anderson, Malcolm; Husein, Andri

2016-03-01

We study numerical solution, especially using 4th order Runge-Kutta method, for solving a twisted Skyrme string equation. We find numerically that the value of minimum energy per unit length of vortex solution for a twisted Skyrmion string is 20.37 × 1060 eV/m.

13. Oscillation of certain higher-order neutral partial functional differential equations.

PubMed

Li, Wei Nian; Sheng, Weihong

2016-01-01

In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

14. Keep Your Distance! Using Second-Order Ordinary Differential Equations to Model Traffic Flow

ERIC Educational Resources Information Center

McCartney, Mark

2004-01-01

A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted. The model can be used as an application of solution techniques taught at first-year undergraduate level and as a motivator to encourage…

15. Student Interpretations of the Terms in First-Order Ordinary Differential Equations in Modelling Contexts

ERIC Educational Resources Information Center

Rowland, David R.; Jovanoski, Zlatko

2004-01-01

A study of first-year undergraduate students' interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function…

16. Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

PubMed Central

Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972

17. Solving Second-Order Ordinary Differential Equations without Using Complex Numbers

ERIC Educational Resources Information Center

Kougias, Ioannis E.

2009-01-01

Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…

18. Factors Affecting Higher Order Thinking Skills of Students: A Meta-Analytic Structural Equation Modeling Study

ERIC Educational Resources Information Center

Budsankom, Prayoonsri; Sawangboon, Tatsirin; Damrongpanit, Suntorapot; Chuensirimongkol, Jariya

2015-01-01

The purpose of the research is to develop and identify the validity of factors affecting higher order thinking skills (HOTS) of students. The thinking skills can be divided into three types: analytical, critical, and creative thinking. This analysis is done by applying the meta-analytic structural equation modeling (MASEM) based on a database of…

19. Efficient High Order Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations: Talk Slides

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Brian R. (Technical Monitor)

2002-01-01

This viewgraph presentation presents information on the attempt to produce high-order, efficient, central methods that scale well to high dimension. The central philosophy is that the equations should evolve to the point where the data is smooth. This is accomplished by a cyclic pattern of reconstruction, evolution, and re-projection. One dimensional and two dimensional representational methods are detailed, as well.

20. Renormalized entropy solutions of the Cauchy problem for a first-order inhomogeneous quasilinear equation

SciTech Connect

Panov, E Yu

2013-10-31

The concept of a renormalized entropy solution of the Cauchy problem for an inhomogeneous quasilinear equation of the first order is introduced. Existence and uniqueness theorems are proved, together with a comparison principle. Connections with generalized entropy solutions are investigated. Bibliography: 10 titles.

1. New modification of Laplace decomposition method for seventh order KdV equation

Kashkari, B. S.; Bakodah, H. O.

2013-10-01

In this paper, we develop a new modification of Laplace decomposition method for solving the seventh order KdV equations. The numerical results show that the method converges rapidly and compared with the Adomian decomposition method. The conservation properties of solution are examined by calculating the first three invariants.

2. Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

PubMed

Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

3. Wave equation for generalized Zener model containing complex order fractional derivatives

Atanacković, Teodor M.; Janev, Marko; Konjik, Sanja; Pilipović, Stevan

2017-03-01

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.

4. A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Ho, Lee-Wing; Maday, Yvon; Patera, Anthony T.; Ronquist, Einar M.

1989-01-01

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

5. Variable step size and order strategy for delay differential equations in PIE(CIE)s mode

Aziz, Nurul Huda Abdul; Majid, Zanariah Abdul

2014-12-01

This article deals with the strategy of variable step size and variable order implementation that has been formulated for solving first order of delay differential equations. This strategy is adapted in PIE(CIE)s mode which is generally based on predictor-corrector scheme in multistep block method of order 4 to 9 with s is for convergence test. The purpose here is to enhance the efficiency of the developed predictor-corrector algorithm in the capability to vary automatically not only for the step size, but the order of the method employed as well. All order and coefficients are stored in the code in order to avoid an expensive computational work. The delay argument would be evaluated using Newton divided-difference interpolation at which the points involved would be similar to the current order of the method. Illustrative examples are included to demonstrate the validity and applicability of the presented strategy and comparison is made with the existing results.

6. Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

PubMed Central

Wang, Gang wei; Xu, Tian zhou; Feng, Tao

2014-01-01

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided. PMID:24523885

7. A reduced-order representation of the Schrödinger equation

Cheng, Ming-C.

2016-09-01

A reduced-order-based representation of the Schrödinger equation is investigated for electron wave functions in semiconductor nanostructures. In this representation, the Schrödinger equation is projected onto an eigenspace described by a small number of basis functions that are generated from the proper orthogonal decomposition (POD). The approach substantially reduces the numerical degrees of freedom (DOF's) needed to numerically solve the Schrödinger equation for the wave functions and eigenstate energies in a quantum structure and offers an accurate solution as detailed as the direct numerical simulation of the Schrödinger equation. To develop such an approach, numerical data accounting for parametric variations of the system are used to perform decomposition in order to generate the POD eigenvalues and eigenvectors for the system. This approach is applied to develop POD models for single and multiple quantum well structure. Errors resulting from the approach are examined in detail associated with the selected numerical DOF's of the POD model and quality of data used for generation of the POD eigenvalues and basis functions. This study investigates the fundamental concepts of the POD approach to the Schrödinger equation and paves a way toward developing an efficient modeling methodology for large-scale multi-block simulation of quantum nanostructures.

8. Existence and multiplicity of solutions to 2mth-order ordinary differential equations

Li, Fuyi; Li, Yuhua; Liang, Zhanping

2007-07-01

In this paper, the existence and multiplicity of solutions are obtained for the 2mth-order ordinary differential equation two-point boundary value problems u(2(m-i))(t)=f(t,u(t)) for all t[set membership, variant][0,1] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, for all i=1,2,...,m. Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

9. Influence of high-order nonlinear fluctuations in the multivariate susceptible-infectious-recovered master equation.

PubMed

Bayati, Basil S; Eckhoff, Philip A

2012-12-01

We perform a high-order analytical expansion of the epidemiological susceptible-infectious-recovered multivariate master equation and include terms up to and beyond single-particle fluctuations. It is shown that higher order approximations yield qualitatively different results than low-order approximations, which is incident to the influence of additional nonlinear fluctuations. The fluctuations can be related to a meaningful physical parameter, the basic reproductive number, which is shown to dictate the rate of divergence in absolute terms from the ordinary differential equations more so than the total number of persons in the system. In epidemiological terms, the effect of single-particle fluctuations ought to be taken into account as the reproductive number approaches unity.

10. A High-Order Direct Solver for Helmholtz Equations with Neumann Boundary Conditions

NASA Technical Reports Server (NTRS)

Sun, Xian-He; Zhuang, Yu

1997-01-01

In this study, a compact finite-difference discretization is first developed for Helmholtz equations on rectangular domains. Special treatments are then introduced for Neumann and Neumann-Dirichlet boundary conditions to achieve accuracy and separability. Finally, a Fast Fourier Transform (FFT) based technique is used to yield a fast direct solver. Analytical and experimental results show this newly proposed solver is comparable to the conventional second-order elliptic solver when accuracy is not a primary concern, and is significantly faster than that of the conventional solver if a highly accurate solution is required. In addition, this newly proposed fourth order Helmholtz solver is parallel in nature. It is readily available for parallel and distributed computers. The compact scheme introduced in this study is likely extendible for sixth-order accurate algorithms and for more general elliptic equations.

11. SIVA/DIVA- INITIAL VALUE ORDINARY DIFFERENTIAL EQUATION SOLUTION VIA A VARIABLE ORDER ADAMS METHOD

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1994-01-01

The SIVA/DIVA package is a collection of subroutines for the solution of ordinary differential equations. There are versions for single precision and double precision arithmetic. These solutions are applicable to stiff or nonstiff differential equations of first or second order. SIVA/DIVA requires fewer evaluations of derivatives than other variable order Adams predictor-corrector methods. There is an option for the direct integration of second order equations which can make integration of trajectory problems significantly more efficient. Other capabilities of SIVA/DIVA include: monitoring a user supplied function which can be separate from the derivative; dynamically controlling the step size; displaying or not displaying output at initial, final, and step size change points; saving the estimated local error; and reverse communication where subroutines return to the user for output or computation of derivatives instead of automatically performing calculations. The user must supply SIVA/DIVA with: 1) the number of equations; 2) initial values for the dependent and independent variables, integration stepsize, error tolerance, etc.; and 3) the driver program and operational parameters necessary for subroutine execution. SIVA/DIVA contains an extensive diagnostic message library should errors occur during execution. SIVA/DIVA is written in FORTRAN 77 for batch execution and is machine independent. It has a central memory requirement of approximately 120K of 8 bit bytes. This program was developed in 1983 and last updated in 1987.

12. High-order all-optical differential equation solver based on microring resonators.

PubMed

Tan, Sisi; Xiang, Lei; Zou, Jinghui; Zhang, Qiang; Wu, Zhao; Yu, Yu; Dong, Jianji; Zhang, Xinliang

2013-10-01

We propose and experimentally demonstrate a feasible integrated scheme to solve all-optical differential equations using microring resonators (MRRs) that is capable of solving first- and second-order linear ordinary differential equations with different constant coefficients. Employing two cascaded MRRs with different radii, an excellent agreement between the numerical simulation and the experimental results is obtained. Due to the inherent merits of silicon-based devices for all-optical computing, such as low power consumption, small size, and high speed, this finding may motivate the development of integrated optical signal processors and further extend optical computing technologies.

13. The fifth-order partial differential equation for the description of the α + β Fermi-Pasta-Ulam model

Kudryashov, Nikolay A.; Volkov, Alexandr K.

2017-01-01

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.

14. Next-to-leading order Balitsky-Kovchegov equation with resummation

SciTech Connect

Lappi, T.; Mantysaari, H.

2016-05-03

Here, we solve the Balitsky-Kovchegov evolution equation at next-to-leading order accuracy including a resummation of large single and double transverse momentum logarithms to all orders. We numerically determine an optimal value for the constant under the large transverse momentum logarithm that enables including a maximal amount of the full NLO result in the resummation. When this value is used, the contribution from the α2s terms without large logarithms is found to be small at large saturation scales and at small dipoles. Close to initial conditions relevant for phenomenological applications, these fixed-order corrections are shown to be numerically important.

15. Next-to-leading order Balitsky-Kovchegov equation with resummation

DOE PAGES

Lappi, T.; Mantysaari, H.

2016-05-03

Here, we solve the Balitsky-Kovchegov evolution equation at next-to-leading order accuracy including a resummation of large single and double transverse momentum logarithms to all orders. We numerically determine an optimal value for the constant under the large transverse momentum logarithm that enables including a maximal amount of the full NLO result in the resummation. When this value is used, the contribution from the α2s terms without large logarithms is found to be small at large saturation scales and at small dipoles. Close to initial conditions relevant for phenomenological applications, these fixed-order corrections are shown to be numerically important.

16. On the study of a nonlinear higher order dispersive wave equation: its mathematical physical structure and anomaly soliton phenomena

Lee, C. T.; Lee, C. C.

2015-04-01

This paper introduces a systematic approach to investigate a higher order nonlinear dispersive wave equation for modeling different wave modes. We present both the conventional KdV-type soliton and anomaly type solitons for the equation. We also show the conservation laws and Hamiltonian structures for the equation. Our results suggest that the underlying equation has more interacting soliton phenomena than one would have known for the classical KdV and Boussinesq equation.

17. GENERAL Pseudopotentials, Lax Pairs and Bäcklund Transformations for Generalized Fifth-Order KdV Equation

Yang, Yun-Qing; Chen, Yong

2011-01-01

Based on the method developed by Nucci, the pseudopotentials, Lax pairs and the singularity manifold equations of the generalized fifth-order KdV equation are derived. By choosing different coefficient, the corresponding results and the Bäcklund transformations can be obtained on three conditioners which include Caudrey—Dodd—Gibbon—Sawada—Kotera equation, the Lax equation and the Kaup-kupershmidt equation.

18. Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

Dai, Qiuyi; Fu, Yuxia

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.

19. Long-time behavior of a finite volume discretization for a fourth order diffusion equation

Maas, Jan; Matthes, Daniel

2016-07-01

We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d≥slant 1 . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

20. Fourier spectral method for higher order space fractional reaction-diffusion equations

2016-11-01

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction-diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.

1. Discrete Kalman filtering equations of second-order form for control-structure interaction simulations

NASA Technical Reports Server (NTRS)

Park, K. C.; Alvin, K. F.; Belvin, W. Keith

1991-01-01

A second-order form of discrete Kalman filtering equations is proposed as a candidate state estimator for efficient simulations of control-structure interactions in coupled physical coordinate configurations as opposed to decoupled modal coordinates. The resulting matrix equation of the present state estimator consists of the same symmetric, sparse N x N coupled matrices of the governing structural dynamics equations as opposed to unsymmetric 2N x 2N state space-based estimators. Thus, in addition to substantial computational efficiency improvement, the present estimator can be applied to control-structure design optimization for which the physical coordinates associated with the mass, damping and stiffness matrices of the structure are needed instead of modal coordinates.

2. Local Discontinuous Galerkin Approximations And Variable Step Size, Variable Order Time Integration For Richards' Equation

Li, H.; Farthing, M. W.; Dawson, C. N.; Miller, C. T.

2004-12-01

Numerical simulation of Richards' equation continues to be difficult. It is highly nonlinear under common constitutive relations and exhibits sharp fronts in both the pressure head and volume fraction for many problems of interest. For a number of multiphase flow problems, the use of variable order and variable step size temporal discretizations has shown some advantages. However, the spatial discretizations commonly used for variably saturated flow are dominated by nonadaptive, low-order finite difference and finite element methods. Discontinuous Galerkin (DG) finite element methods have received significant attention in a number of fields for hyperbolic PDE's and, more recently, for elliptic and parabolic problems. DG approaches like the local discontinuous Galerkin (LDG) method are appealing for modeling subsurface flow since they can lead to velocity fields that are locally mass-conservative without the need for auxiliary variables or alternative meshes. DG discretizations are also inherently local and so better-suited for unstructured meshes and h-p adaption strategies than traditional methods. While some work has been done recently for multiphase subsurface flow, there are a range of issues related to the performance of DG methods for highly nonlinear parabolic problems like Richards' equation that have not been investigated fully. In this work, we consider the combination of higher order adaptive time integration with an LDG spatial discretization for Richards' equation. We compare this approach to standard low-order methods for a series of test problems and consider a number of issues including the methods' relative accuracy and computational efficiency.

3. The projective geometric theory of systems of second-order differential equations: straightening and symmetry theorems

SciTech Connect

Aminova, Asya V; Aminov, Nail' A-M

2010-06-29

In the framework of the projective geometric theory of systems of differential equations, which is being developed by the authors, conditions which ensure that a family of graphs of solutions of a system of m second-order ordinary differential equations y-vector-ddot=f-vector(t,y-vector,y-vector-dot) with m unknown functions y{sup 1}(t),...,y{sup m}(t) can be straightened (that is, transformed into a family of straight lines) by means of a local diffeomorphism of the variables of the system which takes it to the form z-vector''=0 (straightens the system) are investigated. It is shown that the system to be straightened must be cubic with respect to the derivatives of the unknown functions. Necessary and sufficient conditions for straightening the system are found, which have the form of differential equations for the coefficients of the system or are stated in terms of symmetries of the system. For m=1 the system consists of a single equation y-ddot=f-vector(t,y,y-dot), and the tests obtained reduce to the conditions for straightening this equations which were derived by Lie in 1883. Bibliography: 34 titles.

4. Enhanced Modeling of First-Order Plant Equations of Motion for Aeroelastic and Aeroservoelastic Applications

NASA Technical Reports Server (NTRS)

Pototzky, Anthony S.

2010-01-01

A methodology is described for generating first-order plant equations of motion for aeroelastic and aeroservoelastic applications. The description begins with the process of generating data files representing specialized mode-shapes, such as rigid-body and control surface modes, using both PATRAN and NASTRAN analysis. NASTRAN executes the 146 solution sequence using numerous Direct Matrix Abstraction Program (DMAP) calls to import the mode-shape files and to perform the aeroelastic response analysis. The aeroelastic response analysis calculates and extracts structural frequencies, generalized masses, frequency-dependent generalized aerodynamic force (GAF) coefficients, sensor deflections and load coefficients data as text-formatted data files. The data files are then re-sequenced and re-formatted using a custom written FORTRAN program. The text-formatted data files are stored and coefficients for s-plane equations are fitted to the frequency-dependent GAF coefficients using two Interactions of Structures, Aerodynamics and Controls (ISAC) programs. With tabular files from stored data created by ISAC, MATLAB generates the first-order aeroservoelastic plant equations of motion. These equations include control-surface actuator, turbulence, sensor and load modeling. Altitude varying root-locus plot and PSD plot results for a model of the F-18 aircraft are presented to demonstrate the capability.

5. Extension of Low Dissipative High Order Hydrodynamics Schemes for MHD Equations

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sjoegreen, Bjoern; Mansour, Nagi (Technical Monitor)

2002-01-01

The objective of this paper is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD (magnetohydrodynamic) equations. These schemes employed multiresolution wavelets as adaptive numerical dissipation controls to limit the amount and to aid the selection and/or blending of the appropriate types of dissipation to be used. The new scheme is formulated for both the conservative and non-conservative form of the MHD equations in curvi-linear grids. The three features of the present MHD scheme over existing schemes in the open literature are as follows. First, the scheme is constructed for long-time integrations of shock/turbulence/combustion magnetized flows. Available schemes are too diffusive for long-time integrations and/or turbulence/combustion problems. Second, unlike existing schemes for the conservative MHD equations which suffer from ill-conditioned eigen-decompositions, the present scheme makes use of a well-conditioned eigen-decomposition to solve the conservative form of the MHD equations. This is due to, partly. the fact that the divergence of the magnetic field condition is a different type of constraint from its incompressible Navier-Stokes cousin. Third, a new approach to minimize the numerical error of the divergence free magnetic condition for high order scheme is introduced.

6. Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

NASA Technical Reports Server (NTRS)

Walker, K. P.; Freed, A. D.

1991-01-01

New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.

7. Solutions to higher-order anisotropic parabolic equations in unbounded domains

Kozhevnikova, L. M.; Leont'ev, A. A.

2014-01-01

The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder D=(0,\\infty)\\times\\Omega, where \\Omega\\subset R^n, n\\geq 3, is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as t\\to \\infty is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently 'narrow'. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.

8. Maxwell's second- and third-order equations of transfer for non-Maxwellian gases

NASA Technical Reports Server (NTRS)

Baganoff, D.

1992-01-01

Condensed algebraic forms for Maxwell's second- and third-order equations of transfer are developed for the case of molecules described by either elastic hard spheres, inverse-power potentials, or by Bird's variable hard-sphere model. These hardly reduced, yet exact, equations provide a new point of origin, when using the moment method, in seeking approximate solutions in the kinetic theory of gases for molecular models that are physically more realistic than that provided by the Maxwell model. An important by-product of the analysis when using these second- and third-order relations is that a clear mathematical connection develops between Bird's variable hard-sphere model and that for the inverse-power potential.

9. Solutions to higher-order anisotropic parabolic equations in unbounded domains

SciTech Connect

Kozhevnikova, L M; Leont'ev, A A

2014-01-31

The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder D=(0,∞)×Ω, where Ω⊂R{sup n}, n≥3, is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as t→∞ is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently 'narrow'. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.

10. Parallel Implementation of a High Order Implicit Collocation Method for the Heat Equation

NASA Technical Reports Server (NTRS)

Kouatchou, Jules; Halem, Milton (Technical Monitor)

2000-01-01

We combine a high order compact finite difference approximation and collocation techniques to numerically solve the two dimensional heat equation. The resulting method is implicit arid can be parallelized with a strategy that allows parallelization across both time and space. We compare the parallel implementation of the new method with a classical implicit method, namely the Crank-Nicolson method, where the parallelization is done across space only. Numerical experiments are carried out on the SGI Origin 2000.

11. Bilinear form and soliton solutions for the fifth-order Kaup-Kupershmidt equation

Wang, Pan

2017-02-01

In this paper, multi-soliton solutions of the fifth-order Kaup-Kupershmidt (KK) equation have been derived via the auxiliary function in conjunction with the bilinear method. These solutions have not been previously obtained. Propagation and interactions of three solitons have been presented analytically. The direction of the soliton is related to the signs of the parameters aj. The distances of the solitons are related to the values of the parameters aj.

12. B-spline soliton solution of the fifth order KdV type equations

Zahra, W. K.; Ouf, W. A.; El-Azab, M. S.

2013-10-01

In this paper, we develop a numerical solution based on sextic B-spline collocation method for solving the generalized fifth-order nonlinear evolution equations. Applying Von-Neumann stability analysis, the proposed technique is shown to be unconditionally stable. The accuracy of the presented method is demonstrated by a test problem. The numerical results are found to be in good agreement with the exact solution.

13. Chaotic attractors based on unstable dissipative systems via third-order differential equation

Campos-Cantón, E.

2016-07-01

In this paper, we present an approach how to yield 1D, 2D and 3D-grid multi-scroll chaotic systems in R3 based on unstable dissipative systems via third-order differential equation. This class of systems is constructed by a switching control law(SCL) changing the equilibrium point of an unstable dissipative system. The switching control law that governs the position of the equilibrium point varies according to the number of scrolls displayed in the attractor.

14. Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations

Zhou, Gang

2007-05-01

In this paper we consider generalized nonlinear Schrödinger equations with external potentials. We find the expressions for the fourth and the sixth order Fermi golden rules (FGRs), conjectured in Gang and Sigal [Rev. Math. Phys. 17, 1143-1207 (2005); Geom. Funct. Anal. 16, No. 7, 1377-1390 (2006)]. The FGR is a key condition in a study of the asymptotic dynamics of trapped solitons.

15. A Second Order Continuum Theory of Fluids - Beyond the Navier-Stokes Equations

Paolucci, Samuel

2016-11-01

The Navier-Stokes equations have proved very valuable in modeling fluid flows over the last two centuries. However, there are some cases where it has been demonstrated that they do not provide accurate results. In such cases, very large variations in velocity and/or thermal fields occur in the flows. It is recalled that the Navier-Stokes equations result from linear approximations of constitutive quantities. Using continuum mechanics principles, we derive a second order constitutive theory that application of which should provide more accurate results is such cases. One important case is the structure of gas-dynamic shock waves. It has been demonstrated experimentally that the Navier-Stokes formulation yields incorrect shock profiles even at moderate Mach numbers. Current continuum theories, and indeed most statistical mechanics theories, that have been advanced to reconcile such discrepancies have not been fully successful. Thus, application of the second order theory based solely on a continuum formulation provides an excellent test problem. Results of the second-order equations applied to the shock structure are obtained for monatomic and diatomic gases over a large range of Mach numbers and are compared to experimental results.

16. Higher-order numerical solutions using cubic splines. [for partial differential equations

NASA Technical Reports Server (NTRS)

Rubin, S. G.; Khosla, P. K.

1975-01-01

A cubic spline collocation procedure has recently been developed for the numerical solution of partial differential equations. In the present paper, this spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a non-uniform mesh and overall fourth-order accuracy for a uniform mesh. Solutions using both spline procedures, as well as three-point finite difference methods, will be presented for several model problems.-

17. A higher-order split-step Fourier parabolic-equation sound propagation solution scheme.

PubMed

Lin, Ying-Tsong; Duda, Timothy F

2012-08-01

A three-dimensional Cartesian parabolic-equation model with a higher-order approximation to the square-root Helmholtz operator is presented for simulating underwater sound propagation in ocean waveguides. The higher-order approximation includes cross terms with the free-space square-root Helmholtz operator and the medium phase speed anomaly. It can be implemented with a split-step Fourier algorithm to solve for sound pressure in the model. Two idealized ocean waveguide examples are presented to demonstrate the performance of this numerical technique.

18. Landau-type order parameter equation for shear banding in granular Couette flow.

PubMed

Shukla, Priyanka; Alam, Meheboob

2009-08-07

We show that a Landau-type "order-parameter" equation describes the onset of shear-band formation in granular plane Couette flow wherein the flow undergoes an ordering transition into alternate layers of dense and dilute regions of low and high shear rates, respectively, parallel to the flow direction. Even though the linear theory predicts the stability of the homogeneous shear solution in dilute flows, our analytical bifurcation theory suggests that there is a subcritical finite-amplitude instability that is likely to lead to shear-band formation in dilute flows, which is in agreement with previous numerical simulations.

19. Fourth-order partial differential equation noise removal on welding images

SciTech Connect

Halim, Suhaila Abd; Ibrahim, Arsmah; Sulong, Tuan Nurul Norazura Tuan; Manurung, Yupiter HP

2015-10-22

Partial differential equation (PDE) has become one of the important topics in mathematics and is widely used in various fields. It can be used for image denoising in the image analysis field. In this paper, a fourth-order PDE is discussed and implemented as a denoising method on digital images. The fourth-order PDE is solved computationally using finite difference approach and then implemented on a set of digital radiographic images with welding defects. The performance of the discretized model is evaluated using Peak Signal to Noise Ratio (PSNR). Simulation is carried out on the discretized model on different level of Gaussian noise in order to get the maximum PSNR value. The convergence criteria chosen to determine the number of iterations required is measured based on the highest PSNR value. Results obtained show that the fourth-order PDE model produced promising results as an image denoising tool compared with median filter.

20. Improving an estimate of the convergence rate of the seidel method by selecting the optimal order of equations in the system of linear algebraic equations

Borzykh, A. N.

2017-01-01

The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs' book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is 2 n 2 additions and (12) n 2 divisions. Numerical results for random matrices of order 100 are presented that confirm the proposed improvement.

1. A Fourth Order Difference Scheme for the Maxwell Equations on Yee Grid

SciTech Connect

Fathy, Aly E; Wilson, Joshua L

2008-09-01

The Maxwell equations are solved by a long-stencil fourth order finite difference method over a Yee grid, in which different physical variables are located at staggered mesh points. A careful treatment of the numerical values near the boundary is introduced, which in turn leads to a 'symmetric image' formula at the 'ghost' grid points. Such a symmetric formula assures the stability of the boundary extrapolation. In turn, the fourth order discrete curl operator for the electric and magnetic vectors gives a complete set of eigenvalues in the purely imaginary axis. To advance the dynamic equations, the four-stage Runge-Kutta method is utilized, which results in a full fourth order accuracy in both time and space. A stability constraint for the time step is formulated at both the theoretical and numerical levels, using an argument of stability domain. An accuracy check is presented to verify the fourth order precision, using a comparison between exact solution and numerical solutions at a fixed final time. In addition, some numerical simulations of a loss-less rectangular cavity are also carried out and the frequency is measured precisely.

2. A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations

Guo, Yan; Xiong, Tao; Shi, Yufeng

2014-10-01

In this paper, a positivity-preserving fifth-order finite volume compact-WENO scheme is proposed for solving compressible Euler equations. As it is known, conservative compact finite volume schemes have high resolution properties while WENO (Weighted Essentially Non-Oscillatory) schemes are essentially non-oscillatory near flow discontinuities. We extend the idea of WENO schemes to some classical finite volume compact schemes [30], where lower order compact stencils are combined with WENO nonlinear weights to get a higher order finite volume compact-WENO scheme. The newly developed positivity-preserving limiter [43,42] is used to preserve positive density and internal energy for compressible Euler equations of fluid dynamics. The HLLC (Harten, Lax, and van Leer with Contact) approximate Riemann solver [37,4] is used to get the numerical flux at the cell interfaces. Numerical tests are presented to demonstrate the high-order accuracy, positivity-preserving, high-resolution and robustness of the proposed scheme.

3. One-dimensional high-order compact method for solving Euler's equations

Mohamad, M. A. H.; Basri, S.; Basuno, B.

2012-06-01

In the field of computational fluid dynamics, many numerical algorithms have been developed to simulate inviscid, compressible flows problems. Among those most famous and relevant are based on flux vector splitting and Godunov-type schemes. Previously, this system was developed through computational studies by Mawlood [1]. However the new test cases for compressible flows, the shock tube problems namely the receding flow and shock waves were not investigated before by Mawlood [1]. Thus, the objective of this study is to develop a high-order compact (HOC) finite difference solver for onedimensional Euler equation. Before developing the solver, a detailed investigation was conducted to assess the performance of the basic third-order compact central discretization schemes. Spatial discretization of the Euler equation is based on flux-vector splitting. From this observation, discretization of the convective flux terms of the Euler equation is based on a hybrid flux-vector splitting, known as the advection upstream splitting method (AUSM) scheme which combines the accuracy of flux-difference splitting and the robustness of flux-vector splitting. The AUSM scheme is based on the third-order compact scheme to the approximate finite difference equation was completely analyzed consequently. In one-dimensional problem for the first order schemes, an explicit method is adopted by using time integration method. In addition to that, development and modification of source code for the one-dimensional flow is validated with four test cases namely, unsteady shock tube, quasi-one-dimensional supersonic-subsonic nozzle flow, receding flow and shock waves in shock tubes. From these results, it was also carried out to ensure that the definition of Riemann problem can be identified. Further analysis had also been done in comparing the characteristic of AUSM scheme against experimental results, obtained from previous works and also comparative analysis with computational results

4. High-order nite volume WENO schemes for the shallow water equations with dry states

SciTech Connect

Xing, Yulong; Shu, Chi-wang

2011-01-01

The shallow water equations are used to model flows in rivers and coastal areas, and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. These equations have still water steady state solutions in which the flux gradients are balanced by the source term. It is desirable to develop numerical methods which preserve exactly these steady state solutions. Another main difficulty usually arising from the simulation of dam breaks and flood waves flows is the appearance of dry areas where no water is present. If no special attention is paid, standard numerical methods may fail near dry/wet front and produce non-physical negative water height. A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving the shallow water equations. A simple, easy-to-implement positivity-preserving limiter is introduced. One- and two-dimensional numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.

5. First and second order operator splitting methods for the phase field crystal equation

SciTech Connect

Lee, Hyun Geun; Shin, Jaemin; Lee, June-Yub

2015-10-15

In this paper, we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two-phase systems on atomic length and diffusive time scales. A core idea of the methods is to decompose the original equation into linear and nonlinear subequations, in which the linear subequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used. By combining these subequations, we achieve the first- and second-order accuracy in time. We present numerical experiments to show the accuracy and efficiency of the proposed methods.

6. Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism

Steane, Andrew M.

2015-03-01

It is widely believed that classical electromagnetism is either unphysical or inconsistent, owing to pathological behavior when self-force and radiation reaction are non-negligible. We argue that there is no inconsistency as long as it is recognized that certain types of charge distribution are simply impossible, such as, for example, a point particle with finite charge and finite inertia. This is owing to the fact that negative inertial mass is an unphysical concept in classical physics. It remains useful to obtain an equation of motion for small charged objects that describes their motion to good approximation without requiring knowledge of the charge distribution within the object. We give a simple method to achieve this, leading to a reduced-order form of the Abraham-Lorentz-Dirac equation, essentially as proposed by Eliezer, Landau, and Lifshitz and derived by Ford and O'Connell.

7. Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

Garetto, Claudia; Ruzhansky, Michael

2015-07-01

In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.

8. Front and pulse solutions for the complex Ginzburg-Landau equation with higher-order terms.

PubMed

Tian, Huiping; Li, Zhonghao; Tian, Jinping; Zhou, Guosheng

2002-12-01

We investigate one-dimensional complex Ginzburg-Landau equation with higher-order terms and discuss their influences on the multiplicity of solutions. An exact analytic front solution is presented. By stability analysis for the original partial differential equation, we derive its necessary stability condition for amplitude perturbations. This condition together with the exact front solution determine the region of parameter space where the uniformly translating front solution can exist. In addition, stable pulses, chaotic pulses, and attenuation pulses appear generally if the parameters are out of the range. Finally, applying these analysis into the optical transmission system numerically we find that the stable transmission of optical pulses can be achieved if the parameters are appropriately chosen.

9. On p -form theories with gauge invariant second order field equations

Deffayet, Cédric; Mukohyama, Shinji; Sivanesan, Vishagan

2016-04-01

We explore field theories of a single p -form with equations of motions of order strictly equal to 2 and gauge invariance. We give a general method for the classification of such theories which are extensions to the p -forms of the Galileon models for scalars. Our classification scheme allows us to compute an upper bound on the number of different such theories depending on p and on the space-time dimension. We are also able to build a nontrivial Galileon-like theory for a 3-form with gauge invariance and an action which is polynomial into the derivatives of the form. This theory has gauge invariant field equations but an action which is not, like a Chern-Simons theory. Hence the recently discovered no-go theorem stating that there are no nontrivial gauge invariant vector Galileons (which we are also able here to confirm with our method) does not extend to other odd-p cases.

10. First and second order operator splitting methods for the phase field crystal equation

Lee, Hyun Geun; Shin, Jaemin; Lee, June-Yub

2015-10-01

In this paper, we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two-phase systems on atomic length and diffusive time scales. A core idea of the methods is to decompose the original equation into linear and nonlinear subequations, in which the linear subequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used. By combining these subequations, we achieve the first- and second-order accuracy in time. We present numerical experiments to show the accuracy and efficiency of the proposed methods.

11. Collapse for the higher-order nonlinear Schrödinger equation

DOE PAGES

Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; ...

2016-02-01

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data,more » are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.« less

12. Collapse for the higher-order nonlinear Schrödinger equation

SciTech Connect

Achilleos, V.; Diamantidis, S.; Frantzeskakis, D. J.; Horikis, T. P.; Karachalios, N. I.; Kevrekidis, P. G.

2016-02-01

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schr odinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.

13. High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

SciTech Connect

Giraldo, Francis X. . E-mail: giraldo@nrlmry.navy.mil

2006-05-20

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129-150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss-Lobatto quadrature points (order 2N - 1) and a set based on Gauss quadrature points (order 2N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2N integration being the most accurate of the four DG methods studied. The strong advection form with 2N integration performed extremely well even for flows with shock waves. The strong conservation form with 2N - 1 integration yielded results almost as good as those with 2N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

14. A Numerical Solution of the Second-Order-Nonlinear Acoustic Wave Equation in One and in Three Dimensions.

DTIC Science & Technology

1981-01-08

as it propagates over a small interval, and then to correct for absorption. Another nonlinear wave equation of great interest is the Korteweg - DeVries ...acoustics are described by the second-order-nonlinear wave equation , which is derived in this thesis and solved by numerical means. the validity of the...no approximations are made in the second-order-nonlinear acoustic wave equation as it is solved . This represents an advance on the prior art, in which

15. High-order fractional partial differential equation transform for molecular surface construction

PubMed Central

Hu, Langhua; Chen, Duan; Wei, Guo-Wei

2013-01-01

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model

16. High-order fractional partial differential equation transform for molecular surface construction.

PubMed

Hu, Langhua; Chen, Duan; Wei, Guo-Wei

2013-01-01

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model

17. Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

18. Un-Reduction of Systems of Second-Order Ordinary Differential Equations

García-Toraño Andrés, Eduardo; Mestdag, Tom

2016-12-01

In this paper we consider an alternative approach to ''un-reduction''. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) ''primary un-reduced SODE'', and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.

19. Symmetries and generalized higher order conserved vectors of the wave equation on Bianchi I spacetime

Conservation laws of various systems have been studied for decades due to their unparalleled importance in unraveling systems’ intricacies without having to go into microscopic details of the physical process involved. Their association with symmetries has not only had a stupendous impact in the formulation of the fundamental laws of physics, but also open doors to further explorations and unifications of others. In this study, we present the Lie symmetries and nonlinearly self-adjoint classifications of the wave equation on Bianchi I spacetime. For different forms of the metric potentials, generalized higher order non-trivial conserved vectors are constructed. Some exact invariant solutions are also exhibited.

20. Divergence Free High Order Filter Methods for the Compressible MHD Equations

NASA Technical Reports Server (NTRS)

Yea, H. C.; Sjoegreen, Bjoern

2003-01-01

The generalization of a class of low-dissipative high order filter finite difference methods for long time wave propagation of shock/turbulence/combustion compressible viscous gas dynamic flows to compressible MHD equations for structured curvilinear grids has been achieved. The new scheme is shown to provide a natural and efficient way for the minimization of the divergence of the magnetic field numerical error. Standard diver- gence cleaning is not required by the present filter approach. For certain MHD test cases, divergence free preservation of the magnetic fields has been achieved.

1. First Order Solutions for Klein-Gordon-Maxwell Equations in a Specific Curved Manifold Case

SciTech Connect

Murariu, Gabriel

2009-05-22

The aim of this paper is to study the SO(3,1)xU(1) gauge minimally coupled charged spinless field to a spherically symmetric curved space-time. It is derived the first order analytically approximation solution for the system of Klein-Gordon-Maxwell equations. Using these solutions, it evaluated the system electric charge density. The considered space -time manifold generalize an anterior studied one. The chosen space time configuration is of S diagonal type from the MAPLE GRTensor II metrics package.

2. Periodic Folded Wave Patterns for (2+1)-Dimensional Higher-Order Broer Kaup Equation

Huang, Wen-Hua

2008-10-01

A general solution including three arbitrary functions is obtained for the (2+1)-dimensional higher-order Broer Kaup equation by means of WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and their degenerated single folded solitary waves are investigated graphically and are found to be completely elastic.

3. Application of quarter-sweep iteration for first order linear Fredholm integro-differential equations

Aruchunan, Elayaraja; Muthuvalu, Mohana Sundaram; Sulaiman, Jumat

2013-04-01

The main core of this paper is to analyze the application of the quarter-sweep iterative concept on finite difference and composite trapezoidal schemes with Gauss-Seidel iterative method to solve first order linear Fredholm integro-differential equations. The formulation and implementation of the Full-, Half- and Quarter-Sweep Gauss-Seidel methods namely FSGS, HSGS and QSGS respectively are also presented for performance comparison. Furthermore, computational complexity and percentage reduction analysis are also included and integrated with several numerical simulations. Based on numerical results, findings show the proposed QSGS method with the corresponding discretization schemes is superior compared to FSGS and HSGS iterative methods.

4. Numerical simulations with a First order BSSN formulation of Einstein's field equations

Brown, David; Diener, Peter; Field, Scott; Hesthaven, Jan; Herrmann, Frank; Mroue, Abdul; Sarbach, Olivier; Schnetter, Erik; Tiglio, Manuel; Wagman, Michael

2012-03-01

We present a new fully first order strongly hyperbolic representation of the BSSN formulation of Einstein's equations with optional constraint damping terms. In particular, we describe the characteristic fields of the system, discuss its hyperbolicity properties, and present two numerical implementations and simulations: one using finite differences, adaptive mesh refinement and in particular binary black holes, and another one using the discontinuous Galerkin method in spherical symmetry. These results constitute a first step in an effort to combine the robustness of BSSN evolutions with very high accuracy numerical techniques, such as spectral collocation multi-domain or discontinuous Galerkin methods.

5. Numerical simulations with a first-order BSSN formulation of Einstein's field equations

Brown, J. David; Diener, Peter; Field, Scott E.; Hesthaven, Jan S.; Herrmann, Frank; Mroué, Abdul H.; Sarbach, Olivier; Schnetter, Erik; Tiglio, Manuel; Wagman, Michael

2012-04-01

We present a new fully first-order strongly hyperbolic representation of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein’s equations with optional constraint damping terms. We describe the characteristic fields of the system, discuss its hyperbolicity properties, and present two numerical implementations and simulations: one using finite differences, adaptive mesh refinement, and, in particular, binary black holes, and another one using the discontinuous Galerkin method in spherical symmetry. The results of this paper constitute a first step in an effort to combine the robustness of Baumgarte-Shapiro-Shibata-Nakamura evolutions with very high accuracy numerical techniques, such as spectral collocation multidomain or discontinuous Galerkin methods.

6. Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation

NASA Technical Reports Server (NTRS)

1996-01-01

An algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail.

7. A generalization of gauge symmetry, fourth-order gauge field equations and accelerated cosmic expansion

Hsu, Jong-Ping

2014-02-01

A generalization of the usual gauge symmetry leads to fourth-order gauge field equations, which imply a new constant force independent of distances. The force associated with the new U1 gauge symmetry is repulsive among baryons. Such a constant force based on baryon charge conservation gives a field-theoretic understanding of the accelerated cosmic expansion in the observable portion of the universe dominated by baryon galaxies. In consistent with all conservation laws and known forces, a simple rotating "dumbbell model" of the universe is briefly discussed.

8. On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations

Barles, Guy; Chasseigne, Emmanuel

2016-05-01

We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form ut + H (x , t , Du) = 0 in RN × (0 , + ∞) in the case where the idea is to first estimate ut. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an Hölder regularizing effect in space following a result of L.C. Evans and M.R. James.

9. A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

NASA Technical Reports Server (NTRS)

Gerritsen, Margot; Olsson, Pelle

1996-01-01

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

10. Computational Study of Chaotic and Ordered Solutions of the Kuramoto-Sivashinsky Equation

NASA Technical Reports Server (NTRS)

Smyrlis, Yiorgos S.; Papageorgiou, Demetrios T.

1996-01-01

We report the results of extensive numerical experiments on the Kuramoto-Sivashinsky equation in the strongly chaotic regime as the viscosity parameter is decreased and increasingly more linearly unstable modes enter the dynamics. General initial conditions are used and evolving states do not assume odd-parity. A large number of numerical experiments are employed in order to obtain quantitative characteristics of the dynamics. We report on different routes to chaos and provide numerical evidence and construction of strange attractors with self-similar characteristics. As the 'viscosity' parameter decreases the dynamics becomes increasingly more complicated and chaotic. In particular it is found that regular behavior in the form of steady state or steady state traveling waves is supported amidst the time-dependent and irregular motions. We show that multimodal steady states emerge and are supported on decreasing windows in parameter space. In addition we invoke a self-similarity property of the equation, to show that these profiles are obtainable from global fixed point attractors of the Kuramoto-Sivashinsky equation at much larger values of the viscosity.

11. High-order finite-volume methods for the shallow-water equations on the sphere

Ullrich, Paul A.; Jablonowski, Christiane; van Leer, Bram

2010-08-01

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM +-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge-Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM +-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.

12. High order finite difference methods with subcell resolution for advection equations with stiff source terms

SciTech Connect

Wang, Wei; Shu, Chi-Wang; Yee, H.C.; Sjögreen, Björn

2012-01-01

A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.

13. Error analysis of exponential integrators for oscillatory second-order differential equations

Grimm, Volker; Hochbruck, Marlis

2006-05-01

In this paper, we analyse a family of exponential integrators for second-order differential equations in which high-frequency oscillations in the solution are generated by a linear part. Conditions are given which guarantee that the integrators allow second-order error bounds independent of the product of the step size with the frequencies. Our convergence analysis generalizes known results on the mollified impulse method by García-Archilla, Sanz-Serna and Skeel (1998, SIAM J. Sci. Comput. 30 930-63) and on Gautschi-type exponential integrators (Hairer E, Lubich Ch and Wanner G 2002 Geometric Numerical Integration (Berlin: Springer), Hochbruck M and Lubich Ch 1999 Numer. Math. 83 403-26).

14. Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation

Parand, Kourosh; Mazaheri, Pooria; Yousefi, Hossein; Delkhosh, Mehdi

2017-02-01

In this paper, a new method based on Fractional order of Rational Jacobi (FRJ) functions is proposed that utilizes quasilinearization method to solve non-linear singular Thomas-Fermi equation on unbounded interval [0,∞). The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to the sequence of linear ordinary differential equations. Then, by using the FRJs collocation method the equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate.

15. Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

16. Stability of a nonlinear second order equation under parametric bounded noise excitation

Wiebe, Richard; Xie, Wei-Chau

2016-09-01

The motivation for the following work is a structural column under dynamic axial loads with both deterministic (harmonic transmitted forces from the surrounding structure) and random (wind and/or earthquake) loading components. The bounded noise used herein is a sinusoid with an argument composed of a random (Wiener) process deviation about a mean frequency. By this approach, a noise parameter may be used to investigate the behavior through the spectrum from simple harmonic forcing, to a bounded random process with very little harmonic content. The stability of both the trivial and non-trivial stationary solutions of an axially-loaded column (which is modeled as a second order nonlinear equation) under parametric bounded noise excitation is investigated by use of Lyapunov exponents. Specifically the effect of noise magnitude, amplitude of the forcing, and damping on stability of a column is investigated. First order averaging is employed to obtain analytical approximations of the Lyapunov exponents of the trivial solution. For the non-trivial stationary solution however, the Lyapunov exponents are obtained via Monte Carlo simulation as the stability equations become analytically intractable.

17. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

SciTech Connect

Xing, Yulong; Zhang, Xiangxiong; Shu, Chi-wang

2010-01-01

Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas where no water is present, as standard numerical methods may fail in the presence of these areas. These equations also have still water steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. A simple positivity-preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity-preserving property, well-balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.

18. Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

SciTech Connect

Bokhari, Ashfaque H.; Zaman, F. D.; Mahomed, F. M.

2010-05-15

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

19. A High Order, Locally-Adaptive Method for the Navier-Stokes Equations

Chan, Daniel

1998-11-01

I have extended the FOSLS method of Cai, Manteuffel and McCormick (1997) and implemented it within the framework of a spectral element formulation using the Legendre polynomial basis function. The FOSLS method solves the Navier-Stokes equations as a system of coupled first-order equations and provides the ellipticity that is needed for fast iterative matrix solvers like multigrid to operate efficiently. Each element is treated as an object and its properties are self-contained. Only C^0 continuity is imposed across element interfaces; this design allows local grid refinement and coarsening without the burden of having an elaborate data structure, since only information along element boundaries is needed. With the FORTRAN 90 programming environment, I can maintain a high computational efficiency by employing a hybrid parallel processing model. The OpenMP directives provides parallelism in the loop level which is executed in a shared-memory SMP and the MPI protocol allows the distribution of elements to a cluster of SMP's connected via a commodity network. This talk will provide timing results and a comparison with a second order finite difference method.

20. An extension of MacCormack's method for flows with higher-order equations and in different configurations

NASA Technical Reports Server (NTRS)

Ying, S. J.; Liu, V. C.

1978-01-01

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.

1. The most general second-order field equations of bi-scalar-tensor theory in four dimensions

Ohashi, Seiju; Tanahashi, Norihiro; Kobayashi, Tsutomu; Yamaguchi, Masahide

2015-07-01

The Horndeski theory is known as the most general scalar-tensor theory with second-order field equations. In this paper, we explore the bi-scalar extension of the Horndeski theory. Following Horndeski's approach, we determine all the possible terms appearing in the second-order field equations of the bi-scalar-tensor theory. We compare the field equations with those of the generalized multi-Galileons, and confirm that our theory contains new terms that are not included in the latter theory. We also discuss the construction of the Lagrangian leading to our most general field equations.

2. A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

Shishkin, G. I.; Shishkina, L. P.

2015-03-01

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of [InlineMediaObject not available: see fulltext.], where N + 1 and N 0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.

3. High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation

SciTech Connect

Bihari, B L; Brown, P N

2005-03-29

The authors apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb's phenomenon with oscillations of size O(1) and reduces them to O(h{sup r}), where h is the mesh size and r is the order of accuracy. The current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (WENO5) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE's in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, they need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, non-oscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.

4. Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

2017-03-01

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.

5. Second-order structure function scaling derivation from the Euler and magnetohydrodynamic equations.

PubMed

Beronov, Kamen N

2002-06-01

An anomalous scaling paradigm that has recently come to be canonical has two features limiting its range of applicability: The driving and driven fields are separated dyamically and the driving field statistics is prescribed, in terms of the (inertial subrange) scaling of its second-order structure functions and of white-noise statistics in time. Then the spectrum of scaling exponents for the driven field, scalar or vector, depends parametrically on the driving. Here, the coupling of turbulent vorticity to the driving velocity field is considered. Using simple approximations and no white-noise statistics assumption, equations are derived for the evolution of two-point second-order correlations. The turbulent magnetohydrodynamic (MHD) case is treated in an analogous fashion. In the neutral case, the kinematic coupling between vorticity and velocity leads to a unique prediction for the scaling exponent of the second-order structure functions of the two turbulent fields. The velocity scaling exponent estimate is zeta(2)=3(1/2)-1 approximately equal to 0.732, i.e., close to experimental data. Unlike Kolmogorov scaling, this result is systematically derived from the Euler equations. The analogous scaling of MHD fields is now treated beyond the dynamo theory approximation. In contrast to the uniqueness found in the neutral case, predicted MHD scalings depend on one parameter, similar to the "plasma beta" parameter beta(T) relating kinetic to magnetic energy. The nature of predicted dependence of inertial-range scaling exponents on beta(T) agrees with an observed dichotomy between high-beta(T) and low-beta(T) turbulence regimes.

6. Second order Method for Solving 3D Elasticity Equations with Complex Interfaces

PubMed Central

Wang, Bao; Xia, Kelin; Wei, Guo-Wei

2015-01-01

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques has been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson’s ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins.. PMID:25914422

7. Fourth-order algorithms for solving the imaginary-time Gross-Pitaevskii equation in a rotating anisotropic trap

SciTech Connect

Chin, Siu A.; Krotscheck, Eckhard

2005-09-01

By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth-order algorithms for evolving the Gross-Pitaevskii equation in imaginary time. Such fourth-order algorithms are possible only with the use of forward, positive time step factorization schemes. These fourth-order algorithms converge at time-step sizes an order-of-magnitude larger than conventional second-order algorithms. Our use of time-dependent factorization schemes provides a systematic way of devising algorithms for solving this type of nonlinear equations.

8. An arbitrary order diffusion algorithm for solving Schrödinger equations

Chin, S. A.; Janecek, S.; Krotscheck, E.

2009-09-01

We describe a simple and rapidly converging code for solving the local Schrödinger equation in one, two, and three dimensions that is particularly suited for parallel computing environments. Our algorithm uses high-order imaginary time propagators to project out the eigenfunctions. A recently developed multi-product, operator splitting method permits, in principle, convergence to any even order of the time step. We review briefly the theory behind the method and discuss strategies for assessing convergence and accuracy. A forward time step, single product fourth-order factorization of the imaginary time evolution operator can also be used. Our code requires one user defined function which specifies the local external potential. We describe the definition of this function as well as input and output functionalities and convergence criteria. Compared to our previously published code [Computer Physics Communications 178 (2008) 835], the new algorithms can converge at a rate that is only limited by machine precision. Program summaryProgram title: ndsch Catalogue identifier: AEDR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 9282 No. of bytes in distributed program, including test data, etc.: 77 824 Distribution format: tar.gz Programming language: Fortran 90 Computer: Tested on x86, amd64, and Itanium2 architectures. Should run on any architecture providing a Fortran 90 compiler Operating system: So far tested under UNIX/Linux, Mac OSX and Windows. Any OS with a Fortran 90 compiler available should suffice RAM: 2 MB to 16 GB, depending on system size Classification: 6.10 External routines: FFTW3 ( http://www.fftw.org/), Lapack ( http://www.netlib.org/lapack/) Nature of problem: Numerical calculation of the

9. Three-Dimensional High-Order Spectral Volume Method for Solving Maxwell's Equations on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel; Wang, Z. J.

2004-01-01

A three-dimensional, high-order, conservative, and efficient discontinuous spectral volume (SV) method for the solutions of Maxwell's equations on unstructured grids is presented. The concept of discontinuous 2nd high-order loca1 representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) method, but instead of using a Galerkin finite-element formulation, the SV method is based on a finite-volume approach to attain a simpler formulation. Conventional unstructured finite-volume methods require data reconstruction based on the least-squares formulation using neighboring cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In the SV method, one starts with a relatively coarse grid of triangles or tetrahedra, called spectral volumes (SVs), and partition each SV into a number of structured subcells, called control volumes (CVs), that support a polynomial expansion of a desired degree of precision. The unknowns are cell averages over CVs. If all the SVs are partitioned in a geometrically similar manner, the reconstruction becomes universal as a weighted sum of unknowns, and only a few universal coefficients need to be stored for the surface integrals over CV faces. Since the solution is discontinuous across the SV boundaries, a Riemann solver is thus necessary to maintain conservation. In the paper, multi-parameter and symmetric SV partitions, up to quartic for triangle and cubic for tetrahedron, are first presented. The corresponding weight coefficients for CV face integrals in terms of CV cell averages for each partition are analytically determined. These discretization formulas are then applied to the integral form of

10. Painleve Chains for the Study of Integrable Higher Order Differential Equations.

DTIC Science & Technology

1986-12-18

evolution equations , 1,2,3,4, 5 has become of special interest to theoretical physicists. Such equations possess a special type of elementary solution taking...diverse areas of physics including fluid dynamics, ferromagnetism, quantum optics, and crystal dislocations. Solution of important evolution equations ...and the most important evolution equations including the Burgers, Korteweg-de Vries ( KdV ), modified KdV , and Boussinesq equations . The present paper

11. Deriving Lindblad master equations with Keldysh diagrams: Correlated gain and loss in higher order perturbation theory

Müller, Clemens; Stace, Thomas M.

2017-01-01

Motivated by correlated decay processes producing gain, loss, and lasing in driven semiconductor quantum dots [Phys. Rev. Lett. 113, 036801 (2014), 10.1103/PhysRevLett.113.036801; Science 347, 285 (2015), 10.1126/science.aaa2501; Phys. Rev. Lett. 114, 196802 (2015), 10.1103/PhysRevLett.114.196802], we develop a theoretical technique by using Keldysh diagrammatic perturbation theory to derive a Lindblad master equation that goes beyond the usual second-order perturbation theory. We demonstrate the method on the driven dissipative Rabi model, including terms up to fourth order in the interaction between the qubit and both the resonator and environment. This results in a large class of Lindblad dissipators and associated rates which go beyond the terms that have previously been proposed to describe similar systems. All of the additional terms contribute to the system behavior at the same order of perturbation theory. We then apply these results to analyze the phonon-assisted steady-state gain of a microwave field driving a double quantum dot in a resonator. We show that resonator gain and loss are substantially affected by dephasing-assisted dissipative processes in the quantum-dot system. These additional processes, which go beyond recently proposed polaronic theories, are in good quantitative agreement with experimental observations.

12. A quantitative dynamical systems approach to differential learning: self-organization principle and order parameter equations.

PubMed

Frank, T D; Michelbrink, M; Beckmann, H; Schöllhorn, W I

2008-01-01

Differential learning is a learning concept that assists subjects to find individual optimal performance patterns for given complex motor skills. To this end, training is provided in terms of noisy training sessions that feature a large variety of between-exercises differences. In several previous experimental studies it has been shown that performance improvement due to differential learning is higher than due to traditional learning and performance improvement due to differential learning occurs even during post-training periods. In this study we develop a quantitative dynamical systems approach to differential learning. Accordingly, differential learning is regarded as a self-organized process that results in the emergence of subject- and context-dependent attractors. These attractors emerge due to noise-induced bifurcations involving order parameters in terms of learning rates. In contrast, traditional learning is regarded as an externally driven process that results in the emergence of environmentally specified attractors. Performance improvement during post-training periods is explained as an hysteresis effect. An order parameter equation for differential learning involving a fourth-order polynomial potential is discussed explicitly. New predictions concerning the relationship between traditional and differential learning are derived.

13. Direct discontinuous Galerkin method and its variations for second order elliptic equations

DOE PAGES

Huang, Hongying; Chen, Zheng; Li, Jin; ...

2016-08-23

In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662,more » 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.« less

14. Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.

2002-01-01

The rapid increase in available computational power over the last decade has enabled higher resolution flow simulations and more widespread use of unstructured grid methods for complex geometries. While much of this effort has been focused on steady-state calculations in the aerodynamics community, the need to accurately predict off-design conditions, which may involve substantial amounts of flow separation, points to the need to efficiently simulate unsteady flow fields. Accurate unsteady flow simulations can easily require several orders of magnitude more computational effort than a corresponding steady-state simulation. For this reason, techniques for improving the efficiency of unsteady flow simulations are required in order to make such calculations feasible in the foreseeable future. The purpose of this work is to investigate possible reductions in computer time due to the choice of an efficient time-integration scheme from a series of schemes differing in the order of time-accuracy, and by the use of more efficient techniques to solve the nonlinear equations which arise while using implicit time-integration schemes. This investigation is carried out in the context of a two-dimensional unstructured mesh laminar Navier-Stokes solver.

15. Direct discontinuous Galerkin method and its variations for second order elliptic equations

SciTech Connect

Huang, Hongying; Chen, Zheng; Li, Jin; Yan, Jue

2016-08-23

In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.

16. Non-divergence parabolic equations of second order with critical drift in Lebesgue spaces

Chen, Gong

2017-02-01

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \$-u_{t}+Lu=-u_{t}+\\sum_{ij}a_{ij}D_{ij}u+\\sum b_{i}D_{i}u=0\\,(\\geq0,\\,\\leq0)\$ in some domain $\\Omega\\subset \\mathbb{R}^{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L^{p}$ norm of the inhomogeneous term for some number $p 17. COMPARISON OF NUMERICAL METHODS FOR SOLVING THE SECOND-ORDER DIFFERENTIAL EQUATIONS OF MOLECULAR SCATTERING THEORY SciTech Connect Thomas, L.D.; Alexander, M.H.; Johnson, B.R.; Lester Jr., W. A.; Light, J.C.; McLenithan, K.D.; Parker, G.A.; Redmon, M.J.; Schmalz, T.G.; Secrest, D.; Walker, R.B. 1980-07-01 The numerical solution of coupled, second-order differential equations is a fundamental problem in theoretical physics and chemistry. There are presently over 20 commonly used methods. Unbiased comparisons of the methods are difficult to make and few have been attempted. Here we report a comparison of 11 different methods applied to 3 different test problems. The test problems have been constructed to approximate chemical systems of current research interest and to be representative of the state of the art in inelastic molecular collisions. All calculations were done on the same computer and the attempt was made to do all calculations to the same level of accuracy. The results of the initial tests indicated that an improved method might be obtained by using different methods in different integration regions. Such a hybrid program was developed and found to be at least 1.5 to 2.0 times faster than any individual method. 18. Stochastic order parameter equation of isometric force production revealed by drift-diffusion estimates. PubMed Frank, T D; Friedrich, R; Beek, P J 2006-11-01 We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production. 19. Effective Schrödinger equation with general ordering ambiguity position-dependent mass Morse potential NASA Astrophysics Data System (ADS) Ikhdair, Sameer M. 2012-07-01 We solve the parametric generalized effective Schrödinger equation with a specific choice of position-dependent mass function and Morse oscillator potential by means of the Nikiforov-Uvarov method combined with the Pekeris approximation scheme. All bound-state energies are found explicitly and all corresponding radial wave functions are built analytically. We choose the Weyl or Li and Kuhn ordering for the ambiguity parameters in our numerical work to calculate the energy spectrum for a few (H2, LiH, HCl and CO) diatomic molecules with arbitrary vibration n and rotation l quantum numbers and different position-dependent mass functions. Two special cases including the constant mass and the vibration s-wave (l = 0) are also investigated. 20. Upwind methods for the Baer-Nunziato equations and higher-order reconstruction using artificial viscosity NASA Astrophysics Data System (ADS) Fraysse, F.; Redondo, C.; Rubio, G.; Valero, E. 2016-12-01 This article is devoted to the numerical discretisation of the hyperbolic two-phase flow model of Baer and Nunziato. A special attention is paid on the discretisation of intercell flux functions in the framework of Finite Volume and Discontinuous Galerkin approaches, where care has to be taken to efficiently approximate the non-conservative products inherent to the model equations. Various upwind approximate Riemann solvers have been tested on a bench of discontinuous test cases. New discretisation schemes are proposed in a Discontinuous Galerkin framework following the criterion of Abgrall and the path-conservative formalism. A stabilisation technique based on artificial viscosity is applied to the high-order Discontinuous Galerkin method and compared against classical TVD-MUSCL Finite Volume flux reconstruction. 1. High-order accurate solution of the incompressible Navier-Stokes equations on massively parallel computers NASA Astrophysics Data System (ADS) Henniger, R.; Obrist, D.; Kleiser, L. 2010-05-01 The emergence of "petascale" supercomputers requires us to develop today's simulation codes for (incompressible) flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power. In that spirit, we present a massively parallel high-order Navier-Stokes solver for large incompressible flow problems in three dimensions. The governing equations are discretized with finite differences in space and a semi-implicit time integration scheme. This discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers. The iterative solver for the pressure uses a highly efficient commutation-based preconditioner which is robust with respect to grid stretching. The efficiency of the implementation is further enhanced by carefully setting the (adaptive) termination criteria for the different iterative solvers. The computational work is distributed to different processing units by a geometric data decomposition in all three dimensions. This decomposition scheme ensures a low communication overhead and excellent scaling capabilities. The discretization is thoroughly validated. First, we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow. Second, we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria. Third, Orr-Sommerfeld and Squire eigenmodes for plane Poiseuille flow are simulated and compared to analytical results. Fourth, the practical applicability of the implementation is tested for transitional and turbulent channel flow. The results are compared to solutions from a pseudospectral solver. Subsequently, the performance of the commutation-based preconditioner for the pressure iteration is demonstrated. Finally, the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to 2. First and second order numerical methods based on a new convex splitting for phase-field crystal equation NASA Astrophysics Data System (ADS) Shin, Jaemin; Lee, Hyun Geun; Lee, June-Yub 2016-12-01 The phase-field crystal equation derived from the Swift-Hohenberg energy functional is a sixth order nonlinear equation. We propose numerical methods based on a new convex splitting for the phase-field crystal equation. The first order convex splitting method based on the proposed splitting is unconditionally gradient stable, which means that the discrete energy is non-increasing for any time step. The second order scheme is unconditionally weakly energy stable, which means that the discrete energy is bounded by its initial value for any time step. We prove mass conservation, unique solvability, energy stability, and the order of truncation error for the proposed methods. Numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods. Numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase-field crystal equation. 3. Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales NASA Astrophysics Data System (ADS) Han, Zhenlai; Sun, Shurong; Shi, Bao 2007-10-01 By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equationsx[Delta][Delta](t)+p(t)x[gamma]([tau](t))=0 on a time scale ; here [gamma] is a quotient of odd positive integers with p(t) real-valued positive rd-continuous functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1-18] but also unify the oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation. 4. Lax pair, conservation laws and Darboux transformation of the high-order Lax equation in fluid dynamics NASA Astrophysics Data System (ADS) Zheng, Wenxin; Wei, Guangmei 2017-03-01 KdV equation is investigated in fluid dynamics, plasma physics and other fields. By means of poseudopotential procedure, the high-order member of KdV hierarchy, ninth-order Lax's KdV equation in fluid dynamics is studied in this paper. Lax pair in AKNS form are derived from poseudopotential. Based on the Lax pair, an infinite number of conservation laws and Darboux transformation are constructed, and soliton solution is obtained by the Darboux transformation. 5. Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems NASA Astrophysics Data System (ADS) Pan, Hongjing; Xing, Ruixiang 2008-03-01 In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on . These results are not restricted to positive solutions. 6. The dynamics of second-order equations with delayed feedback and a large coefficient of delayed control NASA Astrophysics Data System (ADS) Kashchenko, Sergey A. 2016-12-01 The dynamics of second-order equations with nonlinear delayed feedback and a large coefficient of a delayed equation is investigated using asymptotic methods. Based on special methods of quasi-normal forms, a new construction is elaborated for obtaining the main terms of asymptotic expansions of asymptotic residual solutions. It is shown that the dynamical properties of the above equations are determined mostly by the behavior of the solutions of some special families of parabolic boundary value problems. A comparative analysis of the dynamics of equations with the delayed feedback of three types is carried out. 7. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM. PubMed Singh, Brajesh K; Srivastava, Vineet K 2015-04-01 The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. 8. Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM PubMed Central Singh, Brajesh K.; Srivastava, Vineet K. 2015-01-01 The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639 9. Higher-Order Wave Equation Within the Duffin-Kemmer-Petiau Formalism NASA Astrophysics Data System (ADS) Markov, Yu. A.; Markova, M. A.; Bondarenko, A. I. 2017-03-01 Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism a consistent approach to derivation of the third-order wave equation is suggested. For this purpose, an additional algebraic object, the so-called q-commutator (q is a primitive cubic root of unity) and a new set of matrices ημ instead of the original matrices βμ of the DKP algebra are introduced. It is shown that in terms of these η-matrices, we have succeeded to reduce the procedure of the construction of cubic root of the third-order wave operator to a few simple algebraic transformations and to a certain operation of passage to the limit z → q, where z is some complex deformation parameter entering into the definition of the ημ-matrices. A corresponding generalization of the result obtained to the case of interaction with an external electromagnetic field introduced through the minimal coupling scheme is performed. The application to the problem of construction within the DKP approach of the path integral representation in parasuperspace for the propagator of a massive vector particle in a background gauge field is discussed. 10. A high-order time formulation of the RBC schemes for unsteady compressible Euler equations NASA Astrophysics Data System (ADS) Lerat, A. 2015-12-01 Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5 × 5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock. 11. High-Order Integral Equation Methods for Diffraction Problems Involving Screens and Apertures NASA Astrophysics Data System (ADS) Lintner, Stephane K. This thesis presents a novel approach for the numerical solution of problems of diffraction by infinitely thin screens and apertures. The new methodology relies on combination of weighted versions of the classical operators associated with the Dirichlet and Neumann open-surface problems. In the two-dimensional case, a rigorous proof is presented, establishing that the new weighted formulations give rise to second-kind Fredholm integral equations, thus providing a generalization to open surfaces of the classical closed-surface Calderon formulae. High-order quadrature rules are introduced for the new weighted operators, both in the two-dimensional case as well as the scalar three-dimensional case. Used in conjunction with Krylov subspace iterative methods, these rules give rise to efficient and accurate numerical solvers which produce highly accurate solutions in small numbers of iterations, and whose performance is comparable to that arising from efficient high-order integral solvers recently introduced for closed-surface problems. Numerical results are presented for a wide range of frequencies and a variety of geometries in two- and three-dimensional space, including complex resonating structures as well as, for the first time, accurate numerical solutions of classical diffraction problems considered by the 19th-century pioneers: diffraction of high-frequency waves by the infinitely thin disc, the circular aperture, and the two-hole geometry inherent in Young's experiment. 12. W-transform for exponential stability of second order delay differential equations without damping terms. PubMed Domoshnitsky, Alexander; Maghakyan, Abraham; Berezansky, Leonid 2017-01-01 In this paper a method for studying stability of the equation [Formula: see text] not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation [Formula: see text] is not exponentially stable, the delay equation can be exponentially stable. 13. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations DOE PAGES Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; ... 2013-02-08 The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if andmore » only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.« less 14. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations SciTech Connect Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; Rosin, M. S.; Ricketson, L. F. 2013-02-08 The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration. 15. An adaptive multiblock high-order finite-volume method for solving the shallow-water equations on the sphere DOE PAGES McCorquodale, Peter; Ullrich, Paul; Johansen, Hans; ... 2015-09-04 We present a high-order finite-volume approach for solving the shallow-water equations on the sphere, using multiblock grids on the cubed-sphere. This approach combines a Runge--Kutta time discretization with a fourth-order accurate spatial discretization, and includes adaptive mesh refinement and refinement in time. Results of tests show fourth-order convergence for the shallow-water equations as well as for advection in a highly deformational flow. Hierarchical adaptive mesh refinement allows solution error to be achieved that is comparable to that obtained with uniform resolution of the most refined level of the hierarchy, but with many fewer operations. 16. PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions NASA Astrophysics Data System (ADS) Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P. 2012-10-01 We present a set of software routines in Maple 14 for solving first order ordinary differential equations (FOODEs). The package implements the Prelle-Singer method in its original form together with its extension to include integrating factors in terms of elementary functions. The package also presents a theoretical extension to deal with all FOODEs presenting Liouvillian solutions. Applications to ODEs taken from standard references show that it solves ODEs which remain unsolved using Maple's standard ODE solution routines. New version program summary Program title: PSsolver Catalogue identifier: ADPR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADPR_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2302 No. of bytes in distributed program, including test data, etc.: 31962 Distribution format: tar.gz Programming language: Maple 14 (also tested using Maple 15 and 16). Computer: Intel Pentium Processor P6000, 1.86 GHz. Operating system: Windows 7. RAM: 4 GB DDR3 Memory Classification: 4.3. Catalogue identifier of previous version: ADPR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 144 (2002) 46 Does the new version supersede the previous version?: Yes Nature of problem: Symbolic solution of first order differential equations via the Prelle-Singer method. Solution method: The method of solution is based on the standard Prelle-Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included. Reasons for new version: The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines. Summary 17. A survey on orthogonal matrix polynomials satisfying second order differential equations NASA Astrophysics Data System (ADS) Duran, Antonio J.; Grunbaum, F. Alberto 2005-06-01 The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials.Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949, see (AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143, Dokl. Akad. Nauk SSSR 69(2) (1949) 125). In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n. The aim of this paper is to give an overview of the techniques that have led to these examples, a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples. 18. Analyzing a stochastic time series obeying a second-order differential equation. PubMed Lehle, B; Peinke, J 2015-06-01 The stochastic properties of a Langevin-type Markov process can be extracted from a given time series by a Markov analysis. Also processes that obey a stochastically forced second-order differential equation can be analyzed this way by employing a particular embedding approach: To obtain a Markovian process in 2N dimensions from a non-Markovian signal in N dimensions, the system is described in a phase space that is extended by the temporal derivative of the signal. For a discrete time series, however, this derivative can only be calculated by a differencing scheme, which introduces an error. If the effects of this error are not accounted for, this leads to systematic errors in the estimation of the drift and diffusion functions of the process. In this paper we will analyze these errors and we will propose an approach that correctly accounts for them. This approach allows an accurate parameter estimation and, additionally, is able to cope with weak measurement noise, which may be superimposed to a given time series. 19. High-order virial coefficients and equation of state for hard sphere and hard disk systems. PubMed Hu, Jiawen; Yu, Yang-Xin 2009-11-07 A very simple and accurate approach is proposed to predict the high-order virial coefficients of hard spheres and hard disks. In the approach, the nth virial coefficient B(n) is expressed as the sum of n(D-1) and a remainder, where D is the spatial dimension of the system. When n > or = 3, the remainders of the virials can be accurately expressed with Padé-type functions of n. The maximum deviations of predicted B(5)-B(10) for the two systems are only 0.0209%-0.0044% and 0.0390%-0.0525%, respectively, which are much better than the numerous existing approaches. The virial equation based on the predicted virials diverges when packing fraction eta = 1. With the predicted virials, the compressibility factors of hard sphere system can be predicted very accurately in the whole stable fluid region, and those in the metastable fluid region can also be well predicted up to eta = 0.545. The compressibility factors of hard disk fluid can be predicted very accurately up to eta = 0.63. The simulated B(7) and B(10) for hard spheres are found to be inconsistent with the other known virials and therefore they are modified as 53.2467 and 105.042, respectively. 20. A New Discretization Method of Order Four for the Numerical Solution of One-Space Dimensional Second-Order Quasi-Linear Hyperbolic Equation ERIC Educational Resources Information Center Mohanty, R. K.; Arora, Urvashi 2002-01-01 Three level-implicit finite difference methods of order four are discussed for the numerical solution of the mildly quasi-linear second-order hyperbolic equation A(x, t, u)u[subscript xx] + 2B(x, t, u)u[subscript xt] + C(x, t, u)u[subscript tt] = f(x, t, u, u[subscript x], u[subscript t]), 0 less than x less than 1, t greater than 0 subject to… 1. Higher-order splitting algorithms for solving the nonlinear Schrödinger equation and their instabilities. PubMed Chin, Siu A 2007-11-01 Since the kinetic and potential energy terms of the real-time nonlinear Schrödinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well-known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schrödinger equation, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high-wave-number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for Deltatkmax2 < or =2pi, where kmax=pi/Deltax. 2. Nonlinear waves described by a fifth-order equation derived from the Fermi-Pasta-Ulam system NASA Astrophysics Data System (ADS) Volkov, A. K.; Kudryashov, N. A. 2016-04-01 Nonlinear wave processes described by a fifth-order generalized KdV equation derived from the Fermi-Pasta-Ulam (FPU) model are considered. It is shown that, in contrast to the KdV equation, which demonstrates the recurrence of initial states and explains the FPU paradox, the fifthorder equation fails to pass the Painlevé test, is not integrable, and does not exhibit the recurrence of the initial state. The results of this paper show that the FPU paradox occurs only at an initial stage of a numerical experiment, which is explained by the existence of KdV solitons only on a bounded initial time interval. 3. Singularity confinement for a class of m-th order difference equations of combinatorics. PubMed Adler, Mark; van Moerbeke, Pierre; Vanhaecke, Pol 2008-03-28 In a recent publication, it was shown that a large class of integrals over the unitary group U(n) satisfy nonlinear, non-autonomous difference equations over n, involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again ("singularity confinement"). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painlevé property for the discrete relations is inherited from the Painlevé property of the (continuous) Toeplitz lattice. 4. Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on nonuniform backgrounds NASA Astrophysics Data System (ADS) Chin, Siu A.; Ashour, Omar A.; Nikolić, Stanko N.; Belić, Milivoj R. 2017-01-01 Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. In this work, we use the Darboux transformation to prove, in a unified manner and without knowing the analytical form of the background solution, that the peak height of a high-order breather is just a sum of peak heights of first-order breathers plus that of the background, irrespective of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrödinger equations, such as the Hirota equation, are indicated. 5. Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on nonuniform backgrounds. PubMed Chin, Siu A; Ashour, Omar A; Nikolić, Stanko N; Belić, Milivoj R 2017-01-01 Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. In this work, we use the Darboux transformation to prove, in a unified manner and without knowing the analytical form of the background solution, that the peak height of a high-order breather is just a sum of peak heights of first-order breathers plus that of the background, irrespective of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrödinger equations, such as the Hirota equation, are indicated. 6. Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation. PubMed Yang, Yunqing; Yan, Zhenya; Malomed, Boris A 2015-10-01 We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems. 7. A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations NASA Technical Reports Server (NTRS) Liu, Yen; Kwak, Dochan (Technical Monitor) 1995-01-01 A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind 8. ERKN integrators for systems of oscillatory second-order differential equations NASA Astrophysics Data System (ADS) Wu, Xinyuan; You, Xiong; Shi, Wei; Wang, Bin 2010-11-01 For systems of oscillatory second-order differential equations y+My=f with M∈R, a symmetric positive semi-definite matrix, X. Wu et al. have proposed the multidimensional ARKN methods [X. Wu, X. You, J. Xia, Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250-2257], which are an essential generalization of J.M. Franco's ARKN methods for one-dimensional problems or for systems with a diagonal matrix M=wI [J.M. Franco, Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770-787]. One of the merits of these methods is that they integrate exactly the unperturbed oscillators y+My=0. Regretfully, even for the unperturbed oscillators the internal stages Y of an ARKN method fail to equal the values of the exact solution y(t) at t+ch, respectively. Recently H. Yang et al. proposed the ERKN methods to overcome this drawback [H.L. Yang, X.Y. Wu, Xiong You, Yonglei Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777-1794]. However, the ERKN methods in that paper are only considered for the special case where M is a diagonal matrix with nonnegative entries. The purpose of this paper is to extend the ERKN methods to the general case with M∈R, and the perturbing function f depends only on y. Numerical experiments accompanied demonstrates that the ERKN methods are more efficient than the existing methods for the computation of oscillatory systems. In particular, if M∈R is a symmetric positive semi-definite matrix, it is highly important for the new ERKN integrators to show the energy conservation in the numerical experiments for problems with Hamiltonian H(p,q)=1/2 >pp+1/2 >qMq+V(q) in comparison with the well-known methods in the scientific literature. Those so called separable Hamiltonians arise in many areas of physical sciences, e.g., macromolecular dynamics, astronomy, and classical 9. Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations. PubMed Cooper, F; Hyman, J M; Khare, A 2001-08-01 Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable. 10. Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. Part I NASA Astrophysics Data System (ADS) Hibino, Masaki This article part I and the forthcoming part II are concerned with the study of the Borel summability of divergent power series solutions for singular first-order linear partial differential equations of nilpotent type. Under one restriction on equations, we can divide them into two classes. In this part I, we deal with the one class and obtain the conditions under which divergent solutions are Borel summable. (The other class will be studied in part II.) In order to assure the Borel summability of divergent solutions, global analytic continuation properties for coefficients are required despite of the fact that the domain of the Borel sum is local. 11. A direct multi-step Legendre-Gauss collocation method for high-order Volterra integro-differential equation NASA Astrophysics Data System (ADS) Kajani, M. Tavassoli; Gholampoor, I. 2015-10-01 The purpose of this study is to present a new direct method for the approximate solution and approximate derivatives up to order k to the solution for kth-order Volterra integro-differential equations with a regular kernel. This method is based on the approximation by shifting the original problem into a sequence of subintervals. A Legendre-Gauss-Lobatto collocation method is proposed to solving the Volterra integro-differential equation. Numerical examples show that the approximate solutions have a good degree of accuracy. 12. Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters NASA Astrophysics Data System (ADS) Gaillard, Pierre; Gastineau, Mickaël 2016-06-01 In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the ( x, t) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending on t. This formulation gives in the case of the order 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others. 13. Reflecting Solutions of High Order Elliptic Differential Equations in Two Independent Variables Across Analytic Arcs. Ph.D. Thesis NASA Technical Reports Server (NTRS) Carleton, O. 1972-01-01 Consideration is given specifically to sixth order elliptic partial differential equations in two independent real variables x, y such that the coefficients of the highest order terms are real constants. It is assumed that the differential operator has distinct characteristics and that it can be factored as a product of second order operators. By analytically continuing into the complex domain and using the complex characteristic coordinates of the differential equation, it is shown that its solutions, u, may be reflected across analytic arcs on which u satisfies certain analytic boundary conditions. Moreover, a method is given whereby one can determine a region into which the solution is extensible. It is seen that this region of reflection is dependent on the original domain of difinition of the solution, the arc and the coefficients of the highest order terms of the equation and not on any sufficiently small quantities; i.e., the reflection is global in nature. The method employed may be applied to similar differential equations of order 2n. 14. Solving singular perturbation problem of second order ordinary differential equation using the method of matched asymptotic expansion (MMAE) NASA Astrophysics Data System (ADS) Mohamed, Firdawati binti; Karim, Mohamad Faisal bin Abd 2015-10-01 Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution. 15. A new analytical approach to solve some of the fractional-order partial differential equations NASA Astrophysics Data System (ADS) Manafian, Jalil; Lakestani, Mehrdad 2017-03-01 The aim of the present paper is to present an analytical method for the time fractional biological population model, time fractional Burgers, time fractional Cahn-Hilliard, space-time fractional Whitham-Broer-Kaup, space-time fractional Fokas equations by using the generalized tanh-coth method. The fractional derivative is described in the sense of the modified Riemann-Liouville derivatives. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. We have obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these fractional equations to ordinary differential equations which subsequently resulted into number of exact solutions. 16. Sobolev type equations of time-fractional order with periodical boundary conditions NASA Astrophysics Data System (ADS) Plekhanova, Marina 2016-08-01 The existence of a unique local solution for a class of time-fractional Sobolev type partial differential equations endowed by the Cauchy initial conditions and periodical with respect to every spatial variable boundary conditions on a parallelepiped is proved. General results are applied to study of the unique solvability for the initial boundary value problem to Benjamin-Bona-Mahony-Burgers and Allair partial differential equations. 17. On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences NASA Astrophysics Data System (ADS) Halim, Yacine; Bayram, Mustafa 2016-07-01 This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \\begin{equation*} x_{n+1}=\\frac{\\alpha x_{n-1}+\\beta}{ \\gamma x_{n}x_{n-1}},\\qquad n \\in \\mathbb{N}_{0}, \\end{equation*} where$\\mathbb{N}_{0}=\\mathbb{N}\\cup \\left\\{0\\right\\}$,$\\alpha,\\beta,\\gamma\\in\\mathbb{R}^{+}$, and the initial conditions$x_{-1}$and$x_{0}\$ are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by \\begin{equation*} x_{n+1} = \\frac{\\alpha x_{n-1} + \\beta}{\\gamma y_n x_{n-1}}, \\qquad y_{n+1} = \\frac{\\alpha y_{n-1} +\\beta}{\\gamma x_n y_{n-1}} ,\\qquad n\\in \\mathbb{N}_0, \\end{equation*} and this generalizes the results presented in \\cite{yazlik}

18. A family of fourth-order entropy stable nonoscillatory spectral collocation schemes for the 1-D Navier-Stokes equations

Yamaleev, Nail K.; Carpenter, Mark H.

2017-02-01

High-order numerical methods that satisfy a discrete analog of the entropy inequality are uncommon. Indeed, no proofs of nonlinear entropy stability currently exist for high-order weighted essentially nonoscillatory (WENO) finite volume or weak-form finite element methods. Herein, a new family of fourth-order WENO spectral collocation schemes is developed, that are nonlinearly entropy stable for the one-dimensional compressible Navier-Stokes equations. Individual spectral elements are coupled using penalty type interface conditions. The resulting entropy stable WENO spectral collocation scheme achieves design order accuracy, maintains the WENO stencil biasing properties across element interfaces, and satisfies the summation-by-parts (SBP) operator convention, thereby ensuring nonlinear entropy stability in a diagonal norm. Numerical results demonstrating accuracy and nonoscillatory properties of the new scheme are presented for the one-dimensional Euler and Navier-Stokes equations for both continuous and discontinuous compressible flows.

19. Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrödinger equation and their systematic generation

Chin, Siu A.; Ashour, Omar A.; Nikolić, Stanko N.; Belić, Milivoj R.

2016-10-01

It is well known that Akhmediev breathers of the nonlinear cubic Schrödinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds linearly to form the peak height of the final breather. Using this peak-height formula, we show that at any given periodicity, there exists a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higher-order breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrödinger equation.

20. Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations

Paliathanasis, Andronikos; Tsamparlis, Michael

2016-09-01

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with n independent and m dependent variables (n × m systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation laws. We prove that the point symmetries are generated by the collineations of two (pseudo)metrics, which are defined in the spaces of independent and dependent variables. We demonstrate the general results in two special cases (a) a system of m coupled Laplace equations and (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle. In the second case we determine the complete invariant group of point transformations, and we apply the Lie invariants in order to find invariant solutions of the wave function for a spin-0 particle in the two dimensional hyperbolic space.

1. Instability criteria and pattern formation in the complex Ginzburg-Landau equation with higher-order terms.

PubMed

Mohamadou, Alidou; Ayissi, Bebe Emilienne; Kofané, Timoléon Crépin

2006-10-01

We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings. Modulational instability mediates pattern formation through the lattice. The main feature of the traveling plane waves is its disintegration in pulse train during the propagation through the system.

2. Breathers and rogue waves for an eighth-order nonlinear Schrödinger equation in an optical fiber

Hu, Wen-Qiang; Gao, Yi-Tian; Zhao, Chen; Lan, Zhong-Zhou

2017-02-01

In this paper, an eighth-order nonlinear Schrödinger equation is investigated in an optical fiber, which can be used to describe the propagation of ultrashort nonlinear pulses. Lax pair and infinitely-many conservation laws are derived to verify the integrability of this equation. Via the Darboux transformation and generalized Darboux transformation, the analytic breather and rogue wave solutions are obtained. Influence of the coefficients of operators in this equation, which represent different order nonlinearity, and the spectral parameter on the propagation and interaction of the breathers and rogue waves is also discussed. We find that (i) the periodic of the breathers decreases as the augment of the spectral parameter; (ii) the coefficients of operators change the compressibility and periodic of the breathers, and can affect the interaction range and temporal-spatial distribution of the rogue waves.

3. Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

Ling, Liming; Feng, Bao-Feng; Zhu, Zuonong

2016-07-01

In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N-bright soliton solution in a compact determinant form, the N-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigorously for both the N-soliton and the N-breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.

4. On first- and second-order difference schemes for differential-algebraic equations of index at most two

Bulatov, M. V.; Ming-Gong, Lee; Solovarova, L. S.

2010-11-01

Difference schemes of the Euler and trapezoidal types for the numerical solution of the initial-value problem for linear differential-algebraic equations are examined. These schemes are analyzed for model examples, and their superiority over the familiar first- and second-order implicit methods is shown. Conditions for the convergence of the proposed algorithms are formulated.

5. Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

SciTech Connect

Casas, E.

1999-03-15

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem.

6. Weak interaction for higher-order nonlinear Schrödinger equation: An application of soliton perturbation

Eskandar, S.; Hoseini, S. M.

2017-04-01

Using soliton perturbation theory, we analytically study weak interaction for a higher-order nonlinear Schrödinger equation. An ansatz consists of two well-separate single solitons is considered and slow variation of solitons parameters are found. Twelve different scenarios for when the initial velocities are zero are observed. A good comparison is found between numerical and analytical results.

7. Improved Accuracy of the Asymmetric Second-Order Vegetation Isoline Equation over the RED-NIR Reflectance Space.

PubMed

Miura, Munenori; Obata, Kenta; Taniguchi, Kenta; Yoshioka, Hiroki

2017-02-24

The relationship between two reflectances of different bands is often encountered in cross calibration and parameter retrievals from remotely-sensed data. The asymmetric-order vegetation isoline is one such relationship, derived previously, where truncation error was reduced from the first-order approximated isoline by including a second-order term. This study introduces a technique for optimizing the magnitude of the second-order term and further improving the isoline equation's accuracy while maintaining the simplicity of the derived formulation. A single constant factor was introduced into the formulation to adjust the second-order term. This factor was optimized by simulating canopy radiative transfer. Numerical experiments revealed that the errors in the optimized asymmetric isoline were reduced in magnitude to nearly 1/25 of the errors obtained from the first-order vegetation isoline equation, and to nearly one-fifth of the error obtained from the non-optimized asymmetric isoline equation. The errors in the optimized asymmetric isoline were compared with the magnitudes of the signal-to-noise ratio (SNR) estimates reported for four specific sensors aboard four Earth observation satellites. These results indicated that the error in the asymmetric isoline could be reduced to the level of the SNR by adjusting a single factor.

8. Reduced order feedback control equations for linear time and frequency domain analysis

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1981-01-01

An algorithm was developed which can be used to obtain the equations. In a more general context, the algorithm computes a real nonsingular similarity transformation matrix which reduces a real nonsymmetric matrix to block diagonal form, each block of which is a real quasi upper triangular matrix. The algorithm works with both defective and derogatory matrices and when and if it fails, the resultant output can be used as a guide for the reformulation of the mathematical equations that lead up to the ill conditioned matrix which could not be block diagonalized.

9. Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations.

PubMed

Pandey, Vikash; Holm, Sverre

2016-12-01

The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796-2815 (2000)] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and the shear wave equation derived from the GS model are shown to take the form of the Kelvin-Voigt fractional-derivative wave equation and the fractional diffusion-wave equation, respectively. Therefore, an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the GS model to establish the equivalence of the respective wave equations. Further, a physical interpretation of the characteristic fractional order present in the wave equations is inferred from the GS model. The overall goal is to show that fractional calculus is not just a mathematical framework which can be used to curve-fit the complex behavior of materials. Rather, it can also be derived from real physical processes as illustrated in this work by the example of GS.

10. On the Well-Definedness of the Order of an Ordinary Differential Equation

ERIC Educational Resources Information Center

Dobbs, David E.

2006-01-01

It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical…

11. High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Bryan (Technical Monitor)

2002-01-01

We present the first fifth order, semi-discrete central upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Tadmor-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spacial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.

12. Higher-order rational solitons and rogue-like wave solutions of the (2 + 1)-dimensional nonlinear fluid mechanics equations

Wen, Xiao-Yong; Yan, Zhenya

2017-02-01

The novel generalized perturbation (n, M)-fold Darboux transformations (DTs) are reported for the (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation and its extension by using the Taylor expansion of the Darboux matrix. The generalized perturbation (1 , N - 1) -fold DTs are used to find their higher-order rational solitons and rogue wave solutions in terms of determinants. The dynamics behaviors of these rogue waves are discussed in detail for different parameters and time, which display the interesting RW and soliton structures including the triangle, pentagon, heptagon profiles, etc. Moreover, we find that a new phenomenon that the parameter (a) can control the wave structures of the KP equation from the higher-order rogue waves (a ≠ 0) into higher-order rational solitons (a = 0) in (x, t)-space with y = const . These results may predict the corresponding dynamical phenomena in the models of fluid mechanics and other physically relevant systems.

13. High-order compact ADI method using predictor-corrector scheme for 2D complex Ginzburg-Landau equation

Shokri, Ali; Afshari, Fatemeh

2015-12-01

In this article, a high-order compact alternating direction implicit (HOC-ADI) finite difference scheme is applied to numerical solution of the complex Ginzburg-Landau (GL) equation in two spatial dimensions with periodical boundary conditions. The GL equation has been used as a mathematical model for various pattern formation systems in mechanics, physics, and chemistry. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. To avoid solving the nonlinear system and to increase the accuracy and efficiency of the method, we proposed the predictor-corrector scheme. Validation of the present numerical solutions has been conducted by comparing with the exact and other methods results and evidenced a good agreement.

14. An exact reformulation of the diagonalization step in electronic structure calculations as a set of second order nonlinear equations.

PubMed

2004-06-08

A new formulation of the diagonalization step in self-consistent-field (SCF) electronic structure calculations is presented. It exactly replaces the diagonalization of the effective Hamiltonian with the solution of a set of second order nonlinear equations. The density matrix and/or the new set of occupied orbitals can be directly obtained from the resulting solution. This formulation may offer interesting possibilities for new approaches to efficient SCF calculations. The working equations can be derived either from energy minimization with respect to a Cayley-type parametrization of a unitary matrix, or from a similarity transformation approach.

15. Couple of the variational iteration method and fractional-order Legendre functions method for fractional differential equations.

PubMed

Yin, Fukang; Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid "noise terms" is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique.

16. Some efficient methods for obtaining infinite series solutions of n-th order linear ordinary differential equations

NASA Technical Reports Server (NTRS)

Allen, G.

1972-01-01

The use of the theta-operator method and generalized hypergeometric functions in obtaining solutions to nth-order linear ordinary differential equations is explained. For completeness, the analysis of the differential equation to determine whether the point of expansion is an ordinary point or a regular singular point is included. The superiority of the two methods shown over the standard method is demonstrated by using all three of the methods to work out several examples. Also included is a compendium of formulae and properties of the theta operator and generalized hypergeometric functions which is complete enough to make the report self-contained.

17. L_p-estimates for the nontangential maximal function of the solution to a second-order elliptic equation

Gushchin, A. K.

2016-10-01

The paper is concerned with the properties of the solution to a Dirichlet problem for a homogeneous second-order elliptic equation with L_p-boundary function, p>1. The same conditions are imposed on the coefficients of the equation and the boundary of the bounded domain as were used to establish the solvability of this problem. The L_p-norm of the nontangential maximal function is estimated in terms of the L_p-norm of the boundary value. This result depends on a new estimate, proved below, for the nontangential maximal function in terms of an analogue of the Lusin area integral. Bibliography: 31 titles.

18. A Two Colorable Fourth Order Compact Difference Scheme and Parallel Iterative Solution of the 3D Convection Diffusion Equation

NASA Technical Reports Server (NTRS)

Zhang, Jun; Ge, Lixin; Kouatchou, Jules

2000-01-01

A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it Only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with the Gauss-Seidel type iterative method. This is compared with the known 19 point fourth order compact differenCe scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and the 19 point fourth order compact schemes.

19. A Non-Dissipative Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations

NASA Technical Reports Server (NTRS)

Yefet, Amir; Petropoulos, Peter G.

1999-01-01

We consider a divergence-free non-dissipative fourth-order explicit staggered finite difference scheme for the hyperbolic Maxwell's equations. Special one-sided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is long-time stable, and is fourth-order convergent over complex domains that include dielectric interfaces and perfectly conducting surfaces. We also examine the scheme's behavior near metal surfaces that are not aligned with the grid axes, and compare its accuracy to that obtained by the Yee scheme.

20. High order multi-grid methods to solve the Poisson equation

NASA Technical Reports Server (NTRS)

Schaffer, S.

1981-01-01

High order multigrid methods based on finite difference discretization of the model problem are examined. The following methods are described: (1) a fixed high order FMG-FAS multigrid algorithm; (2) the high order methods; and (3) results are presented on four problems using each method with the same underlying fixed FMG-FAS algorithm.

1. A Study into Discontinuous Galerkin Methods for the Second Order Wave Equation

DTIC Science & Technology

2015-06-01

interpolation does a good job approximating the function as compared to the exact solution (denoted by ∗ to differentiate from the various interpolations). In...200 words) There are numerous numerical methods for solving different types of partial differential equations (PDEs) that describe the physical...of the method through an energy analysis. v THIS PAGE INTENTIONALLY LEFT BLANK vi Table of Contents 1 An Introduction to Solving Partial Differential

2. Exploration of POD-Galerkin Techniques for Developing Reduced Order Models of the Euler Equations

DTIC Science & Technology

2015-07-01

frequencies at the inlet following the FTF/FDF ( Flame Transfer/Describing Function) approach [19]. The perturbation level ε in Eq. (4) is set to 0.1...equations need pre- treatment before the POD eigen-basis calculation. In this section, the four variables (u’, p’, T’ and Y’ox) are normalized by...Merkle, C., and Sankaran, V. "Exploration of POD-Galerkin Method in Developing a Flame Model for Combustion Instability Problems," 7th AIAA

3. Development of High-Order Method for Multi-Physics Problems Governed by Hyperbolic Equations

DTIC Science & Technology

2012-08-01

implicit time marching with large time steps. 4.1 Background The one equation Spalart -Almaras (SA) turbulence model [18-21] in conservative...20] Spalart , P.R., Jou W-H, Strelets, M., Allmaras , S.R.. “Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach,” In...offer significant advantages for the simulation of complex flows and turbulence in non trivial geometries of interest to practical applications. The

4. An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier Stokes equations

Hartmann, Ralf; Houston, Paul

2008-11-01

In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest, such as the lift and drag coefficients of a body immersed in a viscous fluid. With this in mind, the key ingredients in the construction of the method include: (i) an adjoint consistent imposition of the boundary conditions; (ii) an adjoint consistent reformulation of the underlying target functional of practical interest; (iii) design of appropriate interior penalty stabilization terms. Numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the L2-norm, as well as for certain target functionals. Computational comparisons with other discontinuous Galerkin schemes proposed in the literature, including the second scheme of Bassi and Rebay, cf. [F. Bassi, S. Rebay, GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, in: B. Cockburn, G. Karniadakis, C.-W. Shu (Eds.), Discontinuous Galerkin Methods, Lecture Notes in Comput. Sci. Engrg., vol. 11, Springer, Berlin, 2000, pp. 197-208; F. Bassi, S. Rebay, Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 40 (2002) 197-207], the standard SIPG method outlined in [R. Hartmann, P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. I: Method formulation, Int. J. Numer. Anal. Model. 3(1) (2006) 1-20], and an NIPG variant of the new scheme will be undertaken.

5. High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations

DTIC Science & Technology

2008-09-01

Bérenger [11] for the 2-D Maxwell equations. This absorbing layer method surrounds the computational domain with a dispersive medium, defined in such a...no advection or forcing terms. After demon - strating the validity of this prototypical implementation in Section B, we proceed to incorporate the...may improve these results [55], but for the purpose of this dissertation, it is sufficient to demon - strate how to use the auxiliary variable NRBC

6. Developments in the Theory of Nonlinear First-Order Partial Differential Equations.

DTIC Science & Technology

1983-12-01

weakenings of the classical notion of solution lead to nonuniqueness . However, in view of the way these problems arise in applications - in particular...S(ii) If u and v are uniformly continuous and (H3) holds, then u v. (iii) If u and v are Lipschitz continuous, then u = v. This result in fact...special case of viscosity solutions which are Lipschitz continuous (and hence satisfy the equation almost everywhere). Other uniqueness results

7. Variational Multiscale Stabilization of High-Order Spectral Elements for the Convection-Diffusion Equation

DTIC Science & Technology

2012-06-19

Squares [13] for advection- diffusion with a reaction term, or the Unusual Stabilized Finite Element Method (USFEM) [14, 15] are a few examples. In...the underlying numerical scheme [21]. However, Godunov’s theorem [22] implies that the latter property may be violated in the prox - imity of...capturing finite element for- mulations for nonlinear convection-diffusion- reaction equations, Com- put. Methods Appl. Mech. and Engrg. 59 (1986) 307–325

8. Effective equations for matter-wave gap solitons in higher-order transversal states.

PubMed

2013-10-01

We demonstrate that an important class of nonlinear stationary solutions of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) exhibiting nontrivial transversal configurations can be found and characterized in terms of an effective one-dimensional (1D) model. Using a variational approach we derive effective equations of lower dimensionality for BECs in (m,n(r)) transversal states (states featuring a central vortex of charge m as well as n(r) concentric zero-density rings at every z plane) which provides us with a good approximate solution of the original 3D problem. Since the specifics of the transversal dynamics can be absorbed in the renormalization of a couple of parameters, the functional form of the equations obtained is universal. The model proposed finds its principal application in the study of the existence and classification of 3D gap solitons supported by 1D optical lattices, where in addition to providing a good estimate for the 3D wave functions it is able to make very good predictions for the μ(N) curves characterizing the different fundamental families. We have corroborated the validity of our model by comparing its predictions with those from the exact numerical solution of the full 3D GPE.

9. Effective equations for matter-wave gap solitons in higher-order transversal states

2013-10-01

We demonstrate that an important class of nonlinear stationary solutions of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) exhibiting nontrivial transversal configurations can be found and characterized in terms of an effective one-dimensional (1D) model. Using a variational approach we derive effective equations of lower dimensionality for BECs in (m,nr) transversal states (states featuring a central vortex of charge m as well as nr concentric zero-density rings at every z plane) which provides us with a good approximate solution of the original 3D problem. Since the specifics of the transversal dynamics can be absorbed in the renormalization of a couple of parameters, the functional form of the equations obtained is universal. The model proposed finds its principal application in the study of the existence and classification of 3D gap solitons supported by 1D optical lattices, where in addition to providing a good estimate for the 3D wave functions it is able to make very good predictions for the μ(N) curves characterizing the different fundamental families. We have corroborated the validity of our model by comparing its predictions with those from the exact numerical solution of the full 3D GPE.

10. Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed

Canestrelli, Alberto; Dumbser, Michael; Siviglia, Annunziato; Toro, Eleuterio F.

2010-03-01

In this paper, we study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water equations and bed-load transport following a coupled solution strategy. The resulting system of governing equations contains non-conservative products and it is solved simultaneously within each time step. The numerical solution is obtained using a new high-order accurate centered scheme of the finite volume type on unstructured meshes, which is an extension of the one-dimensional PRICE-C scheme recently proposed in Canestrelli et al. (2009) [5]. The resulting first-order accurate centered method is then extended to high order of accuracy in space via a high order WENO reconstruction technique and in time via a local continuous space-time Galerkin predictor method. The scheme is applied to the shallow water equations and the well-balanced properties of the method are investigated. Finally, we apply the new scheme to different test cases with both fixed and movable bed. An attractive future of the proposed method is that it is particularly suitable for engineering applications since it allows practitioners to adopt the most suitable sediment transport formula which better fits the field data.

11. High-order integral equations for electromagnetic problems in layered media with applications in biology and solar cells

Zinser, Brian

We present two distinct mathematical models where high-order integral equations are applied to electromagnetic problems. The first problem is to find the electric potential in and around ion channels and Janus particles. The second problem is to find the electromagnetic scattering caused by a set of simple geometric objects. In biology, we consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. A boundary element method (BEM) for the Poisson-Boltzmann equation based on Muller's hyper-singular second kind integral equation formulation is used to accurately compute electrostatic potentials. The proposed BEM gives O(1) condition numbers and we show that the second order basis converges faster and is more accurate than the first order basis. For solar cells, we develop a Nystrom volume integral equation (VIE) method for calculating the electromagnetic scattering according to the Maxwell equations. The Cauchy principal values (CPVs) that arise from the VIE are computed using a finite size exclusion volume with explicit correction integrals. Outside the exclusion, the hyper-singular integrals are computed using an interpolated quadrature formulae with tensor-product quadrature nodes. We considered cubes, rectangles, cylinders, spheres, and ellipsoids. As the new quadrature weights are pre-calculated and tabulated, the integrals are calculated efficiently at runtime. Simulations with many scatterers demonstrate the efficiency of the interpolated quadrature formulae. We also demonstrate that the resulting VIE has high accuracy and p-convergence.

12. Improved Accuracy of the Asymmetric Second-Order Vegetation Isoline Equation over the RED–NIR Reflectance Space

PubMed Central

Miura, Munenori; Obata, Kenta; Taniguchi, Kenta; Yoshioka, Hiroki

2017-01-01

The relationship between two reflectances of different bands is often encountered in cross calibration and parameter retrievals from remotely-sensed data. The asymmetric-order vegetation isoline is one such relationship, derived previously, where truncation error was reduced from the first-order approximated isoline by including a second-order term. This study introduces a technique for optimizing the magnitude of the second-order term and further improving the isoline equation’s accuracy while maintaining the simplicity of the derived formulation. A single constant factor was introduced into the formulation to adjust the second-order term. This factor was optimized by simulating canopy radiative transfer. Numerical experiments revealed that the errors in the optimized asymmetric isoline were reduced in magnitude to nearly 1/25 of the errors obtained from the first-order vegetation isoline equation, and to nearly one-fifth of the error obtained from the non-optimized asymmetric isoline equation. The errors in the optimized asymmetric isoline were compared with the magnitudes of the signal-to-noise ratio (SNR) estimates reported for four specific sensors aboard four Earth observation satellites. These results indicated that the error in the asymmetric isoline could be reduced to the level of the SNR by adjusting a single factor. PMID:28245566

13. An Accurate Theory and Simple Fourth Order Governing Equations for Orthotropic and Composite Cylindrical Shells.

DTIC Science & Technology

1983-10-01

following basic equations can be deduced for orthotropic circular cylindrical shells. Let a be the radius of the midsurface of the shell, x, y, z the...axial, circumferential and radial coordinates and a, a the dimensionless midsurface coordinates along lines of curvatures (a - , a - . The threea a...8217The components of strain at an arbitrary point of the shell are related to the midsurface displacements by [8,15,16] e ( 1 v , 3 2w e a a a ,2)- 0 a

14. A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation

Beshtokov, M. Kh.

2014-09-01

A nonlocal boundary value problem for a third-order hyperbolic equation with variable coefficients is considered in the one- and multidimensional cases. A priori estimates for the nonlocal problem are obtained in the differential and difference formulations. The estimates imply the stability of the solution with respect to the initial data and the right-hand side on a layer and the convergence of the difference solution to the solution of the differential problem.

15. Kershaw closures for linear transport equations in slab geometry II: High-order realizability-preserving discontinuous-Galerkin schemes

Schneider, Florian

2016-10-01

This paper provides a generalization of the realizability-preserving discontinuous-Galerkin scheme given in [3] to general full-moment models that can be closed analytically. It is applied to the class of Kershaw closures, which are able to provide a cheap closure of the moment problem. This results in an efficient algorithm for the underlying linear transport equation. The efficiency of high-order methods is demonstrated using numerical convergence tests and non-smooth benchmark problems.

16. Boundary and Interface Conditions for High Order Finite Difference Methods Applied to the Euler and Navier-Strokes Equations

NASA Technical Reports Server (NTRS)

Nordstrom, Jan; Carpenter, Mark H.

1998-01-01

Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.

17. A stable high-order finite difference scheme for the compressible Navier Stokes equations: No-slip wall boundary conditions

Svärd, Magnus; Nordström, Jan

2008-05-01

A stable wall boundary procedure is derived for the discretized compressible Navier-Stokes equations. The procedure leads to an energy estimate for the linearized equations. We discretize the equations using high-order accurate finite difference summation-by-parts (SBP) operators. The boundary conditions are imposed weakly with penalty terms. We prove linear stability for the scheme including the wall boundary conditions. The penalty imposition of the boundary conditions is tested for the flow around a circular cylinder at Ma=0.1 and Re=100. We demonstrate the robustness of the SBP-SAT technique by imposing incompatible initial data and show the behavior of the boundary condition implementation. Using the errors at the wall we show that higher convergence rates are obtained for the high-order schemes. We compute the vortex shedding from a circular cylinder and obtain good agreement with previously published (computational and experimental) results for lift, drag and the Strouhal number. We use our results to compare the computational time for a given for a accuracy and show the superior efficiency of the 5th-order scheme.

18. First-order system least-squares for the Helmholtz equation

SciTech Connect

Lee, B.; Manteuffel, T.; McCormick, S.; Ruge, J.

1996-12-31

We apply the FOSLS methodology to the exterior Helmholtz equation {Delta}p + k{sup 2}p = 0. Several least-squares functionals, some of which include both H{sup -1}({Omega}) and L{sup 2}({Omega}) terms, are examined. We show that in a special subspace of [H(div; {Omega}) {intersection} H(curl; {Omega})] x H{sup 1}({Omega}), each of these functionals are equivalent independent of k to a scaled H{sup 1}({Omega}) norm of p and u = {del}p. This special subspace does not include the oscillatory near-nullspace components ce{sup ik}({sup {alpha}x+{beta}y)}, where c is a complex vector and where {alpha}{sub 2} + {beta}{sup 2} = 1. These components are eliminated by applying a non-standard coarsening scheme. We achieve this scheme by introducing {open_quotes}ray{close_quotes} basis functions which depend on the parameter pair ({alpha}, {beta}), and which approximate ce{sup ik}({sup {alpha}x+{beta}y)} well on the coarser levels where bilinears cannot. We use several pairs of these parameters on each of these coarser levels so that several coarse grid problems are spun off from the finer levels. Some extensions of this theory to the transverse electric wave solution for Maxwells equations will also be presented.

19. High precision series solutions of differential equations: Ordinary and regular singular points of second order ODEs

Noreen, Amna; Olaussen, Kåre

2012-10-01

A subroutine for a very-high-precision numerical solution of a class of ordinary differential equations is provided. For a given evaluation point and equation parameters the memory requirement scales linearly with precision P, and the number of algebraic operations scales roughly linearly with P when P becomes sufficiently large. We discuss results from extensive tests of the code, and how one, for a given evaluation point and equation parameters, may estimate precision loss and computing time in advance. Program summary Program title: seriesSolveOde1 Catalogue identifier: AEMW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMW_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 991 No. of bytes in distributed program, including test data, etc.: 488116 Distribution format: tar.gz Programming language: C++ Computer: PC's or higher performance computers. Operating system: Linux and MacOS RAM: Few to many megabytes (problem dependent). Classification: 2.7, 4.3 External routines: CLN — Class Library for Numbers [1] built with the GNU MP library [2], and GSL — GNU Scientific Library [3] (only for time measurements). Nature of problem: The differential equation -s2({d2}/{dz2}+{1-ν+-ν-}/{z}{d}/{dz}+{ν+ν-}/{z2})ψ(z)+{1}/{z} ∑n=0N vnznψ(z)=0, is solved numerically to very high precision. The evaluation point z and some or all of the equation parameters may be complex numbers; some or all of them may be represented exactly in terms of rational numbers. Solution method: The solution ψ(z), and optionally ψ'(z), is evaluated at the point z by executing the recursion A(z)={s-2}/{(m+1+ν-ν+)(m+1+ν-ν-)} ∑n=0N Vn(z)A(z), ψ(z)=ψ(z)+A(z), to sufficiently large m. Here ν is either ν+ or ν-, and Vn(z)=vnz. The recursion is initialized by A(z)=δzν,for n

20. High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities

DTIC Science & Technology

2015-03-31

restoring the design accuracy of the scheme in the presence of singularities at the boundary. While this method is well studied for low order methods...boundary. While this method is well studied for low order methods and for problems in which singularities arise from the geometry (e.g., corners), we adapt...Solution of multiple problems at low cost . . . . . . . . . . . . . . . . . . 56 3.3.2 Parameters of the computational setting

1. Fourth order real space solver for the time-dependent Schrödinger equation with singular Coulomb potential

Majorosi, Szilárd; Czirják, Attila

2016-11-01

We present a novel numerical method and algorithm for the solution of the 3D axially symmetric time-dependent Schrödinger equation in cylindrical coordinates, involving singular Coulomb potential terms besides a smooth time-dependent potential. We use fourth order finite difference real space discretization, with special formulae for the arising Neumann and Robin boundary conditions along the symmetry axis. Our propagation algorithm is based on merging the method of the split-operator approximation of the exponential operator with the implicit equations of second order cylindrical 2D Crank-Nicolson scheme. We call this method hybrid splitting scheme because it inherits both the speed of the split step finite difference schemes and the robustness of the full Crank-Nicolson scheme. Based on a thorough error analysis, we verified both the fourth order accuracy of the spatial discretization in the optimal spatial step size range, and the fourth order scaling with the time step in the case of proper high order expressions of the split-operator. We demonstrate the performance and high accuracy of our hybrid splitting scheme by simulating optical tunneling from a hydrogen atom due to a few-cycle laser pulse with linear polarization.

2. High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

Anderson, R.; Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Quezada de Luna, M.; Rieben, R.; Tomov, V.

2017-04-01

In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

3. Impact of higher-order flows in the moment equations on Pfirsch-Schlüter friction coefficients

SciTech Connect

Honda, M.

2014-09-15

The impact of the higher-order flows in the moment approach on an estimate of the friction coefficients is numerically examined. The higher-order flows are described by the lower-order hydrodynamic flows using the collisional plasma assumption. Their effects have not been consistently taken into account thus far in the widely used neoclassical transport codes based on the moment equations in terms of the Pfirsch-Schlüter flux. Due to numerically solving the friction-flow matrix without using the small-mass ratio expansion, it is clearly revealed that incorporating the higher-order flow effects is of importance especially for plasmas including multiple hydrogenic ions and other lighter species with similar masses.

4. Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation.

PubMed

Kevrekidis, P G; Malomed, Boris A; Chen, Zhigang; Frantzeskakis, D J

2004-11-01

Vortex solitons with the topological charge S=3 , and "quasivortex" (multipole) solitons, which exist instead of the vortices with S=2 and 4, are constructed on a square lattice in the discrete nonlinear Schrödinger equation (true vortices with S=2 were known before, but they are unstable). For each type of solitary wave, its stability interval is found, in terms of the intersite coupling constant. The interval shrinks with increase of S . At couplings above a critical value, oscillatory instabilities set in, resulting in breakup of the vortex or quasivortex into lattice solitons with a lower vorticity. Such localized states may be observed in optical guiding structures, and in Bose-Einstein condensates loaded into optical lattices.

5. Development of High-Order Methods for Multi-Physics Problems Governed by Hyperbolic Equations

DTIC Science & Technology

2010-10-01

the conservative variable state vector: U =  ρ ρu ρv ρE  , and F (U) is the inviscid flux tensor with vector components: f =  ρu ρu2 + p ρuv...ρE + p )u  , g =  ρv ρuv ρv2 + p (ρE + p )v  . The specific energy E is the sum of the specific internal energy e and the kinetic energy...the constitutive relations: e = CV T, p = (γ − 1) [ ρE − ρ 2 (u2 + v2) ] . 0.3 Discretization method The governing equations of fluid motion, given

6. Detection and integration of oscillatory differential equations with initial stepsize, order and method selection

SciTech Connect

Gallivan, K. A.

1980-12-01

Within any general class of problems there typically exist subclasses possessed of characteristics that can be exploited to create techniques more efficient than general methods applied to these subclasses. Two such subclasses of initial-value problems in ordinary differential equations are stiff and oscillatory problems. Indeed, the subclass of oscillatory problems can be further refined into stiff and nonstiff oscillatory problems. This refinement is discussed in detail. The problem of developing a method of detection for nonstiff and stiff oscillatory behavior in initial-value problems is addressed. For this method of detection a control structure is proposed upon which a production code could be based. An experimental code using this control structure is described, and results of numerical tests are presented. 3 figures.

7. Pair formation and global ordering of strongly interacting ferrocolloid mixtures: an integral equation study.

PubMed

Range, Gabriel M; Klapp, Sabine H L

2006-03-21

Using the reference hypernetted chain (RHNC) integral equation theory and an accompanying stability analysis we investigate the structural and phase behaviors of model bidisperse ferrocolloids based on correlations of the homogeneous isotropic high-temperature phase. Our model consists of two species of dipolar hard spheres (DHSs) which dipole moments are proportional to the particle volume. At small packing fractions our results indicate the onset of chain formation, where the (more strongly coupled) A species behaves essentially as a one-component DHS fluid in a background of B particles. At high packing fractions, on the other hand, the RHNC theory indicates the appearance of isotropic-to-ferromagnetic transitions (volume ratios close to one) and demixing transitions (smaller volume ratios). However, contrary with the related case of monodisperse DHS mixtures previously studied by us [Phys. Rev. E 70, 031201 (2004)], none of the present bidisperse systems exhibit demixing within the isotropic phase, rather we observe coupled ferromagnetic/demixing phase transitions.

8. The arbitrary order mixed mimetic finite difference method for the diffusion equation

DOE PAGES

Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco

2016-05-01

Here, we propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux andmore » scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.« less

9. The arbitrary order mixed mimetic finite difference method for the diffusion equation

SciTech Connect

Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco

2016-05-01

Here, we propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.

10. Comparison of diffusion approximation and higher order diffusion equations for optical tomography of osteoarthritis

Yuan, Zhen; Zhang, Qizhi; Sobel, Eric; Jiang, Huabei

2009-09-01

In this study, a simplified spherical harmonics approximated higher order diffusion model is employed for 3-D diffuse optical tomography of osteoarthritis in the finger joints. We find that the use of a higher-order diffusion model in a stand-alone framework provides significant improvement in reconstruction accuracy over the diffusion approximation model. However, we also find that this is not the case in the image-guided setting when spatial prior knowledge from x-rays is incorporated. The results show that the reconstruction error between these two models is about 15 and 4%, respectively, for stand-alone and image-guided frameworks.

11. On an exterior boundary value problem for the Laplace equation with boundary operator of fractional order

Turmetov, B. Kh.

2016-12-01

In the paper in a class of regular harmonic functions we study properties of some integro-differential operators that generalize the operators of fractional differentiation in Hadamard sense. These operators transfer regular harmonic functions to the same function, and are inverse to the regular harmonic functions. Boundary value problem with the boundary operator of fractional order is studied in the exterior of the unit sphere. The considered problem generalizes the well-known Neumann problem on boundary operators of fractional order. We prove a theorem on existence and uniqueness of solutions of the problem. Moreover, an integral representation of the problem solution is obtained.

12. Second- and Higher-Order Virial Coefficients Derived from Equations of State for Real Gases

ERIC Educational Resources Information Center

Parkinson, William A.

2009-01-01

Derivation of the second- and higher-order virial coefficients for models of the gaseous state is demonstrated by employing a direct differential method and subsequent term-by-term comparison to power series expansions. This communication demonstrates the application of this technique to van der Waals representations of virial coefficients.…

13. Conservative method for simulation of a high-order nonlinear Schrödinger equation with a trapped term

Cai, Jia-Xiang; Bai, Chuan-Zhi; Qin, Zhi-Lin

2015-10-01

We propose a new scheme for simulation of a high-order nonlinear Schrödinger equation with a trapped term by using the mid-point rule and Fourier pseudospectral method to approximate time and space derivatives, respectively. The method is proved to be both charge- and energy-conserved. Various numerical experiments for the equation in different cases are conducted. From the numerical evidence, we see the present method provides an accurate solution and conserves the discrete charge and energy invariants to machine accuracy which are consistent with the theoretical analysis. Project supported by the National Natural Science Foundation of China (Grant Nos. 11201169 and 11271195) and the Qing Lan Project of Jiangsu Province, China.

14. Implicit Solution of the Four-field Extended-magnetohydroynamic Equations using High-order High-continuity Finite Elements

SciTech Connect

S.C. Jardin; J.A. Breslau

2004-12-17

Here we describe a technique for solving the four-field extended-magnetohydrodynamic (MHD) equations in two dimensions. The introduction of triangular high-order finite elements with continuous first derivatives (C{sup 1} continuity) leads to a compact representation compatible with direct inversion of the associated sparse matrices. The split semi-implicit method is introduced and used to integrate the equations in time, yielding unconditional stability for arbitrary time step. The method is applied to the cylindrical tilt mode problem with the result that a non-zero value of the collisionless ion skin depth will increase the growth rate of that mode. The effect of this parameter on the reconnection rate and geometry of a Harris equilibrium and on the Taylor reconnection problem is also demonstrated. This method forms the basis for a generalization to a full extended-MHD description of the plasma with six, eight, or more scalar fields.

15. Parallelization of the integral equation formulation of the polarizable continuum model for higher-order response functions.

PubMed

Ferrighi, Lara; Frediani, Luca; Fossgaard, Eirik; Ruud, Kenneth

2006-10-21

We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.

16. A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising from physics and engineering

Mirzaee, Farshid; Bimesl, Saeed

This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.

17. Parallelization of the integral equation formulation of the polarizable continuum model for higher-order response functions

Ferrighi, Lara; Frediani, Luca; Fossgaard, Eirik; Ruud, Kenneth

2006-10-01

We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.

18. A High Order Mixed Vector Finite Element Method for Solving the Time Dependent Maxwell Equations on Unstructured Grids

SciTech Connect

Rieben, R N; Rodrigue, G H; White, D A

2004-03-09

We present a mixed vector finite element method for solving the time dependent coupled Ampere and Faraday laws of Maxwell's equations on unstructured hexahedral grids that employs high order discretization in both space and time. The method is of arbitrary order accuracy in space and up to 5th order accurate in time, making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods. In addition, the method correctly models both the jump discontinuities and the divergence-free properties of the electric and magnetic fields, is charge and energy conserving, conditionally stable, and free of spurious modes. Several computational experiments are performed to demonstrate the accuracy, efficiency and benefits of the method.

19. A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

Christlieb, Andrew J.; Feng, Xiao; Seal, David C.; Tang, Qi

2016-07-01

We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage (i.e., it has no internal stages to store), single-step (i.e., it has no time history that needs to be stored), maintains a discrete divergence-free condition on the magnetic field, and has the capacity to preserve the positivity of the density and pressure. To accomplish this, we use a Taylor discretization of the Picard integral formulation (PIF) of the finite difference WENO method proposed in Christlieb et al. (2015) [23], where the focus is on a high-order discretization of the fluxes (as opposed to the conserved variables). We use the version where fluxes are expanded to third-order accuracy in time, and for the fluid variables space is discretized using the classical fifth-order finite difference WENO discretization. We use constrained transport in order to obtain divergence-free magnetic fields, which means that we simultaneously evolve the magnetohydrodynamic (that has an evolution equation for the magnetic field) and magnetic potential equations alongside each other, and set the magnetic field to be the (discrete) curl of the magnetic potential after each time step. In this work, we compute these derivatives to fourth-order accuracy. In order to retain a single-stage, single-step method, we develop a novel Lax-Wendroff discretization for the evolution of the magnetic potential, where we start with technology used for Hamilton-Jacobi equations in order to construct a non-oscillatory magnetic field. The end result is an algorithm that is similar to our previous work Christlieb et al. (2014) [8], but this time the time stepping is replaced through a Taylor method with the addition of a positivity-preserving limiter. Finally, positivity preservation is realized by introducing a parameterized flux limiter that considers a linear combination of high and low-order numerical fluxes. The choice of the free

20. On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations

Byeon, Jaeyoung; Huh, Hyungjin; Seok, Jinmyoung

2016-07-01

In this paper, we are interested in standing waves with a vortex for the nonlinear Chern-Simons-Schrödinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant λ > 0, representing the strength of the interaction potential, varies. We prove every standing wave is trivial if λ ∈ (0 , 1), every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS if λ = 1 and for every positive integer N, there is a nontrivial standing wave with a vortex point of order N if λ > 1. We also provide some classes of interaction potentials under which the nonexistence of standing waves and the existence of a standing wave with a vortex point of order N are proved.

1. Hard-disk equation of state: first-order liquid-hexatic transition in two dimensions with three simulation methods.

PubMed

Engel, Michael; Anderson, Joshua A; Glotzer, Sharon C; Isobe, Masaharu; Bernard, Etienne P; Krauth, Werner

2013-04-01

We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

2. Time-dependent quantum transport through an interacting quantum dot beyond sequential tunneling: second-order quantum rate equations.

PubMed

Dong, B; Ding, G H; Lei, X L

2015-05-27

A general theoretical formulation for the effect of a strong on-site Coulomb interaction on the time-dependent electron transport through a quantum dot under the influence of arbitrary time-varying bias voltages and/or external fields is presented, based on slave bosons and the Keldysh nonequilibrium Green's function (GF) techniques. To avoid the difficulties of computing double-time GFs, we generalize the propagation scheme recently developed by Croy and Saalmann to combine the auxiliary-mode expansion with the celebrated Lacroix's decoupling approximation in dealing with the second-order correlated GFs and then establish a closed set of coupled equations of motion, called second-order quantum rate equations (SOQREs), for an exact description of transient dynamics of electron correlated tunneling. We verify that the stationary solution of our SOQREs is able to correctly describe the Kondo effect on a qualitative level. Moreover, a comparison with other methods, such as the second-order von Neumann approach and Hubbard-I approximation, is performed. As illustrations, we investigate the transient current behaviors in response to a step voltage pulse and a harmonic driving voltage, and linear admittance as well, in the cotunneling regime.

3. An efficient high-order compact scheme for the unsteady compressible Euler and Navier-Stokes equations

Lerat, A.

2016-10-01

Residual-Based Compact (RBC) schemes approximate the 3-D compressible Euler equations with a 5th- or 7th-order accuracy on a 5 × 5 × 5-point stencil and capture shocks pretty well without correction. For unsteady flows however, they require a costly algebra to extract the time-derivative occurring at several places in the scheme. A new high-order time formulation has been recently proposed [13] for simplifying the RBC schemes and increasing their temporal accuracy. The present paper goes much further in this direction and deeply reconsiders the method. An avatar of the RBC schemes is presented that greatly reduces the computing time and the memory requirements while keeping the same type of successful numerical dissipation. Two and three-dimensional linear stability are analyzed and the method is extended to the 3-D compressible Navier-Stokes equations. The new compact scheme is validated for several unsteady problems in two and three dimension. In particular, an accurate DNS at moderate cost is presented for the evolution of the Taylor-Green Vortex at Reynolds 1600 and Prandtl 0.71. The effects of the mesh size and of the accuracy order in the approximation of Euler and viscous terms are discussed.

4. Decoupling of the Dirac equation correct to the third order for the magnetic perturbation.

PubMed

Ootani, Y; Maeda, H; Fukui, H

2007-08-28

A two-component relativistic theory accurately decoupling the positive and negative states of the Dirac Hamiltonian that includes magnetic perturbations is derived. The derived theory eliminates all of the odd terms originating from the nuclear attraction potential V and the first-order odd terms originating from the magnetic vector potential A, which connect the positive states to the negative states. The electronic energy obtained by the decoupling is correct to the third order with respect to A due to the (2n+1) rule. The decoupling is exact for the magnetic shielding calculation. However, the calculation of the diamagnetic property requires both the positive and negative states of the unperturbed (A=0) Hamiltonian. The derived theory is applied to the relativistic calculation of nuclear magnetic shielding tensors of HX (X=F,Cl,Br,I) systems at the Hartree-Fock level. The results indicate that such a substantially exact decoupling calculation well reproduces the four-component Dirac-Hartree-Fock results.

5. Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability.

PubMed

Wen, Xiao-Yong; Yan, Zhenya; Malomed, Boris A

2016-12-01

An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.

6. Some operational tools for solving fractional and higher integer order differential equations: A survey on their mutual relations

Kiryakova, Virginia S.

2012-11-01

The Laplace Transform (LT) serves as a basis of the Operational Calculus (OC), widely explored by engineers and applied scientists in solving mathematical models for their practical needs. This transform is closely related to the exponential and trigonometric functions (exp, cos, sin) and to the classical differentiation and integration operators, reducing them to simple algebraic operations. Thus, the classical LT and the OC give useful tool to handle differential equations and systems with constant coefficients. Several generalizations of the LT have been introduced to allow solving, in a similar way, of differential equations with variable coefficients and of higher integer orders, as well as of fractional (arbitrary non-integer) orders. Note that fractional order mathematical models are recently widely used to describe better various systems and phenomena of the real world. This paper surveys briefly some of our results on classes of such integral transforms, that can be obtained from the LT by means of "transmutations" which are operators of the generalized fractional calculus (GFC). On the list of these Laplace-type integral transforms, we consider the Borel-Dzrbashjan, Meijer, Krätzel, Obrechkoff, generalized Obrechkoff (multi-index Borel-Dzrbashjan) transforms, etc. All of them are G- and H-integral transforms of convolutional type, having as kernels Meijer's G- or Fox's H-functions. Besides, some special functions (also being G- and H-functions), among them - the generalized Bessel-type and Mittag-Leffler (M-L) type functions, are generating Gel'fond-Leontiev (G-L) operators of generalized differentiation and integration, which happen to be also operators of GFC. Our integral transforms have operational properties analogous to those of the LT - they do algebrize the G-L generalized integrations and differentiations, and thus can serve for solving wide classes of differential equations with variable coefficients of arbitrary, including non-integer order

7. Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: Exact solutions and stability

Wen, Xiao-Yong; Yan, Zhenya; Malomed, Boris A.

2016-12-01

An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.

8. Bilinear forms and solitons for a generalized sixth-order nonlinear Schrödinger equation in an optical fiber

Su, Jing-Jing; Gao, Yi-Tian

2017-01-01

Under investigation in this paper is a generalized sixth-order nonlinear Schrödinger equation, which could describe the attosecond pulses in an optical fiber. Bilinear forms and soliton solutions are derived via the Hirota method. Dynamic behaviors of the solitons are also analyzed. Moreover, we advance a new method, at the heart of which lies the idea that we simplify the limitation of the complex functions to the real ones, to demonstrate that the interaction between the two solitons is elastic and present the mathematical expression of velocity and phase shift of each soliton simultaneously.

9. A Reduced Order Model of the Linearized Incompressible Navier-Strokes Equations for the Sensor/Actuator Placement Problem

NASA Technical Reports Server (NTRS)

Allan, Brian G.

2000-01-01

A reduced order modeling approach of the Navier-Stokes equations is presented for the design of a distributed optimal feedback kernel. This approach is based oil a Krylov subspace method where significant modes of the flow are captured in the model This model is then used in all optimal feedback control design where sensing and actuation is performed oil tile entire flow field. This control design approach yields all optimal feedback kernel which provides insight into the placement of sensors and actuators in the flow field. As all evaluation of this approach, a two-dimensional shear layer and driven cavity flow are investigated.

10. Application of second-order-accurate Total Variation Diminishing (TVD) schemes to the Euler equations in general geometries

NASA Technical Reports Server (NTRS)

Yee, H. C.; Kutler, P.

1983-01-01

A one-parameter family of explicit and implicit second-order-accurate, entropy satisfying, total variation diminishing (TVD) schemes was developed by Harten. These TVD schemes were the property of not generating spurious oscillations for one-dimensional nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. Application of these methods to one- and two-dimensional fluid flows containing shocks (in Cartesian coordinates) yields highly accurate nonoscillatory numerical solutions. The goal of this work is to expand these methods to the multidimensional Euler equations in generalized coordinate systems. Some numerical results of shock waves impinging on cylindrical bodies are compared with MacCormack's method.

11. An improved stability test and stabilisation of linear time-varying systems governed by second-order vector differential equations

Tung, Shen-Lung; Juang, Yau-Tarng; Wu, Wei-Ying; Shieh, Wern-Yarng

2011-12-01

In this article, the problems of exponential stability analysis and stabilisation of linear time-varying systems described by a class of second-order vector differential equations are considered. Using bounding techniques on the trajectories of a linear time-varying system, the stability problem of the time-varying system is transformed to that of a time-invariant system and a new sufficient condition for the exponential stability is obtained. Moreover, the new criterion is proven to be superior to a test presented in the recent literature. Finally, the proposed criterion is applied to the exponential stabilisation problem via state feedback. The results are illustrated by several numerical examples.

12. Lagrange-type modeling of continuous dielectric permittivity variation in double-higher-order volume integral equation method

Chobanyan, E.; Ilić, M. M.; Notaroš, B. M.

2015-05-01

A novel double-higher-order entire-domain volume integral equation (VIE) technique for efficient analysis of electromagnetic structures with continuously inhomogeneous dielectric materials is presented. The technique takes advantage of large curved hexahedral discretization elements—enabled by double-higher-order modeling (higher-order modeling of both the geometry and the current)—in applications involving highly inhomogeneous dielectric bodies. Lagrange-type modeling of an arbitrary continuous variation of the equivalent complex permittivity of the dielectric throughout each VIE geometrical element is implemented, in place of piecewise homogeneous approximate models of the inhomogeneous structures. The technique combines the features of the previous double-higher-order piecewise homogeneous VIE method and continuously inhomogeneous finite element method (FEM). This appears to be the first implementation and demonstration of a VIE method with double-higher-order discretization elements and conformal modeling of inhomogeneous dielectric materials embedded within elements that are also higher (arbitrary) order (with arbitrary material-representation orders within each curved and large VIE element). The new technique is validated and evaluated by comparisons with a continuously inhomogeneous double-higher-order FEM technique, a piecewise homogeneous version of the double-higher-order VIE technique, and a commercial piecewise homogeneous FEM code. The examples include two real-world applications involving continuously inhomogeneous permittivity profiles: scattering from an egg-shaped melting hailstone and near-field analysis of a Luneburg lens, illuminated by a corrugated horn antenna. The results show that the new technique is more efficient and ensures considerable reductions in the number of unknowns and computational time when compared to the three alternative approaches.

13. Second-order correction to the Bigeleisen–Mayer equation due to the nuclear field shift

PubMed Central

Bigeleisen, Jacob

1998-01-01

The nuclear field shift affects the electronic, rotational, and vibrational energies of polyatomic molecules. The theory of the shifts in molecular spectra has been studied by Schlembach and Tiemann [Schlembach, J. & Tiemann, E. (1982) Chem. Phys. 68, 21]; measurements of the electronic and rotational shifts of the diatomic halides of Pb and Tl have been made by Tiemann et al. [Tiemann, E., Knöckel, H. & Schlembach, J. (1982) Ber. Bunsenges. Phys. Chem. 86, 821]. These authors have estimated the relative shifts in the harmonic frequencies of these compounds due to the nuclear field shift to be of the order of 10−6. I have used this estimate of the relative shift in vibrational frequency to calculate the correction to the harmonic oscillator approximation to the isotopic reduced partition-function ratio 208Pb32S/207Pb32S. The correction is 0.3% of the harmonic oscillator value at 300 K. In the absence of compelling evidence to the contrary, it suffices to calculate the nuclear field effect on the total isotopic partition-function ratio from its shift of the electronic zero point energy and the unperturbed molecular vibration. PMID:9560183

14. Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

Amster, Pablo; de Nápoli, Pablo; Pinasco, Juan Pablo

2008-07-01

Let be a time scale with . In this paper we study the asymptotic distribution of eigenvalues of the following linear problem -u[Delta][Delta]=[lambda]u[sigma], with mixed boundary conditions [alpha]u(a)+[beta]u[Delta](a)=0=[gamma]u([rho](b))+[delta]u[Delta]([rho](b)). It is known that there exists a sequence of simple eigenvalues {[lambda]k}k; we consider the spectral counting function , and we seek for its asymptotic expansion as a power of [lambda]. Let d be the Minkowski (or box) dimension of , which gives the order of growth of the number of intervals of length [epsilon] needed to cover , namely . We prove an upper bound of N([lambda]) which involves the Minkowski dimension, , where C is a positive constant depending only on the Minkowski content of (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d=0, infinite Minkowski content), and we show a family of self similar fractal sets where admits two-side estimates.

15. Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

SciTech Connect

Chen, Zheng; Huang, Hongying; Yan, Jue

2015-12-21

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β01) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.

16. Automatic Generation of Analytic Equations for Vibrational and Rovibrational Constants from Fourth-Order Vibrational Perturbation Theory

Matthews, Devin A.; Gong, Justin Z.; Stanton, John F.

2014-06-01

The derivation of analytic expressions for vibrational and rovibrational constants, for example the anharmonicity constants χij and the vibration-rotation interaction constants α^B_r, from second-order vibrational perturbation theory (VPT2) can be accomplished with pen and paper and some practice. However, the corresponding quantities from fourth-order perturbation theory (VPT4) are considerably more complex, with the only known derivations by hand extensively using many layers of complicated intermediates and for rotational quantities requiring specialization to orthorhombic cases or the form of Watson's reduced Hamiltonian. We present an automatic computer program for generating these expressions with full generality based on the adaptation of an existing numerical program based on the sum-over-states representation of the energy to a computer algebra context. The measures taken to produce well-simplified and factored expressions in an efficient manner are discussed, as well as the framework for automatically checking the correctness of the generated equations.

17. Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

DOE PAGES

Chen, Zheng; Huang, Hongying; Yan, Jue

2015-12-21

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β0,β1) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried out to demonstratemore » the accuracy and capability of the maximum-principle-satisfying limiter.« less

18. Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations

Li, Zhi-Hui; Peng, Ao-Ping; Zhang, Han-Xin; Yang, Jaw-Yen

2015-04-01

This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2-20 and Knudsen numbers 0.0001-5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere-cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N-S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive

19. First-Order Acoustic Wave Equation Reverse Time Migration Based on the Dual-Sensor Seismic Acquisition System

You, Jiachun; Liu, Xuewei; Wu, Ru-Shan

2017-03-01

We analyze the mathematical requirements for conventional reverse time migration (RTM) and summarize their rationale. The known information provided by current acquisition system is inadequate for the second-order acoustic wave equations. Therefore, we introduce a dual-sensor seismic acquisition system into the coupled first-order acoustic wave equations. We propose a new dual-sensor reverse time migration called dual-sensor RTM, which includes two input variables, the pressure and vertical particle velocity data. We focus on the performance of dual-sensor RTM in estimating reflection coefficients compared with conventional RTM. Synthetic examples are used for the study of estimating coefficients of reflectors with both dual-sensor RTM and conventional RTM. The results indicate that dual-sensor RTM with two inputs calculates amplitude information more accurately and images structural positions of complex substructures, such as the Marmousi model, more clearly than that of conventional RTM. This shows that the dual-sensor RTM has better accuracy in backpropagation and carries more information in the directivity because of particle velocity injection. Through a simple point-shape model, we demonstrate that dual-sensor RTM decreases the effect of multi-pathing of propagating waves, which is helpful for focusing the energy. In addition, compared to conventional RTM, dual-sensor RTM does not cause extra memory costs. Dual-sensor RTM is, therefore, promising for the computation of multi-component seismic data.

20. High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

PubMed

Banks, H T; Birch, Malcolm J; Brewin, Mark P; Greenwald, Stephen E; Hu, Shuhua; Kenz, Zackary R; Kruse, Carola; Maischak, Matthias; Shaw, Simon; Whiteman, John R

2014-04-13

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685-6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin-Voigt and Maxwell-Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

1. Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions

Lanthaler, S.; Pfefferlé, D.; Graves, J. P.; Cooper, W. A.

2017-04-01

An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre’s parallel velocity via the so-called Baños drift. The correct treatment of the guiding-centre push-forward with the Baños term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.

2. Towards A Fast High-Order Method for Unsteady Incompressible Navier-Stokes Equations using FR/CPR

Cox, Christopher; Liang, Chunlei; Plesniak, Michael

2014-11-01

A high-order compact spectral difference method for solving the 2D incompressible Navier-Stokes equations on unstructured grids is currently being developed. This method employs the gGA correction of Huynh, and falls under the class of methods now refered to as Flux Reconstruction/Correction Procedure via Reconstruction. This method and the artificial compressibility method are integrated along with a dual time-integration scheme to model unsteady incompressible viscous flows. A lower-upper symmetric Gauss-Seidel scheme and a backward Euler scheme are used to efficiently march the solution in pseudo time and physical time, respectively. We demonstrate order of accuracy with steady Taylor-Couette flow at Re = 10. We further validate the solver with steady flow past a NACA0012 airfoil at zero angle of attack at Re = 1850 and unsteady flow past a circle at Re = 100. The implicit time-integration scheme for the pseudo time derivative term is proved efficient and effective for the classical artificial compressibility treatment to achieve the divergence-free condition of the continuity equation. We greatly acknowledge financial support from The George Washington University under the Presidential Merit Fellowship.

3. Parallel High Order Accuracy Methods Applied to Non-Linear Hyperbolic Equations and to Problems in Materials Sciences

SciTech Connect

Jan Hesthaven

2012-02-06

Final report for DOE Contract DE-FG02-98ER25346 entitled Parallel High Order Accuracy Methods Applied to Non-Linear Hyperbolic Equations and to Problems in Materials Sciences. Principal Investigator Jan S. Hesthaven Division of Applied Mathematics Brown University, Box F Providence, RI 02912 Jan.Hesthaven@Brown.edu February 6, 2012 Note: This grant was originally awarded to Professor David Gottlieb and the majority of the work envisioned reflects his original ideas. However, when Prof Gottlieb passed away in December 2008, Professor Hesthaven took over as PI to ensure proper mentoring of students and postdoctoral researchers already involved in the project. This unusual circumstance has naturally impacted the project and its timeline. However, as the report reflects, the planned work has been accomplished and some activities beyond the original scope have been pursued with success. Project overview and main results The effort in this project focuses on the development of high order accurate computational methods for the solution of hyperbolic equations with application to problems with strong shocks. While the methods are general, emphasis is on applications to gas dynamics with strong shocks.

4. High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

PubMed Central

Banks, H T; Birch, Malcolm J; Brewin, Mark P; Greenwald, Stephen E; Hu, Shuhua; Kenz, Zackary R; Kruse, Carola; Maischak, Matthias; Shaw, Simon; Whiteman, John R

2014-01-01

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd. PMID:25834284

5. Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

6. A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrodinger equation.

PubMed

Zhang, Zhongxi; Chen, Liang; Bao, Xiaoyi

2010-04-12

A fourth-order Runge-Kutta in the interaction picture (RK4IP) method is presented for solving the coupled nonlinear Schr odinger equation (CNLSE) that governs the light propagation in optical fibers with randomly varying birefringence. The computational error of RK4IP is caused by the fourth-order Runge-Kutta algorithm, better than the split-step approximation limited by the step size. As a result, the step size of RK4IP can have the same order of magnitude as the dispersion length and/or the nonlinear length of the fiber, provided the birefringence effect is small. For communication fibers with random birefringence, the step size of RK4IP can be orders of magnitude larger than the correlation length and the beating length of the fibers, depending on the interaction between linear and nonlinear effects. Our approach can be applied to the fibers having the general form of local birefringence and treat the Kerr nonlinearity without approximation. Our RK4IP results agree well with those obtained from Manakov-PMD approximation, provided the polarization state can be mixed enough on the Poincar e sphere.

7. Complete group classification of systems of two nonlinear second-Order ordinary differential equations of the form y‧‧ = F(y)

Oguis, G. F.; Moyo, S.; Meleshko, S. V.

2017-03-01

Extensive work has been done on the group classification of systems of equations in the literature. This paper identifies the gap in the literature which concerns the group classification of systems of two nonlinear second-order ordinary differential equations. We provide a complete group classification of systems of two ordinary differential equations of the form, y‧‧ = F(y) , which occur in many physical applications using two approaches which form the essence of this paper.

8. On a compact mixed-order finite element for solving the three-dimensional incompressible Navier-Stokes equations

Wang, Morten M. T.; Sheu, Tony W. H.

1997-09-01

Our work is an extension of the previously proposed multivariant element. We assign this refined element as a compact mixed-order element in the sense that use of this element offers a much smaller bandwidth. The analysis is implemented on quadratic hexahedral elements with a view to analysing a three-dimensional incompressible viscous flow problem using a method formulated within the mixed finite element context. The idea of constructing such a stable element is to bring the marker-and-cell (MAC) grid lay-out to the finite element context. This multivariant element can thus be classified as a discontinuous pressure element. We have several reasons for advocating the proposed multivariant element. The primary advantage gained is its ability to reduce the bandwidth of the matrix equation, as compared with its univariant counterparts, so that it can be effectively stored in a compressed row storage (CRS) format. The resulting matrix equation can be solved efficiently by a multifrontal solver owing to its reduced bandwidth. The coding is, however, complicated by the appearance of restricted degrees of freedom at mid-face nodes. Through analytic study this compact multivariant element has a marked advantage over the multivariant element of Gupta et al. in that both bandwidth and computation time have been drastically reduced.

9. Coupled third-order simplified spherical harmonics and diffusion equation-based fluorescence tomographic imaging of liver cancer

Chen, Xueli; Sun, Fangfang; Yang, Defu; Liang, Jimin

2015-09-01

For fluorescence tomographic imaging of small animals, the liver is usually regarded as a low-scattering tissue and is surrounded by adipose, kidneys, and heart, all of which have a high scattering property. This leads to a breakdown of the diffusion equation (DE)-based reconstruction method as well as a heavy computational burden for the simplified spherical harmonics equation (SPN). Coupling the SPN and DE provides a perfect balance between the imaging accuracy and computational burden. The coupled third-order SPN and DE (CSDE)-based reconstruction method is developed for fluorescence tomographic imaging. This is achieved by doubly using the CSDE for the excitation and emission processes of the fluorescence propagation. At the same time, the finite-element method and hybrid multilevel regularization strategy are incorporated in inverse reconstruction. The CSDE-based reconstruction method is first demonstrated with a digital mouse-based liver cancer simulation, which reveals superior performance compared with the SPN and DE-based methods. It is more accurate than the DE-based method and has lesser computational burden than the SPN-based method. The feasibility of the proposed approach in applications of in vivo studies is also illustrated with a liver cancer mouse-based in situ experiment, revealing its potential application in whole-body imaging of small animals.

10. Coupled third-order simplified spherical harmonics and diffusion equation-based fluorescence tomographic imaging of liver cancer.

PubMed

Chen, Xueli; Sun, Fangfang; Yang, Defu; Liang, Jimin

2015-01-01

For fluorescence tomographic imaging of small animals, the liver is usually regarded as a low-scattering tissue and is surrounded by adipose, kidneys, and heart, all of which have a high scattering property. This leads to a breakdown of the diffusion equation (DE)–based reconstruction method as well as a heavy computational burden for the simplified spherical harmonics equation (SP(N)). Coupling the SP(N) and DE provides a perfect balance between the imaging accuracy and computational burden. The coupled third-order SPN and DE (CSDE)-based reconstruction method is developed for fluorescence tomographic imaging. This is achieved by doubly using the CSDE for the excitation and emission processes of the fluorescence propagation. At the same time, the finite-element method and hybrid multilevel regularization strategy are incorporated in inverse reconstruction. The CSDE-based reconstruction method is first demonstrated with a digital mouse-based liver cancer simulation, which reveals superior performance compared with the SPN and DE-based methods. It is more accurate than the DE-based method and has lesser computational burden than the SPN-based method. The feasibility of the proposed approach in applications of in vivo studies is also illustrated with a liver cancer mouse-based in situ experiment, revealing its potential application in whole-body imaging of small animals.

11. Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits.

PubMed

Kedziora, David J; Ankiewicz, Adrian; Akhmediev, Nail

2012-06-01

We present an explicit analytic form for the two-breather solution of the nonlinear Schrödinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to i, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers.

12. An indirect approach based on Clausius-Clapeyron equation to determine entropy change for the first-order magnetocaloric materials

Xu, Kun; Li, Zhe; Zhang, Yuan-Lei; Jing, Chao

2015-12-01

Taking into account the phase fraction during the structural transition for the first-order magnetocaloric materials, an improved isothermal entropy change (ΔST) determination has been put forward based on the Clausius-Clapeyron (CC) equation. It was found that the ΔST value evaluated by this method is in excellent agreement with those determined from the Maxwell relation (MR) using magnetic measurements for some Heusler alloys with a weak field-induced phase transforming behavior, such as Ni-Mn-Sn Heusler alloys. In comparison with the MR based on isofield magnetization measurements (MRIF), this method is very convenient to obtain the ΔST derived from only few thermomagnetic curves. More importantly, it is quite superior to the MR-based method in eliminating the overestimation of ΔST due to the appearance of the spurious spike derived from MR employing isothermal magnetization measurements (MRIT).

13. Systematics of strongly self-dominant higher-order differential equations based on the Painlevé analysis of their singularities

King, R. B.

1986-04-01

This paper presents a simple way of classifying higher-order differential equations based on the requirements of the Painlevé property, i.e., the presence of no movable critical points. The fundamental building blocks for such equations may be generated by strongly self-dominant differential equations of the type (∂/∂x)nu =(∂/∂xm)[u(m-n+p)/p] in which m and n are positive integers and p is a negative integer. Such differential equations having both a constant degree d and a constant value of the difference n-m form a Painlevé chain; however, only three of the many possible Painlevé chains can have the Painlevé property. Among the three Painlevé chains that can have the Painlevé property, one contains the Burgers' equation; another contains the dominant terms of the first Painlevé transcendent, the isospectral Korteweg-de Vries equation, and the isospectral Boussinesq equation; and the third contains the dominant terms of the second Painlevé transcendent and the isospectral modified (cubic) Korteweg-de Vries equation. Differential equations of the same order and having the same value of the quotient (n-m)/(d-1) can be mixed to generate a new hybrid differential equation. In such cases a hybrid can have the Painlevé property even if only one of its components has the Painlevé property. Such hybridization processes can be used to generate the various fifth-order evolution equations of interest, namely the Caudrey-Dodd-Gibbon, Kuperschmidt, and Morris equations.

14. Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications on non-linear system of fractional order differential equations

Isah, Abdulnasir; Chang, Phang

2016-06-01

In this article we propose the wavelet operational method based on shifted Legendre polynomial to obtain the numerical solutions of non-linear systems of fractional order differential equations (NSFDEs). The operational matrix of fractional derivative derived through wavelet-polynomial transformation are used together with the collocation method to turn the NSFDEs to a system of non-linear algebraic equations. Illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

15. Study of vortex ring dynamics in the nonlinear Schrodinger equation utilizing GPU-accelerated high-order compact numerical integrators

Caplan, Ronald Meyer

We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrodinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics. Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size. The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.

16. The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covariants

Fuchssteiner, Benno; Oevel, Walter

1982-03-01

Using a bi-Hamiltonian formulation we give explicit formulas for the conserved quantities and infinitesimal generators of symmetries for some nonlinear fifth- and seventh-order nonlinear partial differential equations; among them, the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and the Kupershmidt equation. We show that the Lie algebras of the symmetry groups of these equations are of a very special form: Among the C∞ vector fields they are generated from two given commuting vector fields by a recursive application of a single operator. Furthermore, for some higher order equations, those multisoliton solutions, which for ||t||→∞ asymptotically decompose into traveling wave solutions, are characterized as eigenvector decompositions of certain operators.

17. Spectral transverse instabilities and soliton dynamics in the higher-order multidimensional nonlinear Schrödinger equation

Cole, Justin T.; Musslimani, Ziad H.

2015-12-01

Spectral transverse instabilities of one-dimensional solitary wave solutions to the two-dimensional nonlinear Schrödinger (NLS) equation with fourth-order dispersion/diffraction subject to higher-dimensional perturbations are studied. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed using Fourier and finite difference differentiation matrices. It is found that for both signs of the higher-order dispersion coefficient there exists a finite band of unstable transverse modes. In the long wavelength limit we derive an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The time dynamics of a one-dimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. Numerical nonlinear stability analysis is also addressed.

18. Final Report - High-Order Spectral Volume Method for the Navier-Stokes Equations On Unstructured Tetrahedral Grids

SciTech Connect

Wang, Z J

2012-12-06

The overriding objective for this project is to develop an efficient and accurate method for capturing strong discontinuities and fine smooth flow structures of disparate length scales with unstructured grids, and demonstrate its potentials for problems relevant to DOE. More specifically, we plan to achieve the following objectives: 1. Extend the SV method to three dimensions, and develop a fourth-order accurate SV scheme for tetrahedral grids. Optimize the SV partition by minimizing a form of the Lebesgue constant. Verify the order of accuracy using the scalar conservation laws with an analytical solution; 2. Extend the SV method to Navier-Stokes equations for the simulation of viscous flow problems. Two promising approaches to compute the viscous fluxes will be tested and analyzed; 3. Parallelize the 3D viscous SV flow solver using domain decomposition and message passing. Optimize the cache performance of the flow solver by designing data structures minimizing data access times; 4. Demonstrate the SV method with a wide range of flow problems including both discontinuities and complex smooth structures. The objectives remain the same as those outlines in the original proposal. We anticipate no technical obstacles in meeting these objectives.

19. Application of reduced order modeling techniques to problems in heat conduction, isoelectric focusing and differential algebraic equations

Mathai, Pramod P.

This thesis focuses on applying and augmenting 'Reduced Order Modeling' (ROM) techniques to large scale problems. ROM refers to the set of mathematical techniques that are used to reduce the computational expense of conventional modeling techniques, like finite element and finite difference methods, while minimizing the loss of accuracy that typically accompanies such a reduction. The first problem that we address pertains to the prediction of the level of heat dissipation in electronic and MEMS devices. With the ever decreasing feature sizes in electronic devices, and the accompanied rise in Joule heating, the electronics industry has, since the 1990s, identified a clear need for computationally cheap heat transfer modeling techniques that can be incorporated along with the electronic design process. We demonstrate how one can create reduced order models for simulating heat conduction in individual components that constitute an idealized electronic device. The reduced order models are created using Krylov Subspace Techniques (KST). We introduce a novel 'plug and play' approach, based on the small gain theorem in control theory, to interconnect these component reduced order models (according to the device architecture) to reliably and cheaply replicate whole device behavior. The final aim is to have this technique available commercially as a computationally cheap and reliable option that enables a designer to optimize for heat dissipation among competing VLSI architectures. Another place where model reduction is crucial to better design is Isoelectric Focusing (IEF) - the second problem in this thesis - which is a popular technique that is used to separate minute amounts of proteins from the other constituents that are present in a typical biological tissue sample. Fundamental questions about how to design IEF experiments still remain because of the high dimensional and highly nonlinear nature of the differential equations that describe the IEF process as well as

20. Systematics of Strongly Self-Dominant Higher Order Differential Equations Based on the Painleve Analysis of Their Singularities.

DTIC Science & Technology

1985-12-16

evolution equations . 1 ,2 This interest has arisen from the realization that these equations possess a special type of elementary solution which...takes the form of localized disturbances which act somewhat like particles and are therefore known as solitons. Solution of these evolution equations ...these equations u may be regarded as an amplitude, x as a distance, and t as , time. Of particular interest are time-independent solutions where ut=O

1. One and two soliton solutions for seventh-order Caudrey-Dodd-Gibbon and Caudrey-Dodd-Gibbon-KP equations

Wazwaz, Abdul-Majid

2012-08-01

In this work, we explore more applications of the simplified form of the bilinear method to the seventhorder Caudrey-Dodd-Gibbon (CDG) and the Caudrey-Dodd-Gibbon-KP (CDG-KP) equation. We formally derive one and two soliton solutions for each equation. We also show that the two equations do not show resonance.

2. Symmetry analysis of the high-order equations for the description of the Fermi – Pasta – Ulam problem

Kudryashov, N. A.; Volkov, A. K.

2017-01-01

Recently some new nonlinear equations for the description of the Fermi – Pasta – Ulam problem have been derived. The main aim of this work is to use the symmetry test to investigate these equations. We consider equations for the description of the α and α + β Fermi – Pasta – Ulam model. We find the infinitesimal operators and Lie groups, admitted by the equations. Using the groups we find the self-similar variables as well as the reductions to the ordinary differential equations. Some exact solutions are also constructed.

3. A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields

SciTech Connect

St Aubin, J. Keyvanloo, A.; Fallone, B. G.; Vassiliev, O.

2015-02-15

Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm{sup 2} field as well as a smaller 2 × 2 cm{sup 2} field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization

4. Characterization of optical rogue wave based on solitons' eigenvalues of the integrable higher-order nonlinear Schrödinger equation

Weerasekara, Gihan; Maruta, Akihiro

2017-01-01

The dynamics of the optical rogue wave phenomenon in the framework of integrable higher-order nonlinear Schrödinger equation (HNLSE) including the third order dispersion term is presented in this paper. When rogue waves generate through soliton collision, the colliding solitons' eigenvalues of the associated equation of HNLSE should be constant in the vicinity of rogue wave generation. Our results reveal that soliton collision is one of the generation mechanisms of optical rogue waves in anomalous dispersion fiber by taking the third order dispersion into consideration in the HNLSE based model.

5. Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense

Gómez-Aguilar, J. F.; Atangana, Abdon

2017-02-01

In this work the fractional Hunter-Saxton equation applied in the study of diffusion of nematic liquid crystals was done involving partial operators with two fractional orders, α and β, via Atangana-Riemann and Atangana-Caputo with bi-order and via Riemann-Liouville, Caputo-Fabrizio-Riemann and Atangana-Baleanu-Riemann for the space domain. The mathematical equation underpinning this physical phenomenon was solved numerically using an iterative scheme where the numerical approximations for second order were developed. The new approach with two fractional orders is able to consider media with two different layers, scales and properties. The generalization of this equation exhibit different cases of anomalous behavior and the numerical solutions obtained describes the propagation of waves in a nematic liquid cristal.

6. Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

Zhang, Hai-Qiang; Chen, Jian

2016-04-01

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The Nth order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

7. High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium

Barnett, Alex H.; Nelson, Bradley J.; Mahoney, J. Matthew

2015-09-01

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve, the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. (Δ + E +x2) u (x1 ,x2) = 0 where E is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 102 nodes, with an effort that is independent of the frequency parameter E. By combining with a high-order Nyström quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop.

8. Carleman Estimates with No Lower-Order Terms for General Riemann Wave Equations. Global Uniqueness and Observability in One Shot

SciTech Connect

Triggiani, R.; Yao, P.F.

2002-12-19

This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level (H{sup 1} x L{sub 2}) -terms, under essentially minimal smoothness assumptions on the variable (in time and space) coefficients. The paper provides Carleman-type inequalities: first pointwise, for C{sup 2} -solutions, then in integral form for H{sup 1,1}(Q) -solutions. The aim of the present approach is to provide Carleman inequalities which do not contain lower-order terms, a distinguishing feature over most of the literature. Accordingly, global uniqueness results for overdetermined problems as well as Continuous Observability/ Uniform Stabilization inequalities follow in one shot, as a part of the same stream of arguments. Constants in the estimates are, therefore, generally explicit. The paper emphasizes the more challenging pure Neumann B.C. case. The paper is a generalization from the Euclidean to the Riemannian setting of [LTZ] in the more difficult case of purely Neumann B.C., and of [KK1] in the case of Dirichlet B.C. The approach is Riemann geometric, but different from-indeed, more flexible than-the one in [LTY1].

9. Dual-shaped offset reflector antenna designs from solutions of the geometrical optics first-order partial differential equations

NASA Technical Reports Server (NTRS)

Galindo-Israel, V.; Imbriale, W.; Shogen, K.; Mittra, R.

1990-01-01

In obtaining solutions to the first-order nonlinear partial differential equations (PDEs) for synthesizing offset dual-shaped reflectors, it is found that previously observed computational problems can be avoided if the integration of the PDEs is started from an inner projected perimeter and integrated outward rather than starting from an outer projected perimeter and integrating inward. This procedure, however, introduces a new parameter, the main reflector inner perimeter radius p(o), when given a subreflector inner angle 0(o). Furthermore, a desired outer projected perimeter (e.g., a circle) is no longer guaranteed. Stability of the integration is maintained if some of the initial parameters are determined first from an approximate solution to the PDEs. A one-, two-, or three-parameter optimization algorithm can then be used to obtain a best set of parameters yielding a close fit to the desired projected outer rim. Good low cross-polarization mapping functions are also obtained. These methods are illustrated by synthesis of a high-gain offset-shaped Cassegrainian antenna and a low-noise offset-shaped Gregorian antenna.

10. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation

Li, Gongsheng; Zhang, Dali; Jia, Xianzheng; Yamamoto, Masahiro

2013-06-01

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion.

11. A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov

Greenough, J. A.; Rider, W. J.

2004-05-01

A numerical study is undertaken comparing a fifth-order version of the weighted essentially non-oscillatory numerical (WENO5) method to a modern piecewise-linear, second-order, version of Godunov's (PLMDE) method for the compressible Euler equations. A series of one-dimensional test problems are examined beginning with classical linear problems and ending with complex shock interactions. The problems considered are: (1) linear advection of a Gaussian pulse in density, (2) Sod's shock tube problem, (3) the "peak" shock tube problem, (4) a version of the Shu and Osher shock entropy wave interaction and (5) the Woodward and Colella interacting shock wave problem. For each problem and method, run times, density error norms and convergence rates are reported for each method as produced from a common code test-bed. The linear problem exhibits the advertised convergence rate for both methods as well as the expected large disparity in overall error levels; WENO5 has the smaller errors and an enormous advantage in overall efficiency (in accuracy per unit CPU time). For the nonlinear problems with discontinuities, however, we generally see both first-order self-convergence of error as compared to an exact solution, or when an analytic solution is not available, a converged solution generated on an extremely fine grid. The overall comparison of error levels shows some variation from problem to problem. For Sod's shock tube, PLMDE has nearly half the error, while on the peak problem the errors are nearly the same. For the interacting blast wave problem the two methods again produce a similar level of error with a slight edge for the PLMDE. On the other hand, for the Shu-Osher problem, the errors are similar on the coarser grids, but favors WENO by a factor of nearly 1.5 on the finer grids used. In all cases holding mesh resolution constant though, PLMDE is less costly in terms of CPU time by approximately a factor of 6. If the CPU cost is taken as fixed, that is run times are

12. Application of the principal fractional meta-trigonometric functions for the solution of linear commensurate-order time-invariant fractional differential equations.

PubMed

Lorenzo, C F; Hartley, T T; Malti, R

2013-05-13

A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the R-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.

13. Exterior metric approach to a charged axially symmetric celestial body: the fourth-order approximate solutions of Einstein--Maxwell equations

SciTech Connect

Zhou Qi-huang

1988-12-01

Starting with the general expression of a static state axisymmetric metric and using the principle of equivalence and the Maccullagh formula, the Einstein--Maxwell equations of a charged axisymmetric celestial body are obtained. Next, using the method of undetermined coefficients these equations are solved up to fourth-order approximate. These sets of solutions are generally appropriate for all kinds of charged axisymmetric celestial bodies.

14. On the Existence of Non-Oscillatory Phase Functions for Second Order Ordinary Differential Equations in the High-Frequency Regime

DTIC Science & Technology

2014-08-04

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using...solving the linearized equation h2ptq ´ 1 2 r1nptqh1ptq  4λ2 exp prnptqqhptq “ fnptq for all t P R, (15) where fnptq “ ´r2nptq  1 4 pr1nptqq 2 ´ 4λ2...of h1pxptqq is, of course, the same as that of h1pxq. So the solution of the linearized equation (15) can be written as the sum of a nonoscillatory

15. An alternative approach to systems of second-order ordinary differential equations with maximal symmetry. Realizations of sl(n + 2 , R) by special functions

Campoamor-Stursberg, R.

2016-08-01

Using the general solution of the differential equation x¨(t) +g1(t) x˙ +g2(t) x = 0 , a generic basis of the point-symmetry algebra sl(3 , R) is constructed. Deriving the equation from a time-dependent Lagrangian, the basis elements corresponding to Noether symmetries are deduced. The generalized Lewis invariant is constructed explicitly using a linear combination of Noether symmetries. The procedure is generalized to the case of systems of second-order ordinary differential equations with maximal sl(n + 2 , R) -symmetry, and its possible adaptation to the inhomogeneous non-linear case illustrated by an example.

16. Schrödinger and related equations as hamiltonian systems, manifolds of second-order tensors and new ideas of nonlinearity in quantum mechanics

Sławianowski, J. J.; Kovalchuk, V.

2010-01-01

Considered is the Schrödinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting example of "mechanics" with singular Lagrangians, effectively treatable within the framework of Dirac formalism. We discuss also some modified "Schrödinger" equations involving second-order time derivatives and introduce a kind of nondirect, nonperturbative, geometrically-motivated nonlinearity based on making the scalar product a dynamical quantity. There are some reasons to expect that this might be a new way of describing open dynamical systems and explaining some quantum "paradoxes".

17. Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

SciTech Connect

Fike, Jeffrey A.

2013-08-01

The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.

18. Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems.

PubMed

Grima, Ramon

2015-10-01

It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.

19. All-optical 1st- and 2nd-order differential equation solvers with large tuning ranges using Fabry-Pérot semiconductor optical amplifiers.

PubMed

Chen, Kaisheng; Hou, Jie; Huang, Zhuyang; Cao, Tong; Zhang, Jihua; Yu, Yuan; Zhang, Xinliang

2015-02-09

We experimentally demonstrate an all-optical temporal computation scheme for solving 1st- and 2nd-order linear ordinary differential equations (ODEs) with tunable constant coefficients by using Fabry-Pérot semiconductor optical amplifiers (FP-SOAs). By changing the injection currents of FP-SOAs, the constant coefficients of the differential equations are practically tuned. A quite large constant coefficient tunable range from 0.0026/ps to 0.085/ps is achieved for the 1st-order differential equation. Moreover, the constant coefficient p of the 2nd-order ODE solver can be continuously tuned from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps(2) to 0.006205/ps(2). Additionally, a theoretical model that combining the carrier density rate equation of the semiconductor optical amplifier (SOA) with the transfer function of the Fabry-Pérot (FP) cavity is exploited to analyze the solving processes. For both 1st- and 2nd-order solvers, excellent agreements between the numerical simulations and the experimental results are obtained. The FP-SOAs based all-optical differential-equation solvers can be easily integrated with other optical components based on InP/InGaAsP materials, such as laser, modulator, photodetector and waveguide, which can motivate the realization of the complicated optical computing on a single integrated chip.

20. Second-order differential equations for bosons with spin j ≥ 1 and in the bases of general tensor-spinors of rank 2j

Banda Guzmán, V. M.; Kirchbach, M.

2016-09-01

A boson of spin j≥ 1 can be described in one of the possibilities within the Bargmann-Wigner framework by means of one sole differential equation of order twice the spin, which however is known to be inconsistent as it allows for non-local, ghost and acausally propagating solutions, all problems which are difficult to tackle. The other possibility is provided by the Fierz-Pauli framework which is based on the more comfortable to deal with second-order Klein-Gordon equation, but it needs to be supplemented by an auxiliary condition. Although the latter formalism avoids some of the pathologies of the high-order equations, it still remains plagued by some inconsistencies such as the acausal propagation of the wave fronts of the (classical) solutions within an electromagnetic environment. We here suggest a method alternative to the above two that combines their advantages while avoiding the related difficulties. Namely, we suggest one sole strictly D^{(j,0)oplus (0,j)} representation specific second-order differential equation, which is derivable from a Lagrangian and whose solutions do not violate causality. The equation under discussion presents itself as the product of the Klein-Gordon operator with a momentum-independent projector on Lorentz irreducible representation spaces constructed from one of the Casimir invariants of the spin-Lorentz group. The basis used is that of general tensor-spinors of rank 2 j.

1. Dispersive properties of high-order Nédélec/edge element approximation of the time-harmonic Maxwell equations.

PubMed

Ainsworth, Mark

2004-03-15

The dispersive behaviour of high-order Nédélec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency omega on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where omega h is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1< omega h, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly.

2. Application of the hybrid method with constant coefficients to solving the integro-differential equations of first order

Mehdiyeva, Galina; Imanova, Mehriban; Ibrahimov, Vagif

2012-11-01

As is well known investigation of many processes of natural sciences reduce to the solving of initial value problem for integro-differential equations which are one of the priority areas of modern mathematics. To define the exact solution of such problems is not always possible. Therefore the scientists constructed approximate methods for solving them. There are a number of papers devoted to finding approximate solutions of integro-differential equations. Unlike at papers investigated, here the numerical solution of initial value problem for Volterra integro-differential equations by the hybrid methods, constructed concrete methods with the degree p ≤ 6 and suggested algorithm for using them.

3. A Lagrangian model for soil water dynamics: can we step beyond Richard's equation while preserving capillarity as first order control?

2016-04-01

Water storage in the unsaturated zone is controlled by capillary forces which increase nonlinearly with decreasing pore size, because water acts as a wetting fluid in soil. The standard approach to represent capillary and gravity controlled soil water dynamics is the Darcy-Richards equation in combination with suitable soil water characteristics. This continuum model essentially assumes capillarity controlled diffusive fluxes to dominate soil water dynamics under local thermodynamic equilibrium conditions. Today we know that the assumptions of local equilibrium conditions e.g. and a mainly diffusive flow are often not appropriate, particularly during rainfall events in structured soils. Rapid or preferential flow imply a strong local disequilibrium and imperfect mixing between a fast fraction of soil water, traveling in interconnected coarse pores or non-capillary macropores, and the slower diffusive flow in finer fractions of the pore space. Although various concepts have been proposed to overcome the inability of the Darcy - Richards concept to cope with not-well mixed preferential flow, we still lack an approach that is commonly accepted. Notwithstanding the listed short comings, one should not mistake the limitations of the Richards equation with non-importance of capillary forces in soil. Without capillarity infiltrating rainfall would drain into groundwater bodies, leaving an empty soil as the local equilibrium state - there would be no soil water dynamics at all, probably even no terrestrial vegetation without capillary forces. Better alternatives for the Darcy-Richards approach are thus highly desirable, as long they preserve the grain of "truth" about capillarity as first order control. Here we propose such an alternative approach to simulate soil moisture dynamics in a stochastic and yet physical way. Soil water is represented by particles of constant mass, which travel according to the Itô form of the Fokker Planck equation. The model concept builds on

4. Stability analysis and investigation of higher order Schrödinger equation for strongly dispersive ion-acoustic wave in plasma

Gogoi, R.; Kalita, L.; Devi, N.

2010-02-01

Much interest was shown towards the studies on nonlinear stability in the late sixties. Plasma instabilities play an important role in plasma dynamics. More attention has been given towards stability analysis after recognizing that they are one of the principal obstacles in the way of a successful resolution of the problem of controlled thermonuclear fusion. Nonlinearity and dispersion are the two important characteristics of plasma instabilities. Instabilities and nonlinearity are the two important and interrelated terms. In our present work, the continuity and momentum equations for both ions and electrons together with the Poisson equation are considered as cold plasma model. Then we have adopted the modified reductive perturbation technique (MRPT) from Demiray [1] to derive the higher order equation of the Nonlinear Schrödinger equation (NLSE). In this work, detailed mathematical expressions and calculations are done to investigate the changing character of the modulation of ion acoustic plasma wave through our derived equation. Thus we have extended the application of MRPT to derive the higher order equation. Both progressive wave solutions as well as steady state solutions are derived and they are plotted for different plasma parameters to observe dark/bright solitons. Interesting structures are found to exist for different plasma parameters.

5. Accuracy of Weak-Form Discretisation and Extention of Recursive Transfer Method for Scattering Problems Governed by Fourth-Order Differential Equation

Kato, Hatsuhiro; Kato, Hatsuyoshi

2016-05-01

We proposed a new discretisation scheme for deriving a second-order difference equation from any system being formulated with the weak-form theory framework. The proposed scheme enables us to extend the application range of the recursive transfer method (RTM) and to express perfectly matching conditions for port boundaries in a discrete fashion under the RTM framework. To evaluate the accuracy and demonstrate the validity of the proposed scheme, we discussed the scattering problem governed by the fourth-order differential equation that was hitherto outside the RTM application range. The difference equation can play an important role in maintaining the balance of the bending moment and the shear force at the interface of two segments. Using the new port boundary condition, a quasi-localised wave was extracted and found to be related to the phase shift due to Fano resonance.

6. Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

Majumder, D. P.; Dhar, A. K.

2015-08-01

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

7. Modelling the flow of a second order fluid through and over a porous medium using the volume averages. I. The generalized Brinkman's equation

Minale, Mario

2016-02-01

In this paper, the generalized Brinkman's equation for a viscoelastic fluid is derived using the volume averages. Darcy's generalised equation is consequently obtained neglecting the first and the second Brinkman's correction with respect to the drag term. The latter differs from the Newtonian drag because of an additional term quadratic in the velocity and inversely proportional to a "viscoelastic" permeability defined in the paper. The viscoelastic permeability tensor can be calculated by solving a boundary value problem, but it must be in fact experimentally measured. To isolate the elastic contribution, the constitutive equation of the second order fluid of Coleman and Noll is chosen because, in simple shear at steady state, second order fluids show a constant viscosity and first and second normal stress differences quadratic in the shear rate. The model predictions are compared with data of the literature obtained in a Darcy's experiment and the agreement is good.

8. Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed

Canestrelli, Alberto; Siviglia, Annunziato; Dumbser, Michael; Toro, Eleuterio F.

2009-06-01

This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263-91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103-34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300-21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.

9. Robust Optimal Stopping-Time Control for Nonlinear Systems

SciTech Connect

Ball, J.A.; Chudoung, J.; Day, M.V.

2002-10-01

We formulate a robust optimal stopping-time problem for a state-space system and give the connection between various notions of lower value function for the associated games (and storage function for the associated dissipative system) with solutions of the appropriate variational inequality (VI) (the analogue of the Hamilton-Jacobi-Bellman-Isaacs equation for this setting). We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense and a positive definite supersolution of the VI can be used for stability analysis.

10. An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations

Pan, Liang; Xu, Kun; Li, Qibing; Li, Jiequan

2016-12-01

For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21]. In this paper, based on the same time-stepping method and the second-order GKS flux function [42], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge-Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function

11. Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials.

PubMed

Chen, Yong; Yan, Zhenya

2016-03-22

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields.

12. Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials

PubMed Central

Chen, Yong; Yan, Zhenya

2016-01-01

Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schrödinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time -symmetry was introduced in 1998. In this paper, we present novel bright solitons of the NLS equation with third-order dispersion in some complex -symmetric potentials (e.g., physically relevant -symmetric Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear modes even if the respective linear -symmetric phases are broken. Moreover, we also use the adiabatic changes of the control parameters to excite the initial modes related to exact solitons to reach stable nonlinear modes. The elastic interactions of two solitons are exhibited in the third-order NLS equation with -symmetric potentials. Our results predict the dynamical phenomena of soliton equations in the presence of third-order dispersion and -symmetric potentials arising in nonlinear fiber optics and other physically relevant fields. PMID:27002543

13. Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order.

PubMed

Petrović, Nikola Z; Belić, Milivoj; Zhong, Wei-Ping

2011-02-01

We obtain exact traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients and polynomial Kerr nonlinearity of an arbitrarily high order. Exact solutions, given in terms of Jacobi elliptic functions, are presented for the special cases of cubic-quintic and septic models. We demonstrate that the widely used method for finding exact solutions in terms of Jacobi elliptic functions is not applicable to the nonlinear Schrödinger equation with saturable nonlinearity.

14. A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants.

PubMed

Lin, Ying-Tsong; Collis, Jon M; Duda, Timothy F

2012-11-01

An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

15. Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line

Hashemi, M. S.; Baleanu, D.

2016-07-01

We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate ϑ is imposed onto the problem in order to transform the dependent variable u (x , t) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation.

16. Bound-state solitons for the coupled variable-coefficient higher-order nonlinear Schrödinger equations in the inhomogeneous optical fiber

Liu, De-Yin; Tian, Bo; Xie, Xi-Yang

2017-03-01

Bound-state vector soliton solutions for the coupled variable-coefficient higher-order nonlinear Schrödinger equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fiber, are investigated. Introducing auxiliary functions, we derive the bilinear forms and corresponding constraints on the variable coefficients. Through symbolic computation, we construct the one- and two-soliton solutions. We see that the variable coefficients in the equations affect the soliton structures. With different choices of the variable coefficients, we obtain the cubic, periodic, and parabolic solitons. Bound-state solitons and interactions are analyzed graphically.

17. Estimates of the stabilization rate as t{yields}{infinity} of solutions of the first mixed problem for a quasilinear system of second-order parabolic equations

SciTech Connect

Kozhevnikova, L M; Mukminov, F Kh

2000-02-28

A quasilinear system of parabolic equations with energy inequality is considered in a cylindrical domain {l_brace}t>0{r_brace}x{omega}. In a broad class of unbounded domains {omega} two geometric characteristics of a domain are identified which determine the rate of convergence to zero as t{yields}{infinity} of the L{sub 2}-norm of a solution. Under additional assumptions on the coefficients of the quasilinear system estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation.

18. Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps

Ren, Yong; Sakthivel, R.

2012-07-01

In this paper, we study a class of second-order neutral stochastic evolution equations with infinite delay and Poisson jumps (SNSEEIPs), in which the initial value belongs to the abstract space B. We establish the existence and uniqueness of mild solutions for SNSEEIPs under non-Lipschitz condition with Lipschitz condition being considered as a special case by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial data by means of a corollary of the Bihari inequality. An application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given to illustrate the theory.

19. Comment on: "Traveling wave solutions for fifth-order KdV type equations with time-dependent coefficients" [Commun Nonlinear Sci Numer Simulat 19 (2014) 404-408

Sarkar, Tanmay

2015-06-01

In this paper, we demonstrate that previously reported traveling wave solutions for the fifth order KdV type equations with time dependent coefficients (Triki and Wazwaz, 2014) are incorrect. We present the corrected traveling wave solutions for fifth order KdV type equations using sine-cosine method. In addition, we provide traveling wave solutions for the Kawahara equation and Kaup-Kupershmidt equation as an application.

20. A Partially-ordered-set Based Approach to the Dirac Equation in 3+1 space-time

Earle, Keith; Knuth, Kevin

2012-02-01

Recent work by Knuth and co-workers has shown how insights into Einstein's Theory of Special Relativity may be obtained by careful reasoning about consistent quantification of a poset. The Feynman Chessboard problem in 1+1 spacetime can be treated from this perspective, for example. Alternative methods of solution based on techniques borrowed from statistical mechanics have also been developed over the years to solve the Feynman Chessboard model in 1+1 spacetime. One particularly intriguing solution is based on a master-equation approach developed by McKeon and Ord for 1+1 spacetime. We will show how this model may be extended to 3+1 spacetime using techniques developed by Bialynicki-Birula, thus providing an alternative derivation of the Dirac equation. An external electromagnetic field can be accommodated very naturally in the formalism from which a pleasing pictorial representation of electromagnetic interactions in the lattice picture emerges.

1. A study of infrasound propagation based on high-order finite difference solutions of the Navier-Stokes equations.

PubMed

Marsden, O; Bogey, C; Bailly, C

2014-03-01

The feasibility of using numerical simulation of fluid dynamics equations for the detailed description of long-range infrasound propagation in the atmosphere is investigated. The two dimensional (2D) Navier Stokes equations are solved via high fidelity spatial finite differences and Runge-Kutta time integration, coupled with a shock-capturing filter procedure allowing large amplitudes to be studied. The accuracy of acoustic prediction over long distances with this approach is first assessed in the linear regime thanks to two test cases featuring an acoustic source placed above a reflective ground in a homogeneous and weakly inhomogeneous medium, solved for a range of grid resolutions. An atmospheric model which can account for realistic features affecting acoustic propagation is then described. A 2D study of the effect of source amplitude on signals recorded at ground level at varying distances from the source is carried out. Modifications both in terms of waveforms and arrival times are described.

2. The Picard–Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system

SciTech Connect

Fedorov, Yuri E-mail: Chara.Pantazi@upc.edu; Pantazi, Chara E-mail: Chara.Pantazi@upc.edu

2014-03-15

We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard–Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard–Fuchs equations can be used to study various properties of the actions.

3. Kinetics of solute adsorption at solid/aqueous interfaces: searching for the theoretical background of the modified pseudo-first-order kinetic equation.

PubMed

2008-05-20

It is shown that the modified pseudo-first-order (MPFO) kinetic equation proposed recently by Yang and Al-Duri simulates well the behavior of the kinetics governed by the rate of surface reaction and described by our general kinetic equation, based on the statistical rate theory. The linear representation with respect to time, offered by the MPFO equation seems to be a convenient tool for distinguishing between the surface reaction and the diffusional kinetics. Also, a method of distinguishing between the surface reaction and the intraparticle diffusion model based on analyzing the initial kinetic isotherms of sorption is proposed. The applicability of these procedures is demonstrated by the analysis of adsorption kinetics of the reactive yellow dye onto an activated carbon.

4. Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation.

PubMed

Wang, Lei; Zhang, Jian-Hui; Wang, Zi-Qi; Liu, Chong; Li, Min; Qi, Feng-Hua; Guo, Rui

2016-01-01

We study the nonlinear waves on constant backgrounds of the higher-order generalized nonlinear Schrödinger (HGNLS) equation describing the propagation of ultrashort optical pulse in optical fibers. We derive the breather, rogue wave, and semirational solutions of the HGNLS equation. Our results show that these three types of solutions can be converted into the nonpulsating soliton solutions. In particular, we present the explicit conditions for the transitions between breathers and solitons with different structures. Further, we investigate the characteristics of the collisions between the soliton and breathers. Especially, based on the semirational solutions of the HGNLS equation, we display the novel interactions between the rogue waves and other nonlinear waves. In addition, we reveal the explicit relation between the transition and the distribution characteristics of the modulation instability growth rate.

5. Vlasov equation eigenvalues and eigenvectors for Fourier-Hermite dispersion matrices of order greater than 1,000

NASA Technical Reports Server (NTRS)

Grant, F. C.

1972-01-01

The connection between the Van Kampen and Landau representations of the Vlasov equations has been extended to Fourier-Hermite expansions containing more than 1000 terms by taking advantage of the properties of tridiagonal matrices. These numerical results are regarded as conclusive indications of the nonuniformly convergent behavior of the approximation curve in the limit of an infinite number of terms and represent an extension of work begun by Grant (1967) and by Grant and Feix (1967).

6. Analytical and Numerical Problems Associtated with Extending the Use of the Second order Even-parity Transport Equation.

DTIC Science & Technology

1981-06-01

including the ground was demonstrated by Straker (Ref. 2) who showed a significant effect on the atmospheric neutron distribution due to the presence of...regions of the spatial domain. This set of equations was solved by finite difference methods. The angular dependence of the neutron distribution in...known as flux synthesis (Refs. 19-22) and provides a method of intro- ducing a priori knowledge of the particle distribution into the trial solution

7. Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations

Otto, Felix; Ramos, Fabio

2010-12-01

We derive upper bounds for the infinite-time and space average of the L 1-norm of the Littlewood-Paley decomposition of weak solutions of the 3 D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, {mathbf{U}_kappa˜ɛ^{1/3} kappa^{-1/3}} , holds as an upper bound for a region of wavenumbers near the dissipative cutoff.

8. High-order compact MacCormack scheme for two-dimensional compressible and non-hydrostatic equations of the atmosphere

2016-09-01

This study is devoted to application of the fourth-order compact MacCormack scheme to spatial differencing of the conservative form of two-dimensional and non-hydrostatic equation of a dry atmosphere. To advance the solution in time a four-stage Runge-Kutta method is used. To perform the simulations, two test cases including evolution of a warm bubble and a cold bubble in a neutral atmosphere with open and rigid boundaries are employed. In addition, the second-order MacCormack and the standard fourth-order compact MacCormack schemes are used to perform the simulations. Qualitative and quantitative assessment of the numerical results for different test cases exhibit the superiority of the fourth-order compact MacCormack scheme on the second-order method.

9. Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.

2016-08-01

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

10. Conformal kernel for the next-to-leading-order BFKL equation in N=4 super Yang-Mills theory

SciTech Connect

Balitsky, Ian; Chirilli, Giovanni A.

2009-02-01

Using the requirement of Moebius invariance of N=4 super Yang-Mills amplitudes in the Regge limit, we restore the explicit form of the conformal next-to-leading-order Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel out of the eigenvalues known from the forward next-to-leading-order BFKL result.

11. High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations

Antoine, Xavier; Besse, Christophe; Rispoli, Vittorio

2016-12-01

The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one- and two-components systems of Gross-Pitaevskii equations.

12. Exact Soliton Solutions for the (2+1)-Dimensional Coupled Higher-Order Nonlinear Schrödinger Equations in Birefringent Optical-Fiber Communication

Cai, Yue-Jin; Bai, Cheng-Lin; Luo, Qing-Long

2017-03-01

In birefringent optical fibers, the propagation of femtosecond soliton pulses is described by coupled higher-order nonlinear Schrödinger equations. In this paper, we will investigate the bright and dark soliton solutions of (2+1)-dimensional coupled higher-order nonlinear Schrödinger equations, with the aid of symbolic computation and the Hirota method. On the basis of soliton solutions, we test and discuss the interactions graphically between the solitons in the x-z, x-t, and z-t planes. Supported by the National Natural Science Foundation of China under Grant No. 61671227 and the Natural Science Foundation of Shandong Province under Grant No. ZR2014AM018

13. Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential.

PubMed

Wen, Xiao-Yong; Yan, Zhenya; Yang, Yunqing

2016-06-01

The integrable nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential [M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 110, 064105 (2013)] is investigated, which is an integrable extension of the standard nonlinear Schrödinger equation. Its novel higher-order rational solitons are found using the nonlocal version of the generalized perturbation (1,N-1)-fold Darboux transformation. These rational solitons illustrate abundant wave structures for the distinct choices of parameters (e.g., the strong and weak interactions of bright and dark rational solitons). Moreover, we also explore the dynamical behaviors of these higher-order rational solitons with some small noises on the basis of numerical simulations.

14. Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

Dhar, A. K.; Mondal, J.

2015-05-01

Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

15. Regularity of the Solution of Elliptic Problems with Piecewise Analytic Data. Part 1. Boundary Value Problems for Linear Ellilptic Equation of Second Order.

DTIC Science & Technology

1986-05-01

10O. PROGRAM ELEMENT. 11RO1JECT. TASK * Institute for Physical Science and Technology AREA & WORK UNIT NUMBERS ! ~University of Maryland 1% College...ELLIPTIC EQUATION OF SECOND ORDER I. Babuvka I Institute for Physical Science and Technology University of Maryland B. Guo 2 Department of Mathematics...neighborhood of the Program PROBE of Noetic Technologies, St. Louis. corners of the domain, place where the type of the boundary condition changes, etc

16. Improving the accuracy of mass-lumped finite-elements in the first-order formulation of the wave equation by defect correction

Shamasundar, R.; Mulder, W. A.

2016-10-01

Finite-element discretizations of the acoustic wave equation in the time domain often employ mass lumping to avoid the cost of inverting a large sparse mass matrix. For the second-order formulation of the wave equation, mass lumping on Legendre-Gauss-Lobatto points does not harm the accuracy. Here, we consider a first-order formulation of the wave equation. In that case, the numerical dispersion for odd-degree polynomials exhibits super-convergence with a consistent mass matrix but mass lumping destroys that property. We consider defect correction as a means to restore the accuracy, in which the consistent mass matrix is approximately inverted using the lumped one as preconditioner. For the lowest-degree element on a uniform mesh, fourth-order accuracy in 1D can be obtained with just a single iteration of defect correction. The numerical dispersion curve describes the error in the eigenvalues of the discrete set of equations. However, the error in the eigenvectors also play a role, in two ways. For polynomial degrees above one and when considering a 1-D mesh with constant element size and constant material properties, a number of modes, equal to the maximum polynomial degree, are coupled. One of these is the correct physical mode that should approximate the true eigenfunction of the operator, the other are spurious and should have a small amplitude when the true eigenfunction is projected onto them. We analyze the behaviour of this error as a function of the normalized wavenumber in the form of the leading terms in its series expansion and find that this error exceeds the dispersion error, except for the lowest degree where the eigenvector error is zero. Numerical 1-D tests confirm this behaviour. We briefly analyze the 2-D case, where the lowest-degree polynomial also appears to provide fourth-order accuracy with defect correction, if the grid of squares or triangles is highly regular and material properties are constant.

17. Higher-order Peregrine combs and Peregrine walls for the variable-coefficient Lenells-Fokas equation

Wang, Zi-Qi; Wang, Xin; Wang, Lei; Sun, Wen-Rong; Qi, Feng-Hua

2017-02-01

In this paper, we study the variable-coefficient Lenells-Fokas (LF) model. Under large periodic modulations in the variable coefficients of the LF model, the generalized Akhmediev breathers develop into the breather multiple births (BMBs) from which we obtain the Peregrine combs (PCs). The PCs can be considered as the limiting case of the BMBs and be transformed into the Peregrine walls (PWs) with a specific amplitude of periodic modulation. We further investigate the spatiotemporal characteristics of the PCs and PWs analytically. Based on the second-order breather and rogue-wave solutions, we derive the corresponding higher-order structures (higher-order PCs and PWs) under proper periodic modulations. What is particularly noteworthy is that the second-order PC can be converted into the Peregrine pyramid which exhibits the higher amplitude and thickness. Our results could be helpful for the design of experiments in the optical fiber communications.

18. Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms

Fujioka, J.; Espinosa, A.

2015-11-01

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.

19. Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms.

PubMed

Fujioka, J; Espinosa, A

2015-11-01

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.

20. Higher-order split operator schemes for solving the Schrödinger equation in the time-dependent wave packet method: applications to triatomic reactive scattering calculations.

PubMed

Sun, Zhigang; Yang, Weitao; Zhang, Dong H

2012-02-14

The efficiency of the numerical propagators for solving the time-dependent Schrödinger equation in the wave packet approach to reactive scattering is of vital importance. In this Perspective, we first briefly review the propagators used in quantum reactive scattering calculations and their applications to triatomic reactions. Then we present a detailed comparison of about thirty higher-order split operator propagators for solving the Schrödinger equation with their applications to the wave packet evolution within a one-dimensional Morse potential, and the total reaction probability calculations for the H + HD, H + NH, H + O(2), and F + HD reactions. These four triatomic reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these higher-order propagation methods for describing reactive scattering dynamics. Our calculations reveal that the most often used second-order split operator method is typically more efficient for a direct reaction, particularly for those involving flat potential energy surfaces. However, the optimal higher-order split operator methods are more suitable for a reaction with resonances and intermediate complexes or a reaction experiencing potential energy surface with fluctuations of considerable amplitude. Three 4th-order and one 6th-order split operator methods, which are most efficient for solving reactive scattering in various conditions among the tested ones, are recommended for general applications. In addition, a brief discussion on the relative performance between the Chebyshev real wave packet method and the split operator method is given. The results in this Perspective are expected to stimulate more applications of (high-order) split operators to the quantum reactive scattering calculation and other related problems.

1. A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

SciTech Connect

Wieselquist, William A.; Anistratov, Dmitriy Y.; Morel, Jim E.

2014-09-15

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h{sup 2}) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

2. Higher-order nonlinear equations for the electron-acoustic waves in a nonextensive electron-positron-ion plasma

Rafat, A.; Rahman, M. M.; Alam, M. S.; Mamun, A. A.

2015-07-01

A precise theoretical investigation has been made on electron-acoustic (EA) Gardner solitons (GSs) and double layers (DLs) in a four-component plasma system consisting of nonextensive hot electrons and positrons, inertial cold electrons, and immobile positive ions. The well-known reductive perturbation method has been used to derive the Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and Gardner equations along with their solitary wave as well as double layer solutions. It has been found that depending on the plasma parameters, the K-dV solitons and GSs are either compressive or rarefactive, whereas the mK-dV solitons are only compressive, and Gardner DLs are only rarefactive. The analytical comparison among the K-dV solitons, mK-dV solitons, and GSs are also investigated. It has been identified that the basic properties of such EA solitons and EA DLs are significantly modified due to the effects of nonextensivity and other plasma parameters related to plasma particle number densities and to temperature of different plasma species. The results of our present investigation can be helpful for understanding the nonlinear electrostatic structures associated with EA waves in various interstellar space plasma environments and cosmological scenarios (viz. quark-gluon plasma, protoneutron stars, stellar polytropes, hadronic matter, dark-matter halos, etc.)

3. High-order rogue waves of the coupled nonlinear Schrödinger equations with negative coherent coupling in an isotropic medium

Sun, Wen-Rong; Tian, Bo; Xie, Xi-Yang; Chai, Jun; Jiang, Yan

2016-10-01

High-order rogue waves of the coupled nonlinear Schrödinger equations with negative coherent coupling, which describe the propagation of orthogonally polarized optical waves in an isotropic medium, are reported in this paper. Key point lies in the introduction of a limit process in the Darboux transformation, with which we obtain a family of the first- and second-order rational solutions for the purpose of modelling the rogue waves. We observe that the double-hump rogue wave in the course of evolution turns into the one-hump rogue wave, and that the dark rogue wave with four valleys in the course of evolution turns into the bright rogue wave. It is found that the second-order rogue wave can split up, giving birth to the multiple rogue waves.

4. Second-order rogue wave breathers in the nonlinear Schrödinger equation with quadratic potential modulated by a spatially-varying diffraction coefficient.

PubMed

Zhong, Wei-Ping; Belić, Milivoj; Zhang, Yiqi

2015-02-09

Nonlinear Schrödinger equation with simple quadratic potential modulated by a spatially-varying diffraction coefficient is investigated theoretically. Second-order rogue wave breather solutions of the model are constructed by using the similarity transformation. A modal quantum number is introduced, useful for classifying and controlling the solutions. From the solutions obtained, the behavior of second order Kuznetsov-Ma breathers (KMBs), Akhmediev breathers (ABs), and Peregrine solitons is analyzed in particular, by selecting different modulation frequencies and quantum modal parameter. We show how to generate interesting second order breathers and related hybrid rogue waves. The emergence of true rogue waves - single giant waves that are generated in the interaction of KMBs, ABs, and Peregrine solitons - is explicitly displayed in our analytical solutions.

5. A functional realization of 𝔰𝔩(3, ℝ) providing minimal Vessiot-Guldberg-Lie algebras of nonlinear second-order ordinary differential equations as proper subalgebras

Campoamor-Stursberg, R.

2016-06-01

A functional realization of the Lie algebra s l (" separators=" 3 , R) as a Vessiot-Guldberg-Lie algebra of second order differential equation (SODE) Lie systems is proposed. It is shown that a minimal Vessiot-Guldberg-Lie algebra L V G is obtained from proper subalgebras of s l (" separators=" 3 , R) for each of the SODE Lie systems of this type by particularization of one functional and two scalar parameters of the s l (" separators=" 3 , R) -realization. The relation between the various Vessiot-Guldberg-Lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of SODE Lie systems.

6. Improved fully-implicit spherical harmonics methods for first and second order forms of the transport equation using Galerkin Finite Element

Laboure, Vincent Matthieu

In this dissertation, we focus on solving the linear Boltzmann equation -- or transport equation -- using spherical harmonics (PN) expansions with fully-implicit time-integration schemes and Galerkin Finite Element spatial discretizations within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The presentation is composed of two main ensembles. On one hand, we study the first-order form of the transport equation in the context of Thermal Radiation Transport (TRT). This nonlinear application physically necessitates to maintain a positive material temperature while the PN approximation tends to create oscillations and negativity in the solution. To mitigate these flaws, we provide a fully-implicit implementation of the Filtered PN (FPN) method and investigate local filtering strategies. After analyzing its effect on the conditioning of the system and showing that it improves the convergence properties of the iterative solver, we numerically investigate the error estimates derived in the linear setting and observe that they hold in the non-linear case. Then, we illustrate the benefits of the method on a standard test problem and compare it with implicit Monte Carlo (IMC) simulations. On the other hand, we focus on second-order forms of the transport equation for neutronics applications. We mostly consider the Self-Adjoint Angular Flux (SAAF) and Least-Squares (LS) formulations, the former being globally conservative but void incompatible and the latter having -- in all generality -- the opposite properties. We study the relationship between these two methods based on the weakly-imposed LS boundary conditions. Equivalences between various parity-based PN methods are also established, in particular showing that second-order filters are not an appropriate fix to retrieve void compatibility. The importance of global conservation is highlighted on a heterogeneous multigroup k-eigenvalue test problem. Based on these considerations, we propose a new

7. Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN

Liang, Sihua; Zhang, Jihui

2016-11-01

In this paper, we deal with the existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical nonlinearity: -ε4 Δ2 u + ε4 a + b ∫ ℝN |∇ u|)2 d x Δ u + V ( x ) u = |u| 2* * - 2 u + h ( x , u ) , (t, x) ∈ ℝ × ℝN. By using Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.

8. A high-order solver for unsteady incompressible Navier-Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping

Cox, Christopher; Liang, Chunlei; Plesniak, Michael W.

2016-06-01

We report development of a high-order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. This compact high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids.

9. A high-order solver for unsteady incompressible Navier-Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping

Cox, Christopher; Liang, Chunlei; Plesniak, Michael

2015-11-01

This paper reports development of a high-order compact method for solving unsteady incompressible flow on unstructured grids with implicit time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ the classical artificial compressibility treatment, where dual time stepping is needed to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time-stepping scheme. Three-dimensional results computed on many processing elements will be presented. The high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. Financial support provided under the GW Presidential Merit Fellowship.

10. Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

Achilleos, V.; Bishop, A. R.; Diamantidis, S.; Frantzeskakis, D. J.; Horikis, T. P.; Karachalios, N. I.; Kevrekidis, P. G.

2016-07-01

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections or to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, and chaotic spatiotemporal structures. Suitable low-dimensional phase-space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

11. Parallelization of the Red-Black Algorithm on Solving the Second-Order PN Transport Equation with the Hybrid Finite Element Method

SciTech Connect

Yaqi Wang; Cristian Rabiti; Giuseppe Palmiotti

2011-06-01

The Red-Black algorithm has been successfully applied on solving the second-order parity transport equation with the PN approximation in angle and the Hybrid Finite Element Method (HFEM) in space, i.e., the Variational Nodal Method (VNM) [1,2,3,4,5]. Any transport solving techniques, including the Red-Black algorithm, need to be parallelized in order to take the advantage of the development of supercomputers with multiple processors for the advanced modeling and simulation. To our knowledge, an attempt [6] was done to parallelize it, but it was devoted only to the z axis plans in three-dimensional calculations. General parallelization of the Red-Black algorithm with the spatial domain decomposition has not been reported in the literature. In this summary, we present our implementation of the parallelization of the Red-Black algorithm and its efficiency results.

12. Soliton excitations and interactions for the three-coupled fourth-order nonlinear Schrödinger equations in the alpha helical proteins

Sun, Wen-Rong; Tian, Bo; Wang, Yu-Feng; Zhen, Hui-Ling

2015-06-01

Three-coupled fourth-order nonlinear Schrödinger equations describe the dynamics of alpha helical proteins with the interspine coupling at the higher order. Through symbolic computation and binary Bell-polynomial approach, bilinear forms and N-soliton solutions for such equations are constructed. Key point lies in the introduction of auxiliary functions in the Bell-polynomial expression. Asymptotic analysis is applied to investigate the elastic interaction between the two solitons: two solitons keep their original amplitudes, energies and velocities invariant after the interaction except for the phase shifts. Soliton amplitudes are related to the energy distributed in the solitons of the three spines. Overtaking interaction, head-on interaction and bound-state solitons of two solitons are given. Bound states of three bright solitons arise when all of them propagate in parallel. Elastic interaction between the bound-state solitons and one bright soliton is shown. Increase of the lattice parameter can lead to the increase of the soliton velocity, that is, the interaction period becomes shorter. The solitons propagating along the neighbouring spines are found to interact elastically. Those solitons, exhibited in this paper, might be viewed as a possible carrier of bio-energy transport in the protein molecules.

13. Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

Jia, Shu-Liang; Gao, Yi-Tian; Zhao, Chen; Lan, Zhong-Zhou; Feng, Yu-Jie

2017-01-01

Under investigation in this paper is a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber. Through the Darboux transformation (DT) and generalized DT, we obtain the multi-soliton solutions, breathers and rogue waves. Choosing different values of α( x), β( x), γ( x) and δ( x), which are the coefficients of the third-, fourth-, fifth- and sixth-order dispersions, respectively, we investigate their effects on those solutions, where x is the scaled propagation variable. When α( x), β( x), γ( x) and δ( x) are chosen as the linear, parabolic and periodic functions, we obtain the parabolic, cubic and quasi-periodic solitons, respectively. Head-on and overtaking interactions between the two solitons are presented, and the interactions are elastic. Besides, with certain values of the spectral parameter λ, a shock region between the two solitons appears, and the interaction is inelastic. Interactions between two kinds of the breathers are also studied, and we find that the interaction regions are similar to those of the second-order rogue waves. Rogue waves are split into some first-order rogue waves when α( x), β( x), γ( x) and δ( x) are the periodic or odd-numbered functions.

14. Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects

Wang, Lei; Zhang, Jian-Hui; Liu, Chong; Li, Min; Qi, Feng-Hua

2016-06-01

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W -shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W -shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

15. Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects.

PubMed

Wang, Lei; Zhang, Jian-Hui; Liu, Chong; Li, Min; Qi, Feng-Hua

2016-06-01

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

16. A problem with inverse time for a singularly perturbed integro-differential equation with diagonal degeneration of the kernel of high order

Bobodzhanov, A. A.; Safonov, V. F.

2016-04-01

We consider an algorithm for constructing asymptotic solutions regularized in the sense of Lomov (see [1], [2]). We show that such problems can be reduced to integro-differential equations with inverse time. But in contrast to known papers devoted to this topic (see, for example, [3]), in this paper we study a fundamentally new case, which is characterized by the absence, in the differential part, of a linear operator that isolates, in the asymptotics of the solution, constituents described by boundary functions and by the fact that the integral operator has kernel with diagonal degeneration of high order. Furthermore, the spectrum of the regularization operator A(t) (see below) may contain purely imaginary eigenvalues, which causes difficulties in the application of the methods of construction of asymptotic solutions proposed in the monograph [3]. Based on an analysis of the principal term of the asymptotics, we isolate a class of inhomogeneities and initial data for which the exact solution of the original problem tends to the limit solution (as \\varepsilon\\to+0) on the entire time interval under consideration, also including a boundary-layer zone (that is, we solve the so-called initialization problem). The paper is of a theoretical nature and is designed to lead to a greater understanding of the problems in the theory of singular perturbations. There may be applications in various applied areas where models described by integro-differential equations are used (for example, in elasticity theory, the theory of electrical circuits, and so on).

17. Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations

Balajewicz, Maciej; Tezaur, Irina; Dowell, Earl

2016-09-01

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier-Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier-Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. The reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

18. Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

DOE PAGES

Balajewicz, Maciej; Tezaur, Irina; Dowell, Earl

2016-05-25

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and themore » Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. As a result, the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.« less

19. Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

SciTech Connect

Balajewicz, Maciej; Tezaur, Irina; Dowell, Earl

2016-05-25

For a projection-based reduced order model (ROM) of a fluid flow to be stable and accurate, the dynamics of the truncated subspace must be taken into account. This paper proposes an approach for stabilizing and enhancing projection-based fluid ROMs in which truncated modes are accounted for a priori via a minimal rotation of the projection subspace. Attention is focused on the full non-linear compressible Navier–Stokes equations in specific volume form as a step toward a more general formulation for problems with generic non-linearities. Unlike traditional approaches, no empirical turbulence modeling terms are required, and consistency between the ROM and the Navier–Stokes equation from which the ROM is derived is maintained. Mathematically, the approach is formulated as a trace minimization problem on the Stiefel manifold. As a result, the reproductive as well as predictive capabilities of the method are evaluated on several compressible flow problems, including a problem involving laminar flow over an airfoil with a high angle of attack, and a channel-driven cavity flow problem.

20. Computation of stochastic wellhead protection zones by combining the first-order second-moment method and Kolmogorov backward equation analysis

Kunstmann, H.; Kinzelbach, W.

2000-11-01

Input parameters of groundwater models are usually poorly known and model results suffer from uncertainty. When conservative decisions have to be drawn, the quantification of uncertainties is necessary. Monte Carlo techniques are suited for this analysis but usually require a huge computational effort. An alternative and computationally efficient approach is the first-order second-moment (FOSM) analysis which directly propagates the uncertainty originating from input parameters into the result. We apply the FOSM method to both the groundwater flow and solute transport equations. It is shown how calibration on the basis of measured heads yields the "Principle of Interdependent Uncertainty" that correlates the uncertainties of feasible transmissivities and recharge rates. The method is used to compute the uncertainty of steady state heads and of steady state solute concentrations. The second-moment analysis of solute concentrations is combined with the Kolmogorov backward equations and applied to the stochastic computation of wellhead protection zones for a pumping well group in Gambach (Germany). Unconditional and conditional simulation results are compared to corresponding Monte Carlo simulations. The unconditioned FOSM method reveals a computational advantage of a factor of 5-10 against the Monte Carlo method in terms of CPU time requirements. Conditioned FOSM shows an even larger advantage with a factor of 50-100 against the usual inverse stochastic modeling method based on Monte Carlo techniques.

1. Conservation laws, bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber

SciTech Connect

Chai, Jun; Tian, Bo Zhen, Hui-Ling; Sun, Wen-Rong

2015-08-15

Under investigation in this paper is a fifth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on the Lax pair, infinitely-many conservation laws are derived. With the aid of auxiliary functions, bilinear forms, one-, two- and three-soliton solutions in analytic forms are generated via the Hirota method and symbolic computation. Soliton velocity varies linearly with the coefficients of the high-order terms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons as well as the bound state are depicted. For the interactions among the three solitons, two head-on and one overtaking interactions, three overtaking interactions, an interaction between a bound state and a single soliton and the bound state are displayed. Graphical analysis shows that the interactions between the two solitons are elastic, and interactions among the three solitons are pairwise elastic. Stability analysis yields the modulation instability condition for the soliton solutions.

2. NLSEmagic: Nonlinear Schrödinger equation multi-dimensional Matlab-based GPU-accelerated integrators using compact high-order schemes

Caplan, R. M.

2013-04-01

We present a simple to use, yet powerful code package called NLSEmagic to numerically integrate the nonlinear Schrödinger equation in one, two, and three dimensions. NLSEmagic is a high-order finite-difference code package which utilizes graphic processing unit (GPU) parallel architectures. The codes running on the GPU are many times faster than their serial counterparts, and are much cheaper to run than on standard parallel clusters. The codes are developed with usability and portability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with the MEX-compiler interface. The packages are freely distributed, including user manuals and set-up files. Catalogue identifier: AEOJ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOJ_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 124453 No. of bytes in distributed program, including test data, etc.: 4728604 Distribution format: tar.gz Programming language: C, CUDA, MATLAB. Computer: PC, MAC. Operating system: Windows, MacOS, Linux. Has the code been vectorized or parallelized?: Yes. Number of processors used: Single CPU, number of GPU processors dependent on chosen GPU card (max is currently 3072 cores on GeForce GTX 690). Supplementary material: Setup guide, Installation guide. RAM: Highly dependent on dimensionality and grid size. For typical medium-large problem size in three dimensions, 4GB is sufficient. Keywords: Nonlinear Schröodinger Equation, GPU, high-order finite difference, Bose-Einstien condensates. Classification: 4.3, 7.7. Nature of problem: Integrate solutions of the time-dependent one-, two-, and three-dimensional cubic nonlinear Schrödinger equation. Solution method: The integrators utilize a fully-explicit fourth-order Runge-Kutta scheme in time

3. Generalized semi-analytical solutions to multispecies transport equation coupled with sequential first-order reaction network with spatially or temporally variable transport and decay coefficients

Suk, Heejun

2016-08-01

This paper presents a semi-analytical procedure for solving coupled the multispecies reactive solute transport equations, with a sequential first-order reaction network on spatially or temporally varying flow velocities and dispersion coefficients involving distinct retardation factors. This proposed approach was developed to overcome the limitation reported by Suk (2013) regarding the identical retardation values for all reactive species, while maintaining the extensive capability of the previous Suk method involving spatially variable or temporally variable coefficients of transport, general initial conditions, and arbitrary temporal variable inlet concentration. The proposed approach sequentially calculates the concentration distributions of each species by employing only the generalized integral transform technique (GITT). Because the proposed solutions for each species' concentration distributions have separable forms in space and time, the solution for subsequent species (daughter species) can be obtained using only the GITT without the decomposition by change-of-variables method imposing the limitation of identical retardation values for all the reactive species by directly substituting solutions for the preceding species (parent species) into the transport equation of subsequent species (daughter species). The proposed solutions were compared with previously published analytical solutions or numerical solutions of the numerical code of the Two-Dimensional Subsurface Flow, Fate and Transport of Microbes and Chemicals (2DFATMIC) in three verification examples. In these examples, the proposed solutions were well matched with previous analytical solutions and the numerical solutions obtained by 2DFATMIC model. A hypothetical single-well push-pull test example and a scale-dependent dispersion example were designed to demonstrate the practical application of the proposed solution to a real field problem.

4. Optimized equation of the state of the square-well fluid of variable range based on a fourth-order free-energy expansion

Espíndola-Heredia, Rodolfo; del Río, Fernando; Malijevsky, Anatol

2009-01-01

The free energy of square-well (SW) systems of hard-core diameter σ with ranges 1≤λ≤3 is expanded in a perturbation series. This interval covers most ranges of interest, from short-ranged SW fluids (λ ≃1.2) used in modeling colloids to long ranges (λ ≃3) where the van der Waals classic approximation holds. The first four terms are evaluated by means of extensive Monte Carlo simulations. The calculations are corrected for the thermodynamic limit and care is taken to evaluate and to control the various sources of error. The results for the first two terms in the series confirm well-known independent results but have an increased estimated accuracy and cover a wider set of well ranges. The results for the third- and fourth-order terms are novel. The free-energy expansion for systems with short and intermediate ranges, 1≤λ≤2, is seen to have properties similar to those of systems with longer ranges, 2≤λ≤3. An equation of state (EOS) is built to represent the free-energy data. The thermodynamics given by this EOS, confronted against independent computer simulations, is shown to predict accurately the internal energy, pressure, specific heat, and chemical potential of the SW fluids considered and for densities 0≤ρσ3≤0.9 including subcritical temperatures. This fourth-order theory is estimated to be accurate except for a small region at high density, ρσ3≈0.9, and low temperature where terms of still higher order might be needed.

5. Kinetics of solute adsorption at solid/solution interfaces: a theoretical development of the empirical pseudo-first and pseudo-second order kinetic rate equations, based on applying the statistical rate theory of interfacial transport.

PubMed

2006-08-24

For practical applications of solid/solution adsorption processes, the kinetics of these processes is at least as much essential as their features at equilibrium. Meanwhile, the general understanding of this kinetics and its corresponding theoretical description are far behind the understanding and the level of theoretical interpretation of adsorption equilibria in these systems. The Lagergren empirical equation proposed at the end of 19th century to describe the kinetics of solute sorption at the solid/solution interfaces has been the most widely used kinetic equation until now. This equation has also been called the pseudo-first order kinetic equation because it was intuitively associated with the model of one-site occupancy adsorption kinetics governed by the rate of surface reaction. More recently, its generalization for the two-sites-occupancy adsorption was proposed and called the pseudo-second-order kinetic equation. However, the general use and the wide applicability of these empirical equations during more than one century have not resulted in a corresponding fundamental search for their theoretical origin. Here the first theoretical development of these equations is proposed, based on applying the new fundamental approach to kinetics of interfacial transport called the Statistical Rate Theory. It is shown that these empirical equations are simplified forms of a more general equation developed here, for the case when the adsorption kinetics is governed by the rate of surface reactions. The features of that general equation are shown by presenting exhaustive model investigations, and the applicability of that equation is tested by presenting a quantitative analysis of some experimental data reported in the literature.

6. Three-phase compositional modeling of CO2 injection by higher-order finite element methods with CPA equation of state for aqueous phase

Moortgat, Joachim; Li, Zhidong; Firoozabadi, Abbas

2012-12-01

Most simulators for subsurface flow of water, gas, and oil phases use empirical correlations, such as Henry's law, for the CO2 composition in the aqueous phase, and equations of state (EOS) that do not represent the polar interactions between CO2and water. Widely used simulators are also based on lowest-order finite difference methods and suffer from numerical dispersion and grid sensitivity. They may not capture the viscous and gravitational fingering that can negatively affect hydrocarbon (HC) recovery, or aid carbon sequestration in aquifers. We present a three-phase compositional model based on higher-order finite element methods and incorporate rigorous and efficient three-phase-split computations for either three HC phases or water-oil-gas systems. For HC phases, we use the Peng-Robinson EOS. We allow solubility of CO2in water and adopt a new cubic-plus-association (CPA) EOS, which accounts for cross association between H2O and CO2 molecules, and association between H2O molecules. The CPA-EOS is highly accurate over a broad range of pressures and temperatures. The main novelty of this work is the formulation of a reservoir simulator with new EOS-based unique three-phase-split computations, which satisfy both the equalities of fugacities in all three phases and the global minimum of Gibbs free energy. We provide five examples that demonstrate twice the convergence rate of our method compared with a finite difference approach, and compare with experimental data and other simulators. The examples consider gravitational fingering during CO2sequestration in aquifers, viscous fingering in water-alternating-gas injection, and full compositional modeling of three HC phases.

7. Discontinuous isogeometric analysis methods for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation

Owens, A. R.; Welch, J. A.; Kópházi, J.; Eaton, M. D.

2016-06-01

In this paper two discontinuous Galerkin isogeometric analysis methods are developed and applied to the first-order form of the neutron transport equation with a discrete ordinate (SN) angular discretisation. The discontinuous Galerkin projection approach was taken on both an element level and the patch level for a given Non-Uniform Rational B-Spline (NURBS) patch. This paper describes the detailed dispersion analysis that has been used to analyse the numerical stability of both of these schemes. The convergence of the schemes for both smooth and non-smooth solutions was also investigated using the method of manufactured solutions (MMS) for multidimensional problems and a 1D semi-analytical benchmark whose solution contains a strongly discontinuous first derivative. This paper also investigates the challenges posed by strongly curved boundaries at both the NURBS element and patch level with several algorithms developed to deal with such cases. Finally numerical results are presented both for a simple pincell test problem as well as the C5G7 quarter core MOX/UOX small Light Water Reactor (LWR) benchmark problem. These numerical results produced by the isogeometric analysis (IGA) methods are compared and contrasted against linear and quadratic discontinuous Galerkin finite element (DGFEM) SN based methods.

8. First-Order System Least Squares for Velocity-Vorticity-Pressure Form of the Stokes Equations, with Application to Linear Elasticity

NASA Technical Reports Server (NTRS)

Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F.

1996-01-01

In this paper, we study the least-squares method for the generalized Stokes equations (including linear elasticity) based on the velocity-vorticity-pressure formulation in d = 2 or 3 dimensions. The least squares functional is defined in terms of the sum of the L(exp 2)- and H(exp -1)-norms of the residual equations, which is weighted appropriately by by the Reynolds number. Our approach for establishing ellipticity of the functional does not use ADN theory, but is founded more on basic principles. We also analyze the case where the H(exp -1)-norm in the functional is replaced by a discrete functional to make the computation feasible. We show that the resulting algebraic equations can be uniformly preconditioned by well-known techniques.

9. Solving Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

10. Aerospace Power Scholarly Research Program. Delivery Order 0013: Volume 2 - Flux Equations for a Direct Methanol Fuel Cell Solid Polymer Electrolyte Membrane

DTIC Science & Technology

2005-07-01

iii) concentrated solution theory based on the Onsager irreversible thermodynamics approach to transport processes. In very dilute to very...Stefan-Maxwell, Onsager 16. SECURITY CLASSIFICATION OF: 19a. NAME OF RESPONSIBLE PERSON (Monitor) a. REPORT Unclassified b. ABSTRACT...Flux Equations using Onsager Thermodynamics................................................................................ 39 4.1.3.1

11. Aerospace Power Scholarly Research Program. Delivery Order 0013: Volume 1. Development of Performance/Design Equations for a Direct Methanol Fuel Cell

DTIC Science & Technology

2005-07-01

gas mixture if the gas pressure is greater than 10 bar). Substituting the above result into Eq. (17) leads to: Δ= )()( gigi μμ (@ T...partial pressures of CH3OH, A gOHCHP )(3 , H2O, A gOHP )(2 may be obtained from the following equation: satili t gigi PxPyP

12. The Pendulum Equation

ERIC Educational Resources Information Center

Fay, Temple H.

2002-01-01

We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

13. Parametrically defined differential equations

Polyanin, A. D.; Zhurov, A. I.

2017-01-01

The paper deals with nonlinear ordinary differential equations defined parametrically by two relations. It proposes techniques to reduce such equations, of the first or second order, to standard systems of ordinary differential equations. It obtains the general solution to some classes of nonlinear parametrically defined ODEs dependent on arbitrary functions. It outlines procedures for the numerical solution of the Cauchy problem for parametrically defined differential equations.

14. Higher order nonlinear equations for the dust-acoustic waves in a dusty plasma with two temperature-ions and nonextensive electrons

SciTech Connect

Emamuddin, M.; Yasmin, S.; Mamun, A. A.

2013-04-15

The nonlinear propagation of dust-acoustic waves in a dusty plasma whose constituents are negatively charged dust, Maxwellian ions with two distinct temperatures, and electrons following q-nonextensive distribution, is investigated by deriving a number of nonlinear equations, namely, the Korteweg-de-Vries (K-dV), the modified Korteweg-de-Vries (mK-dV), and the Gardner equations. The basic characteristics of the hump (positive potential) and dip (negative potential) shaped dust-acoustic (DA) Gardner solitons are found to exist beyond the K-dV limit. The effects of two temperature ions and electron nonextensivity on the basic features of DA K-dV, mK-dV, and Gardner solitons are also examined. It has been observed that the DA Gardner solitons exhibit negative (positive) solitons for qq{sub c}) (where q{sub c} is the critical value of the nonextensive parameter q). The implications of our results in understanding the localized nonlinear electrostatic perturbations existing in stellar polytropes, quark-gluon plasma, protoneutron stars, etc. (where ions with different temperatures and nonextensive electrons exist) are also briefly addressed.

15. Uniform and C^1-approximability of functions on compact subsets of \\mathbb R^2 by solutions of second-order elliptic equations

Paramonov, P. V.; Fedorovskii, K. Yu

1999-02-01

Several necessary and sufficient conditions for the existence of uniform or C^1-approximation of functions on compact subsets of \\mathbb R^2 by solutions of elliptic systems of the form c_{11}u_{x_1x_1}+2c_{12}u_{x_1x_2}+c_{22}u_{x_2x_2}=0 with constant complex coefficients c_{11}, c_{12} and c_{22} are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by "gluing together" some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.

16. Discontinuous finite element space-angle treatment of the first order linear Boltzmann transport equation with magnetic fields: Application to MRI-guided radiotherapy

SciTech Connect

St Aubin, J.; Keyvanloo, A.; Fallone, B. G.

2016-01-15

Purpose: The advent of magnetic resonance imaging (MRI) guided radiotherapy systems demands the incorporation of the magnetic field into dose calculation algorithms of treatment planning systems. This is due to the fact that the Lorentz force of the magnetic field perturbs the path of the relativistic electrons, hence altering the dose deposited by them. Building on the previous work, the authors have developed a discontinuous finite element space-angle treatment of the linear Boltzmann transport equation to accurately account for the effects of magnetic fields on radiotherapy doses. Methods: The authors present a detailed description of their new formalism and compare its accuracy to GEANT4 Monte Carlo calculations for magnetic fields parallel and perpendicular to the radiation beam at field strengths of 0.5 and 3 T for an inhomogeneous 3D slab geometry phantom comprising water, bone, and air or lung. The accuracy of the authors’ new formalism was determined using a gamma analysis with a 2%/2 mm criterion. Results: Greater than 98.9% of all points analyzed passed the 2%/2 mm gamma criterion for the field strengths and orientations tested. The authors have benchmarked their new formalism against Monte Carlo in a challenging radiation transport problem with a high density material (bone) directly adjacent to a very low density material (dry air at STP) where the effects of the magnetic field dominate collisions. Conclusions: A discontinuous finite element space-angle approach has been proven to be an accurate method for solving the linear Boltzmann transport equation with magnetic fields for cases relevant to MRI guided radiotherapy. The authors have validated the accuracy of this novel technique against GEANT4, even in cases of strong magnetic field strengths and low density air.

17. Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers.

PubMed

Wang, Lei; Zhu, Yu-Jie; Qi, Feng-Hua; Li, Min; Guo, Rui

2015-06-01

In this paper, the nonautonomous Lenells-Fokas (LF) model is investigated. The modulational instability analysis of the solutions with variable coefficients in the presence of a small perturbation is studied. Higher-order soliton, breather, earthwormon, and rogue wave solutions of the nonautonomous LF model are derived via the n-fold variable-coefficient Darboux transformation. The solitons and earthwormons display the elastic collisions. It is found that the nonautonomous LF model admits the higher-order periodic rogue waves, composite rogue waves (rogue wave pair), and oscillating rogue waves, whose dynamics can be controlled by the inhomogeneous nonlinear parameters. Based on the second-order rogue wave, a diamond structure consisting of four first-order rogue waves is observed. In addition, the semirational solutions (the mixed rational-exponential solutions) of the nonautonomous LF model are obtained, which can be used to describe the interactions between the rogue waves and breathers. Our results could be helpful for the design of experiments in the optical fiber communications.

18. Low-energy physics of the t -J model in d =∞ using extremely correlated Fermi liquid theory: Cutoff second-order equations

Shastry, B. Sriram; Perepelitsky, Edward

2016-07-01

We present the results for the low-energy properties of the infinite-dimensional t -J model with J =0 , using O (λ2) equations of the extremely correlated Fermi liquid formalism. The parameter λ ∈[0 ,1 ] is analogous to the inverse spin parameter 1 /(2 S ) in quantum magnets. The present analytical scheme allows us to approach the physically most interesting regime near the Mott insulating state n ≲1 . It overcomes the limitation to low densities n ≲0.7 of earlier calculations, by employing a variant of the skeleton graph expansion, and a high-frequency cutoff that is essential for maintaining the known high-T entropy. The resulting quasiparticle weight Z , the low ω ,T self-energy, and the resistivity are reported. These are quite close at all densities to the exact numerical results of the U =∞ Hubbard model, obtained using the dynamical mean field theory. The present calculation offers the advantage of generalizing to finite T rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising T . The present scheme is generalizable to finite dimensions and a nonvanishing J .

19. Uniqueness of Maxwell's Equations.

ERIC Educational Resources Information Center

Cohn, Jack

1978-01-01

Shows that, as a consequence of two feasible assumptions and when due attention is given to the definition of charge and the fields E and B, the lowest-order equations that these two fields must satisfy are Maxwell's equations. (Author/GA)

20. Reduced Braginskii equations

SciTech Connect

Yagi, M.; Horton, W. )

1994-07-01

A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.

1. Assimilation of Along-track Altimetry Data Into An Eddy-permitting Primitive-equation Model of The North and Tropical Atlantic Ocean Using Isopycnal-eof Order-reduction

Faucher, P.; de Mey, P.; Gavart, M.

We present and discuss altimetric assimilation experiments into a primitive-equation model of the North Atlantic using isopycnal EOFs to propagate the altimeter sig- nal downwards and to the other model variables. Faucher, Gavart and De Mey (JGR, 2002) showed from a set of historical hydrographic data that the dominant isopycnal EOF accounts for most of the surface dynamic height variability in the North Atlantic ocean. In addition the reduced-order observability problem for altimetry is more nat- urally studied in isopycnal coordinates because the displacement of isopycnals is the largest contribution of deep ocean dynamics to the sea-level changes. The 1/3 degree ocean model from the CLIPPER and MERCATOR projects (based on OPA 8.1 code developped at LODYC, Paris) was used to solve the primitive equations from 20S to 70N. The assimilation experiments were performed with the combined along-track TOPEX-POSEIDON and ERS-1 data sets between 1 january 1993 and 31 decem- ber 1993. We implemented a multivariate reduced-order optimal interpolation method (SOFA: De Mey and Benkiran, 2002) with a vertical projection of altimetry data using data-based isopycnal EOFs. This paper will show and discuss compared results from several approaches in different regions of the North Atlantic.

2. Noncommutativity and the Friedmann Equations

SciTech Connect

Sabido, M.; Socorro, J.; Guzman, W.

2010-07-12

In this paper we study noncommutative scalar field cosmology, we find the noncommutative Friedmann equations as well as the noncommutative Klein-Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutitive parameter.

3. Fractional chemotaxis diffusion equations.

PubMed

Langlands, T A M; Henry, B I

2010-05-01

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.

4. Elliptic Equations of Higher Stochastic Order

DTIC Science & Technology

2009-01-01

stochastic spaces, such as Hida or Kondratiev spaces [11, 12], or even larger exponential spaces [16]. The traditional approach [17, 20, 21, etc.] has to...2, 384–408. [7] T. Hida , H-H. Kuo, J. Potthoff, and L. Sreit, White noise, Kluwer Academic Publishers, Boston, 1993. [8] H. Holden, B. Øksendal, J

5. Integrable fourth-order difference equations

2010-06-01

In this paper an attempt is made to find four-dimensional analogs of two-dimensional Quispel, Roberts and Thompson mappings and identified four distinct cases have been identified. The obtained mappings are measure preserving. The integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible.

6. An investigation on the effect of second-order additional thickness distributions to the upper surface of an NACA 64 sub 1-212 airfoil. [using flow equations and a CDC 7600 digital computer

NASA Technical Reports Server (NTRS)

Hague, D. S.; Merz, A. W.

1975-01-01

An investigation was conducted on a CDC 7600 digital computer to determine the effects of additional thickness distributions to the upper surface of an NACA 64 sub 1 - 212 airfoil. Additional thickness distributions employed were in the form of two second-order polynomial arcs which have a specified thickness at a given chordwise location. The forward arc disappears at the airfoil leading edge, the aft arc disappears at the airfoil trailing edge. At the juncture of the two arcs, x = x, continuity of slope is maintained. The effect of varying the maximum additional thickness and its chordwise location on airfoil lift coefficient, pitching moment, and pressure distribution was investigated. Results were obtained at a Mach number of 0.2 with an angle-of-attack of 6 degrees on the basic NACA 64 sub 1 - 212 airfoil, and all calculations employ the full potential flow equations for two dimensional flow. The relaxation method of Jameson was employed for solution of the potential flow equations.

7. An investigation on the effect of second-order additional thickness distributions to the upper surface of an NACA 64-206 airfoil. [using flow equations and a CDC 7600 digital computer

NASA Technical Reports Server (NTRS)

Merz, A. W.; Hague, D. S.

1975-01-01

An investigation was conducted on a CDC 7600 digital computer to determine the effects of additional thickness distributions to the upper surface of an NACA 64-206 airfoil. Additional thickness distributions employed were in the form of two second-order polynomial arcs which have a specified thickness at a given chordwise location. The forward arc disappears at the airfoil leading edge, the aft arc disappears at the airfoil trailing edge. At the juncture of the two arcs, x = x, continuity of slope is maintained. The effect of varying the maximum additional thickness and its chordwise location on airfoil lift coefficient, pitching moment, and pressure distribution was investigated. Results were obtained at a Mach number of 0.2 with an angle-of-attack of 6 degrees on the basic NACA 64-206 airfoil, and all calculations employ the full potential flow equations for two dimensional flow. The relaxation method of Jameson was employed for solution of the potential flow equations.

8. Nonlinear equations of 'variable type'

Larkin, N. A.; Novikov, V. A.; Ianenko, N. N.

In this monograph, new scientific results related to the theory of equations of 'variable type' are presented. Equations of 'variable type' are equations for which the original type is not preserved within the entire domain of coefficient definition. This part of the theory of differential equations with partial derivatives has been developed intensively in connection with the requirements of mechanics. The relations between equations of the considered type and the problems of mathematical physics are explored, taking into account quasi-linear equations, and models of mathematical physics which lead to equations of 'variable type'. Such models are related to transonic flows, problems involving a separation of the boundary layer, gasdynamics and the van der Waals equation, shock wave phenomena, and a combustion model with a turbulent diffusion flame. Attention is also given to nonlinear parabolic equations, and nonlinear partial differential equations of the third order.

9. Bilinear forms and soliton solutions for a fourth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein

Yang, Jin-Wei; Gao, Yi-Tian; Wang, Qi-Min; Su, Chuan-Qi; Feng, Yu-Jie; Yu, Xin

2016-01-01

In this paper, a fourth-order variable-coefficient nonlinear Schrödinger equation is studied, which might describe a one-dimensional continuum anisotropic Heisenberg ferromagnetic spin chain with the octuple-dipole interaction or an alpha helical protein with higher-order excitations and interactions under continuum approximation. With the aid of auxiliary function, we derive the bilinear forms and corresponding constraints on the variable coefficients. Via the symbolic computation, we obtain the Lax pair, infinitely many conservation laws, one-, two- and three-soliton solutions. We discuss the influence of the variable coefficients on the solitons. With different choices of the variable coefficients, we obtain the parabolic, cubic, and periodic solitons, respectively. We analyse the head-on and overtaking interactions between/among the two and three solitons. Interactions between a bound state and a single soliton are displayed with different choices of variable coefficients. We also derive the quasi-periodic formulae for the three cases of the bound states.

10. Nonlinear differential equations

SciTech Connect

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

11. Equation poems

Prentis, Jeffrey J.

1996-05-01

One of the most challenging goals of a physics teacher is to help students see that the equations of physics are connected to each other, and that they logically unfold from a small number of basic ideas. Derivations contain the vital information on this connective structure. In a traditional physics course, there are many problem-solving exercises, but few, if any, derivation exercises. Creating an equation poem is an exercise to help students see the unity of the equations of physics, rather than their diversity. An equation poem is a highly refined and eloquent set of symbolic statements that captures the essence of the derivation of an equation. Such a poetic derivation is uncluttered by the extraneous details that tend to distract a student from understanding the essential physics of the long, formal derivation.

12. Penetration equations

SciTech Connect

Young, C.W.

1997-10-01

In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.

13. Conservational PDF Equations of Turbulence

NASA Technical Reports Server (NTRS)

Shih, Tsan-Hsing; Liu, Nan-Suey

2010-01-01

Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application

14. Accuracy of perturbative master equations.

PubMed

Fleming, C H; Cummings, N I

2011-03-01

We consider open quantum systems with dynamics described by master equations that have perturbative expansions in the system-environment interaction. We show that, contrary to intuition, full-time solutions of order-2n accuracy require an order-(2n+2) master equation. We give two examples of such inaccuracies in the solutions to an order-2n master equation: order-2n inaccuracies in the steady state of the system and order-2n positivity violations. We show how these arise in a specific example for which exact solutions are available. This result has a wide-ranging impact on the validity of coupling (or friction) sensitive results derived from second-order convolutionless, Nakajima-Zwanzig, Redfield, and Born-Markov master equations.

15. Riemann equations'' in bidifferential calculus

Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.

2015-10-01

We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

16. Beautiful equations

Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul

2014-07-01

In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).

17. Kepler Equation solver

NASA Technical Reports Server (NTRS)

Markley, F. Landis

1995-01-01

Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.

18. Obtaining Maxwell's equations heuristically

Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger

2013-02-01

Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.

19. Marcus equation

DOE R&D Accomplishments Database

1998-09-21

In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.

20. On systems of Boolean equations