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Sample records for order bellman-isaacs equations

  1. Supersymmetric fifth order evolution equations

    SciTech Connect

    Tian, K.; Liu, Q. P.

    2010-03-08

    This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.

  2. On third order integrable vector Hamiltonian equations

    NASA Astrophysics Data System (ADS)

    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.

  3. Linearizability for third order evolution equations

    NASA Astrophysics Data System (ADS)

    Basarab-Horwath, P.; Güngör, F.

    2017-08-01

    The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations is also reviewed in this framework.

  4. Next-order structure-function equations

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  6. Tachyons and Higher Order Wave Equations

    NASA Astrophysics Data System (ADS)

    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.

  7. Higher Order Equations and Constituent Fields

    NASA Astrophysics Data System (ADS)

    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.

  8. Nonlocal diffusion second order partial differential equations

    NASA Astrophysics Data System (ADS)

    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.

  9. Local dynamics for high-order semilinear hyperbolic equations

    NASA Astrophysics Data System (ADS)

    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 \

  10. First-order partial differential equations in classical dynamics

    NASA Astrophysics Data System (ADS)

    Smith, B. R.

    2009-12-01

    Carathèodory's classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton-Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.

  11. 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.

  12. 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

  13. 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…

  14. Numerical solution of distributed order fractional differential equations

    NASA Astrophysics Data System (ADS)

    Katsikadelis, John T.

    2014-02-01

    In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

  15. 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.

  16. An explicit high order method for fractional advection diffusion equations

    NASA Astrophysics Data System (ADS)

    Sousa, Ercília

    2014-12-01

    We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1<α≤2. This operator is defined by a combination of the left and right Riemann-Liouville fractional derivatives. We study the convergence of the numerical method through consistency and stability. The order of convergence varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. The analysis is confirmed by numerical examples.

  17. Numerical approach to differential equations of fractional order

    NASA Astrophysics Data System (ADS)

    Momani, Shaher; Odibat, Zaid

    2007-10-01

    In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.

  18. Third order equations of motion and the Ostrogradsky instability

    NASA Astrophysics Data System (ADS)

    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.

  19. Vector order parameter in general relativity: Covariant equations

    NASA Astrophysics Data System (ADS)

    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.

  20. 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.

  1. 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.

  2. Second order evolution equations which describe pseudospherical surfaces

    NASA Astrophysics Data System (ADS)

    Catalano Ferraioli, D.; de Oliveira Silva, L. A.

    2016-06-01

    Second order evolution differential equations that describe pseudospherical surfaces are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K = - 1, and can be seen as the compatibility condition of an associated sl (2 , R) -valued linear problem, also referred to as a zero curvature representation. Under the assumption that the linear problem is defined by 1-forms ωi =fi1 dx +fi2 dt, i = 1 , 2 , 3, with fij depending on (x , t , z ,z1 ,z2) and such that f21 = η, η ∈ R, we give a complete and explicit classification of equations of the form zt = A (x , t , z) z2 + B (x , t , z ,z1) . According to the classification, these equations are subdivided in three main classes (referred to as Types I-III) together with the corresponding linear problems. Explicit examples of differential equations of each type are determined by choosing certain arbitrary differentiable functions. Svinolupov-Sokolov equations admitting higher weakly nonlinear symmetries, Boltzmann equation and reaction-diffusion equations like Murray equation are some known examples of such equations. Other explicit examples are presented, as well.

  3. On homogeneous second order linear general quantum difference equations.

    PubMed

    Faried, Nashat; Shehata, Enas M; El Zafarani, Rasha M

    2017-01-01

    In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations [Formula: see text] [Formula: see text], in a neighborhood of the unique fixed point [Formula: see text] of the strictly increasing continuous function β, defined on an interval [Formula: see text]. These equations are based on the general quantum difference operator [Formula: see text], which is defined by [Formula: see text], [Formula: see text]. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.

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

    NASA Astrophysics Data System (ADS)

    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.

  5. Perfectly matched layers for Maxwell's equations in second order formulation

    SciTech Connect

    Sjogreen, B; Petersson, A

    2004-07-26

    We consider the two-dimensional Maxwell's equations in domains external to perfectly conducting objects of complex shape. The equations are discretized using a node-centered finite-difference scheme on a Cartesian grid and the boundary condition are discretized to second order accuracy employing an embedded technique which does not suffer from a ''small-cell'' time-step restriction in the explicit time-integration method. The computational domain is truncated by a perfectly matched layer (PML). We derive estimates for both the error due to reflections at the outer boundary of the PML, and due to discretizing the continuous PML equations. Using these estimates, we show how the parameters of the PML can be chosen to make the discrete solution of the PML equations converge to the solution of Maxwell's equations on the unbounded domain, as the grid size goes to zero. Several numerical examples are given.

  6. Second-order envelope equation of graphene electrons

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  8. On the solutions of fractional order of evolution equations

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  10. A New Low Dissipative High Order Schemes for MHD Equations

    NASA Technical Reports Server (NTRS)

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

    2002-01-01

    The goal of this talk is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD 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 curvilinear grids.

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

    NASA Astrophysics Data System (ADS)

    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.

  12. Absorbing boundary conditions for second-order hyperbolic equations

    NASA Technical Reports Server (NTRS)

    Jiang, Hong; Wong, Yau Shu

    1990-01-01

    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.

  13. 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.

  14. 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.

  15. High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

    DOE PAGES

    Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.

    2017-09-01

    High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemannmore » problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.« less

  16. 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.

  17. 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.

  18. 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…

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

    NASA Astrophysics Data System (ADS)

    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.

  20. 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.

  1. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  4. On fractional Langevin equation involving two fractional orders

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  6. A fourth order accurate adaptive mesh refinement method forpoisson's equation

    SciTech Connect

    Barad, Michael; Colella, Phillip

    2004-08-20

    We present a block-structured adaptive mesh refinement (AMR) method for computing solutions to Poisson's equation in two and three dimensions. It is based on a conservative, finite-volume formulation of the classical Mehrstellen methods. This is combined with finite volume AMR discretizations to obtain a method that is fourth-order accurate in solution error, and with easily verifiable solvability conditions for Neumann and periodic boundary conditions.

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

    NASA Astrophysics Data System (ADS)

    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.

  8. 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.

  9. 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

  10. 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.

  11. 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.

  12. 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

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

    NASA Astrophysics Data System (ADS)

    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.

  14. 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

  15. 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.

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

    NASA Astrophysics Data System (ADS)

    Adam, C.; Santamaria, F.

    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.

  17. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  20. 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.

  1. Chaos in the fractional order nonlinear Bloch equation with delay

    NASA Astrophysics Data System (ADS)

    Baleanu, Dumitru; Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha

    2015-08-01

    The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ = 0 , we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α = 0.8548 with subsequent period doubling that leads to chaos at α = 0.9436 . A periodic window is observed for the range 0.962 < α < 0.9858 , with chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value α = 0.8532 , and the transition from two to four cycles at α = 0.9259 . With further increases in the fractional order, period doubling continues until at α = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α = 0.8441 , and α = 0.8635 , respectively. However, the system exhibits chaos at much lower values of α (α = 0.8635). A periodic window is observed in the interval 0.897 < α < 0.9341 , with chaos again appearing for larger values of α . In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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).

  4. High-order regularization in lattice-Boltzmann equations

    NASA Astrophysics Data System (ADS)

    Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.

    2017-04-01

    A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.

  5. On the stability and convergence of the time-fractional variable order telegraph equation

    NASA Astrophysics Data System (ADS)

    Atangana, Abdon

    2015-07-01

    In this work, we have generalized the time-fractional telegraph equation using the concept of derivative of fractional variable order. The generalized equation is called time-fractional variable order telegraph equation. This new equation was solved numerically via the Crank-Nicholson scheme. Stability and convergence of the numerical solution were presented in details. Numerical simulations of the approximate solution of the time-fractional variable order telegraph equation were presented for different values of the grid point.

  6. 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.

  7. 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.

  8. 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

  9. 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.

  10. Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

    NASA Astrophysics Data System (ADS)

    Pradip, Roul

    2013-09-01

    The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy—perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.

  11. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  13. 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.

  14. 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…

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

    PubMed

    Ali, Ayyaz; Iqbal, Muhammad Asad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad; Mohyud-Din, Syed Tauseef

    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.

  16. Properties of solutions of third-order trinomial difference equations with deviating arguments

    NASA Astrophysics Data System (ADS)

    Gleska, Alina; Migda, Małgorzata

    2017-07-01

    The third-order nonlinear trinomial difference equation of the form Δ3xn+pnΔ xn +1+qnf (xσ (n ))=0 is studied. Rewriting this equation as a binomial third-order difference equation we establish a classification of all nonoscillatory solutions, criteria for oscillation of bounded solutions and sufficient conditions for the existence of certain types of nonoscillatory solutions of the above equation.

  17. 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.

  18. Polynomial Solutions of Nth Order Non-Homogeneous Differential Equations

    ERIC Educational Resources Information Center

    Levine, Lawrence E.; Maleh, Ray

    2002-01-01

    It was shown by Costa and Levine that the homogeneous differential equation (1-x[superscript N])y([superscript N]) + A[subscript N-1]x[superscript N-1)y([superscript N-1]) + A[subscript N-2]x[superscript N-2])y([superscript N-2]) + ... + A[subscript 1]xy[prime] + A[subscript 0]y = 0 has a finite polynomial solution if and only if [for…

  19. 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.

  20. Equations of condition for high order Runge-Kutta-Nystrom formulae

    NASA Technical Reports Server (NTRS)

    Bettis, D. G.

    1974-01-01

    Derivation of the equations of condition of order eight for a general system of second-order differential equations approximated by the basic Runge-Kutta-Nystrom algorithm. For this general case, the number of equations of condition is considerably larger than for the special case where the first derivative is not present. Specifically, it is shown that, for orders two through eight, the number of equations for each order is 1, 1, 1, 2, 3, 5, and 9 for the special case and is 1, 1, 2, 5, 13, 34, and 95 for the general case.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    Maldonado, Mercedes; Molinero, María Celeste; Pickering, Andrew; Prada, Julia

    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.

  3. 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.

  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 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.

  5. Comparison of additional second-order terms in finite-difference Euler equations and regularized fluid dynamics equations

    NASA Astrophysics Data System (ADS)

    Ovsyannikov, V. M.

    2017-05-01

    In recent years, an area of research in computational mathematics has emerged that is associated with the numerical solution of fluid flow problems based on regularized fluid dynamics equations involving additional terms with velocity, pressure, and body force. The inclusion of these functions in the additional terms has been physically substantiated only for pressure and body force. In this paper, the continuity equation obtained geometrically by Euler is shown to involve second-order terms in time that contain Jacobians of the velocity field and are consistent with some of the additional terms in the regularized fluid dynamics equations. The same Jacobians are contained in the inhomogeneous right-hand side of the wave equation and generate waves of pressure, density, and sound. Physical interpretations of the additional terms used in the regularized fluid dynamics equations are given.

  6. A not so short note on the Klein Gordon equation at second order

    NASA Astrophysics Data System (ADS)

    Malik, Karim A.

    2007-03-01

    We give the governing equations for multiple scalar fields in a flat Friedmann Robertson Walker (FRW) background spacetime on all scales, allowing for metric and field perturbations up to second order. We then derive the Klein Gordon equation at second order in closed form in terms of gauge-invariant perturbations of the fields in the uniform curvature gauge. We also give a simplified form of the Klein Gordon equation using the slow-roll approximation.

  7. 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…

  8. 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…

  9. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  12. 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.

  13. Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

    NASA Astrophysics Data System (ADS)

    Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

    2017-04-01

    This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

  14. 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.

  15. 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).

  16. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  18. 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.

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

    NASA Astrophysics Data System (ADS)

    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).

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  2. 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)

  3. Painlevé analysis and exact solutions of the fourth-order equation for description of nonlinear waves

    NASA Astrophysics Data System (ADS)

    Kudryashov, Nikolay A.

    2015-11-01

    The fourth-order equation for description of nonlinear waves is considered. A few variants of this equation are studied. Painlevé test is applied to investigate integrability of these equations. We show that all these equations are not integrable, but some exact solutions of these equations exist. Analytic solutions in closed-form of the equations are found.

  4. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order

    NASA Astrophysics Data System (ADS)

    Johnston, S. J.; Jafari, H.; Moshokoa, S. P.; Ariyan, V. M.; Baleanu, D.

    2016-01-01

    The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

  5. 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.

  6. Solitary wave solution to a singularly perturbed generalized Gardner equation with nonlinear terms of any order

    NASA Astrophysics Data System (ADS)

    Zhou, J. B.; Xu, J.; Wei, J. D.; Yang, X. Q.

    2017-04-01

    This paper is concerned with the existence of travelling wave solutions to a singularly perturbed generalized Gardner equation with nonlinear terms of any order. By using geometric singular perturbation theory and based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of solitary wave solutions of this equation is proved when the perturbation parameter is sufficiently small. The numerical simulations verify our theoretical analysis.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  9. On Differences between Fractional and Integer Order Differential Equations for Dynamical Games

    NASA Astrophysics Data System (ADS)

    Ahmed, Elsayed M. E.; Elgazzar, Ahmed S.; Shehata, Mohamed I.

    2008-04-01

    We argue that fractional order differential equations are more suitable to model complex adaptive systems. Hence they are applied in replicator equations for non-cooperative games. Rock-scissors-paper game is discussed. It is known that its integer order model does not have a stable equilibrium. Its fractional order model is shown to have a locally asymptotically stable internal solution. A fractional order asymmetric game is shown to have a locally asymptotically stable internal solution. This is not the case for its integer order counterpart.

  10. New explicit global asymptotic stability criteria for higher order difference equations

    NASA Astrophysics Data System (ADS)

    El-Morshedy, Hassan A.

    2007-12-01

    New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.

  11. Solving linear fractional-order differential equations via the enhanced homotopy perturbation method

    NASA Astrophysics Data System (ADS)

    Naseri, E.; Ghaderi, R.; Ranjbar N, A.; Sadati, J.; Mahmoudian, M.; Hosseinnia, S. H.; Momani, S.

    2009-10-01

    The linear fractional differential equation is solved using the enhanced homotopy perturbation method (EHPM). In this method, the convergence has been provided by selecting a stabilizing linear part. The most significant features of this method are its simplicity and its excellent accuracy and convergence for the whole range of fractional-order differential equations.

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

    NASA Astrophysics Data System (ADS)

    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.

  13. Numerical algorithm for the third-order partial differential equation with local boundary conditions

    NASA Astrophysics Data System (ADS)

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

    2017-09-01

    Three-step difference schemes generated by Taylor's decomposition on four points for the approximate solution of the local boundary-value problems for a third order partial differential equation are presented. Results of numerical experiments are provided.

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

    NASA Astrophysics Data System (ADS)

    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.

  15. 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.

  16. Existence of solutions with a single semicycle for a general second-order rational difference equation

    NASA Astrophysics Data System (ADS)

    Li, Xianyi

    2007-10-01

    By making use of inclusion theorem, we show in this paper the existence of solutions with a single semicycle for a general second-order rational difference equation. As a corollary, our results positively confirm Conjectures 4.8.3 and 5.4.6 in [M.R.S. Kulenovic, G. Ladas, Dynamics of Second-Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002].

  17. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

  20. Nonoscillation and oscillation of second order half-linear differential equations

    NASA Astrophysics Data System (ADS)

    Kong, Qingkai

    2007-08-01

    We study the oscillation problems for the second order half-linear differential equation [p(t)[Phi](x')]'+q(t)[Phi](x)=0, where [Phi](u)=ur-1u with r>0, 1/p and q are locally integrable on ; p>0, q[greater-or-equal, slanted]0 a.e. on , and . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p[identical to]1, our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712-723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180-188] on linear equations to the half-linear case for all r>0. These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39-47] on nonoscillation when 01. The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363-373].

  1. 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

  2. On the global behavior of a high-order rational difference equation

    NASA Astrophysics Data System (ADS)

    Dehghan, Mehdi; Rastegar, Narges

    2009-06-01

    In this paper, we consider the (k+1)-order rational difference equation y={p+qy+ry}/{1+y},n=0,1,2,… where k∈{1,2,3,…}, and the initial conditions y,…,y,y and the parameters p, q and r are non-negative. We investigate the global stability, the periodic character and the boundedness nature of solutions of the above mentioned difference equation. In particular, our results solve the open problem introduced by Kulenovic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002].

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  5. Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations

    SciTech Connect

    McKinney, W.R.

    1989-01-01

    Fully discrete approximations to 1-periodic solutions of the Generalized Korteweg de-Vries and the Cahn-Hilliard equations are analyzed. These approximations are generated by an Implicit Runge-Kutta method for the temporal discretization and a Galerkin Finite Element method for the spatial discretization. Furthermore, these approximations may be of arbitrarily high order. In particular, it is shown that the well-known order reduction phenomenon afflicting Implicit Runge Kutta methods does not occur. Numerical results supporting these optimal error estimates for the Korteweg-de Vries equation and indicating the existence of a slow motion manifold for the Cahn-Hilliard equation are also provided.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  8. 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.

  9. 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.

  10. 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.

  11. Existence and Mann iterative approximations of nonoscillatory solutions of nth-order neutral delay differential equations

    NASA Astrophysics Data System (ADS)

    Liu, Zeqing; Gao, Haiyan; Kang, Shin Min; Shim, Soo Hak

    2007-05-01

    In this paper we consider the following nth-order neutral delay differential equation: where n is a positive integer, , [tau]>0, [sigma]i>0 for i=1,...,k, and . By employing the contraction mapping principle, we obtain several existence results of nonoscillatory solutions for the above equation, construct a few Mann-type iterative approximation schemes for these nonoscillatory solutions and establish several error estimates between the approximate solutions and the nonoscillatory solutions. In addition, we obtain some sufficient conditions for the existence of infinitely many nonoscillatory solutions. These results presented in this paper extend, improve and unify many known results due to Cheng and Annie [J.F. Cheng, Z. Annie, Existence of nonoscillatory solution to second order linear neutral delay equation, J. Systems Sci. Math. Sci. 24 (2004) 389-397 (in Chinese)], Graef, Yang and Zhang [J.R. Graef, B. Yang, B.G. Zhang, Existence of nonoscillatory and oscillatory solutions of neutral differential equations with positive and negative coefficients, Math. Bohem. 124 (1999) 87-102], Kulenovic and Hadziomerspahic [M.R.S. Kulenovic, S. Hadziomerspahic, Existence of nonoscillatory solution of second order linear neutral delay equation, J. Math. Anal. Appl. 228 (1998) 436-448; M.R.S. Kulenovic, S. Hadziomerspahic, Existence of nonoscillatory solution for linear neutral delay equation, Fasc. Math. 32 (2001) 61-72], Zhang and Yu [B.G. Zhang, J.S. Yu, On the existence of asymptotically decaying positive solutions of second order neutral differential equations, J. Math. Anal. Appl. 166 (1992) 1-11], Zhang [B.G. Zhang, On the positive solutions of a kind of neutral equations, Acta Math. Appl. Sinica 19 (1996) 222-230] and Zhou and Zhang [Y. Zhou, B.G. Zhang, Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients, Appl. Math. Lett. 15 (2002) 867-874] and others. Some nontrivial examples are given to

  12. A second-order projection method for the incompressible Navier Stokes equations on quadrilateral grids

    SciTech Connect

    Bell, J.B.; Solomon, J.M.; Szymczak, W.G.

    1989-04-01

    This paper describes a second-order projection method for the incompressible Navier-Stokes equations on a logically-rectangular quadrilateral grid. The method uses a second-order fractional step scheme in which one first solves diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. The spatial discretization of the diffusion-convection equations is accomplished by formally transforming the equations to a uniform computational space. The diffusion terms are then discretized using standard finite-difference approximations. The convection terms are discretized using a second-order Godunov method that provides a robust discretization of these terms at high Reynolds number. The projection is approximated using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented illustrating the performance of the method. 13 refs., 5 figs.

  13. Second-order perturbation theory: a covariant approach involving a barotropic equation of state

    NASA Astrophysics Data System (ADS)

    Osano, Bob

    2017-06-01

    We present a covariant and gauge-invariant formalism suited to the study of second-order effects associated with higher order tensor perturbations. The analytical method we have developed enables us to characterize pure second-order tensor perturbations about the FLRW model having different kinds of equations of state. Our analysis of the radiation case suggests that it may be feasible to examine the CMB polarization arising from higher order perturbations.

  14. 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…

  15. Diagonally implicit block backward differentiation formula for solving linear second order ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Zainuddin, N.; Ibrahim, Z. B.; Othman, K. I.

    2014-10-01

    The three point block method for solving second order ordinary differential equations (ODEs) directly using constant step size is derived. The reliability of this new method is verified in the numerical results with the improved performance in terms of computation time while maintaining the accuracy. The comparison is presented between the new method and classical backward differentiation formulas (BDF) of order 3.

  16. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  18. A New Approach to Model Order Reduction of the Navier-Stokes Equations

    NASA Astrophysics Data System (ADS)

    Balajewicz, Maciej

    A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier-Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

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

    NASA Astrophysics Data System (ADS)

    Morgado, M. L.; Rebelo, M.

    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.

  20. 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.

  1. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  3. Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation

    PubMed Central

    Wang, Yin; Zhang, Jun

    2010-01-01

    We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19-point fourth order discretization scheme to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equation with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact scheme (SOC), compared with the previously published fourth order compact scheme (FOC). PMID:21151737

  4. 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.

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

    NASA Astrophysics Data System (ADS)

    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. 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.

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

    NASA Astrophysics Data System (ADS)

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

    NASA Astrophysics Data System (ADS)

    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.

  9. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  12. High-order accurate difference schemes for the Hodgkin-Huxley equations

    NASA Astrophysics Data System (ADS)

    Amsallem, David; Nordström, Jan

    2013-11-01

    A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

  13. 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.

  14. A High-Order Accurate Parallel Solver for Maxwell's Equations on Overlapping Grids

    SciTech Connect

    Henshaw, W D

    2005-09-23

    A scheme for the solution of the time dependent Maxwell's equations on composite overlapping grids is described. The method uses high-order accurate approximations in space and time for Maxwell's equations written as a second-order vector wave equation. High-order accurate symmetric difference approximations to the generalized Laplace operator are constructed for curvilinear component grids. The modified equation approach is used to develop high-order accurate approximations that only use three time levels and have the same time-stepping restriction as the second-order scheme. Discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed. The implementation is optimized for component grids that are Cartesian, resulting in a fast and efficient method. The solver runs on parallel machines with each component grid distributed across one or more processors. Numerical results in two- and three-dimensions are presented for the fourth-order accurate version of the method. These results demonstrate the accuracy and efficiency of the approach.

  15. Von mises- and crocco-type hydrodynamical transformations: Order reduction of nonlinear equations, construction of Bäcklund transformations and of new integrable equations

    NASA Astrophysics Data System (ADS)

    Fedotov, I. A.; Polyanin, A. D.

    2011-09-01

    Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Bäcklund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.

  16. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  18. 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.

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

    NASA Astrophysics Data System (ADS)

    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. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  2. A family of solutions of a higher order PVI equation near a regular singularity

    NASA Astrophysics Data System (ADS)

    Shimomura, Shun

    2006-09-01

    Restriction of the N-dimensional Garnier system to a complex line yields a system of second-order nonlinear differential equations, which may be regarded as a higher order version of the sixth Painlevé equation. Near a regular singularity of the system, we present a 2N-parameter family of solutions expanded into convergent series. These solutions are constructed by iteration, and their convergence is proved by using a kind of majorant series. For simplicity, we describe the proof in the case N = 2.

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

    NASA Astrophysics Data System (ADS)

    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.

  4. 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.

  5. 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.

  6. Efficient parametric analysis of the chemical master equation through model order reduction.

    PubMed

    Waldherr, Steffen; Haasdonk, Bernard

    2012-07-02

    Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.

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

    NASA Astrophysics Data System (ADS)

    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.

  8. Effective quadrature formula in solving linear integro-differential equations of order two

    NASA Astrophysics Data System (ADS)

    Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

    2017-08-01

    In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

  9. 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).

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

    NASA Astrophysics Data System (ADS)

    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.

  11. High-Order Compact Difference Scheme for the Numerical Solution of Time Fractional Heat Equations

    PubMed Central

    Karatay, Ibrahim; Bayramoglu, Serife R.

    2014-01-01

    A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space derivative. The spectral stability and the Fourier stability analysis of the difference scheme are shown. Finally a detailed numerical analysis, including tables, figures, and error comparison, is given to demonstrate the theoretical results and high accuracy of the proposed scheme. PMID:24696040

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  14. Second-order equation of motion for electromagnetic radiation back-reaction

    NASA Astrophysics Data System (ADS)

    Matolcsi, T.; Fülöp, T.; Weiner, M.

    2017-09-01

    We take the viewpoint that the physically acceptable solutions of the Lorentz-Dirac equation for radiation back-reaction are actually determined by a second-order equation of motion, the self-force being given as a function of spacetime location and velocity. We propose three different methods to obtain this self-force function. For two example systems, we determine the second-order equation of motion exactly in the non-relativistic regime via each of these three methods, leading to the same result. We reveal that, for both systems considered, back-reaction induces a damping proportional to velocity and, in addition, it decreases the effect of the external force.

  15. Diffusion through ordered force fields in nanopores represented by Smoluchowski equation

    SciTech Connect

    Wang, F.Y.; Zhu, Z.H.; Rudolph, V.

    2009-06-15

    The classical Einstein or Fick diffusion equation was developed in random force fields. When the equation is applied to gas transport through coal, significant discrepancies are observed between experimental and simulation results. The explanation may be that the random force field assumption is violated. In this article, we analyze molecular transport driven by both random and ordered (directional) forces in nanopores. When applied to CO{sub 2} transport through cone-shaped carbon nano-tubes (CNTs) and Li{sup +} doped graphite pores, computational results show that directional force fields may significantly affect porous media flow. Directional forces may be generated by potential gradients arising from a range of non-uniform characteristics, such as variations in the pore-sizes and in local surface compositions. On the basis of the simulation and experimental results, the Smoluchowski and Fokker-Planck equations, which account for the directional force fields, are recommended for diffusion through ordered force fields in nanopores.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  1. 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…

  2. 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…

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

    NASA Astrophysics Data System (ADS)

    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.

  4. 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.

  5. 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…

  6. 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

  7. 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…

  8. 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.

  9. 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.

  10. 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…

  11. Nonlocal Symmetries and Finite Transformations of the Fifth-Order KdV Equation

    NASA Astrophysics Data System (ADS)

    Hao, Xiazhi; Liu, Yinping; Tang, Xiaoyan; Li, Zhibin

    2017-05-01

    The nth finite transformations of the fifth-order KdV equation are obtained from the Lie point symmetry approach via localisation of nonlocal symmetries to local ones of the enlarged system. Through the obtained transformations, some periodic and soliton solutions are derived.

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

    NASA Astrophysics Data System (ADS)

    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. Strategic Competence as a Fourth-Order Factor Model: A Structural Equation Modeling Approach

    ERIC Educational Resources Information Center

    Phakiti, Aek

    2008-01-01

    This article reports on an empirical study that tests a fourth-order factor model of strategic competence through the use of structural equation modeling (SEM). The study examines the hierarchical relationship of strategic competence to (a) strategic knowledge of cognitive and metacognitive strategy use in general (i.e., trait) and (b) strategic…

  14. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  16. 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

  17. Lie symmetry analysis and explicit solutions of the time fractional fifth-order KdV equation.

    PubMed

    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.

  18. A reduced-order representation of the Schrödinger equation

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  20. 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.

  1. Phase dynamics of periodic wavetrains leading to the 5th order KP equation

    NASA Astrophysics Data System (ADS)

    Ratliff, Daniel J.

    2017-09-01

    Using the previous approach outlined in Ratliff and Bridges (2016, 2015), a novel method is presented to derive the fifth order Kadomtsev-Petviashvili(KP) equation from periodic wavetrains. As a result, the coefficients and criterion for the fifth order KP to emerge take a universal form that can be determined a-priori, relating to the system's conservation laws and the termination of a Jordan chain. Moreover, the analysis reveals that generically a mixed dispersive term qXXXY appears within the final phase equation. The theory presented here is complimented by an example from the context of flexural gravity waves in shallow water and a higher order Nonlinear Schrödinger model relevant in plasma physics, demonstrating how the coefficients in this model are determined via elementary calculations.

  2. 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.

  3. 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.

  4. Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

    NASA Astrophysics Data System (ADS)

    Tessarotto, Massimo; Asci, Claudio

    2017-05-01

    In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N ≡1/ε ≫ 1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε.

  5. 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.

  6. Travelling wave solutions for higher-order wave equations of kdv type (iii).

    PubMed

    Li, Jibin; Rui, Weigou; Long, Yao; He, Bin

    2006-01-01

    By using the theory of planar dynamical systems to the travelling wave equation of a higher order nonlinear wave equations of KdV type, the existence of smooth solitary wave, kink wave and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact explicit parametric representations of these waves are obtain.

  7. 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.

  8. 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.

  9. A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation

    DOE PAGES

    Devendran, Dharshi; Graves, Daniel; Johansen, Hans; ...

    2017-05-08

    In this paper, we present a fourth-order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

  10. 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.

  11. 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.

  12. Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

    NASA Astrophysics Data System (ADS)

    Magdziarz, M.; Teuerle, M.

    2017-05-01

    In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type.

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

    NASA Astrophysics Data System (ADS)

    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. Classical seventh-, sixth-, and fifth-order Runge-Kutta-Nystrom formulas with stepsize control for general second-order differential equations

    NASA Technical Reports Server (NTRS)

    Fehlberg, E.

    1974-01-01

    Runge-Kutta-Nystrom formulas of the seventh, sixth, and fifth order were derived for the general second order (vector) differential equation written as the second derivative of x = f(t, x, the first derivative of x). The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the local truncation error in x, and they require no more evaluations per step than the earlier Runge-Kutta formulas for the first derivative of x = f(t, x). The developed formulas are expected to be time saving in comparison to the Runge-Kutta formulas for first-order differential equations, since it is not necessary to convert the second-order differential equations into twice as many first-order differential equations. The examples shown saved from 25 percent to 60 percent more computer time than the earlier formulas for first-order differential equations, and are comparable in accuracy.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    Pindza, Edson; Owolabi, Kolade M.

    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.

  19. Reduced-order-model based feedback control of the Modified Hasegawa-Wakatani equations

    NASA Astrophysics Data System (ADS)

    Goumiri, Imene; Rowley, Clarence; Ma, Zhanhua; Gates, David; Parker, Jeffrey; Krommes, John

    2012-10-01

    In this study, we demonstrate the development of model-based feedback control for stabilization of an unstable equilibrium obtained in the Modified Hasegawa-Wakatani (MHW) equations, a classic model in plasma turbulence. First, a balanced truncation is applied; a model reduction technique that has been proved successful in flow control design problems, to obtain a low dimensional model of the linearized MHW equation. A model-based feedback controller is then designed for the reduced order model using linear quadratic regulators (LQR) then a linear quadratic gaussian (LQG) control. The controllers are then applied on the original linearized and nonlinear MHW equations to stabilize the equilibrium and suppress the transition to drift-wave induced turbulences.

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

    NASA Astrophysics Data System (ADS)

    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.

  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

    NASA Astrophysics Data System (ADS)

    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. Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

    NASA Astrophysics Data System (ADS)

    Du, Yanwei; Liu, Yang; Li, Hong; Fang, Zhichao; He, Siriguleng

    2017-09-01

    In this article, a fully discrete local discontinuous Galerkin (LDG) method with high-order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time-fractional fourth-order partial differential equation (PDE). In the temporal direction, for approximating the fractional derivative with order α ∈ (0 , 1), the weighted and shifted Grünwald difference (WSGD) scheme with second-order convergence rate is introduced and for approximating the integer time derivative, two step backward Euler method with second-order convergence rate is used. For the spatial direction, the LDG method is used. For the numerical theories, the stability is derived and a priori error results are proved. Further, some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  11. 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.

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

    NASA Astrophysics Data System (ADS)

    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. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  16. On the Fractional-Order Logistic Equation with Two Different Delays

    NASA Astrophysics Data System (ADS)

    El-Sayed, Ahmed M. A.; El-Saka, Hala A. A.; El-Maghrabi, Esam M.

    2011-04-01

    The fractional-order logistic equation with the two different delays r1, r2 > 0, Dα x(t) = ρx(t - r1)[1-x(t -r2)], t > 0 and ρ > 0, with the initial data x(t) = x0, t ≤ 0 are considered. The existence of a unique uniformly stable solution is studied and the Adams-type predictor-corrector method is applied to obtain the numerical solution.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

  20. 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.-

  1. 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.

  2. 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.-

  3. 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.

  4. 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

    NASA Astrophysics Data System (ADS)

    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.

  5. A high order multi-resolution solver for the Poisson equation with application to vortex methods

    NASA Astrophysics Data System (ADS)

    Hejlesen, Mads Mølholm; Spietz, Henrik Juul; Walther, Jens Honore

    2015-11-01

    A high order method is presented for solving the Poisson equation subject to mixed free-space and periodic boundary conditions by using fast Fourier transforms (FFT). The high order convergence is achieved by deriving mollified Green's functions from a high order regularization function which provides a correspondingly smooth solution to the Poisson equation. The high order regularization function may be obtained analogous to the approximate deconvolution method used in turbulence models and strongly relates to deblurring algorithms used in image processing. At first we show that the regularized solver can be combined with a short range particle-particle correction for evaluating discrete particle interactions in the context of a particle-particle particle-mesh (P3M) method. By a similar approach we extend the regularized solver to handle multi-resolution patches in continuum field simulations by super-positioning an inter-mesh correction. For sufficiently smooth vector fields this multi-resolution correction can be achieved without the loss of convergence rate. An implementation of the multi-resolution solver in a two-dimensional re-meshed particle-mesh based vortex method is presented and validated.

  6. 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.

  7. Differential quadrature solutions of eighth-order boundary-value differential equations

    NASA Astrophysics Data System (ADS)

    Liu, G. R.; Wu, T. Y.

    2002-08-01

    Special cases of linear eighth-order boundary-value problems have been solved using polynomial splines. However, divergent results were obtained at points adjacent to boundary points. This paper presents an accurate and general approach to solve this class of problems, utilizing the generalized differential quadrature rule (GDQR) proposed recently by the authors. Explicit weighting coefficients are formulated to implement the GDQR for eighth-order differential equations. A mathematically important by-product of this paper is that a new kind of Hermite interpolation functions is derived explicitly for the first time. Linear and non-linear illustrations are given to show the practical usefulness of the approach developed. Using Frechet derivatives, non-linear eighth-order problems are also solved for the first time. Numerical results obtained using even only seven sampling points are of excellent accuracy and convergence in an entire domain. The present GDQR has shown clear advantages over the existing methods and demonstrated itself as a general, stable, and accurate numerical method to solve high-order differential equations.

  8. 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.

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

    NASA Astrophysics Data System (ADS)

    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

    NASA Astrophysics Data System (ADS)

    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. Numerical approximation of Newell-Whitehead-Segel equation of fractional order

    NASA Astrophysics Data System (ADS)

    Kumar, Devendra; Sharma, Ram Prakash

    2016-06-01

    The aim of the present work is to propose a user friendly approach based on homotopy analysis method combined with Sumudu transform method to drive analytical and numerical solutions of the fractional Newell-Whitehead-Segel amplitude equation which describes the appearance of the stripe patterns in 2-dimensional systems. The coupling of homotopy analysis method with Sumudu transform algorithm makes the calculation very easy. The proposed technique gives an analytic solution in the form of series which converge very fastly. The analytical and numerical results reveal that the coupling of homotopy analysis technique with Sumudu transform algorithm is very easy to apply and highly accuratewhen apply to non-linear differential equation of fractional order.

  12. 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.

  13. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  16. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

    NASA Astrophysics Data System (ADS)

    Jensen, Robert

    1988-03-01

    We prove that viscosity solutions in W 1,∞ of the second order, fully nonlinear, equation F( D 2 u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.

  17. 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

  18. 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.

  19. 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.

  20. 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

  1. 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

  2. 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

  3. High-order accurate methods for solving the time-harmonic Maxwell's equations

    NASA Astrophysics Data System (ADS)

    Wilcox, Lucas Charles

    Maxwell's equations are the partial differential equations describing electromagnetism. They can be used to model electric and magnetic fields in different materials from light in fiber optic cables to radar waves bouncing off a stealth fighter jet. In problems with electromagnetic radiation of a single frequency Maxwell's equations may be reduced to their time-harmonic form. Further simplifying the problem a multilayer boundary variation method for the forward modeling of multilayered diffraction optics is presented. This approach enables fast and high-order accurate modeling of periodic transmission optics consisting of an arbitrary number of materials and interfaces of general shape subject to plane wave illumination or, by solving a sequence of problems, illumination by beams. The key developments of the algorithm are discussed as are details of an efficient implementation. Numerous comparisons with exact solutions and highly accurate direct solutions confirm the accuracy, versatility, and efficiency of the proposed method. The high accuracy of the method is leveraged to solve an application involving the in-coupling process for grating-coupled planar optical waveguide sensors. For more general solutions of the time-harmonic Maxwell's equations an hp-adaptive discontinuous Galerkin finite element method is studied. The discontinuous Galerkin finite element method is a general method for solving partial differential equations that has had success with time evolution problems. The application to time-harmonic problems is a new and developing area of research. As a first step, an overlapping Schwarz method for the discontinuous Galerkin discretization of the indefinite Helmholtz equation is examined. For an hp-adaptive method to be successful an error indicator is required to determine the areas of the computational domain that need increased resolution. The use of adjoint based error indicators is explored through solving the time-harmonic Maxwell's equations for

  4. Globally Hyperbolic Moment Model of Arbitrary Order for One-Dimensional Special Relativistic Boltzmann Equation

    NASA Astrophysics Data System (ADS)

    Kuang, Yangyu; Tang, Huazhong

    2017-06-01

    This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors, hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.

  5. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  10. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  12. 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.

  13. High Order Filter Methods for the Non-ideal Compressible MHD Equations

    NASA Technical Reports Server (NTRS)

    Yee, 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 divergence cleaning is not required by the present filter approach. For certain non-ideal MHD test cases, divergence free preservation of the magnetic fields has been achieved.

  14. The Initial Value Problem for Fractional Order Differential Equations with Constant Coefficients. 2nd Edition

    DTIC Science & Technology

    1989-09-30

    eigenfunctions needing m initial conditions for a unique solution. These eigenfunctions will be cast in terms of Mittag -Leffler functions (16), long...modified basis equations. These solutions take the form (-(at)i) h (t ) = Yh (0 ) (26) which is a special case of the beta order Mittag -Leffler...function defined as (16.102) (x pE (x) = lYpf- (27) p.,o In Mittag -Leffler notation the homogeneous solution is Yh(t) = Yh(O) E)[-(at)A, (28) 19 where this

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

    NASA Technical Reports Server (NTRS)

    Abarbanel, Saul; Ditkowski, Adi

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    Abdulwahhab, Muhammad Alim; Jhangeer, Adil

    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.

  19. 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.

  20. 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.

  1. Nonlinear gravitational self-force: Second-order equation of motion

    NASA Astrophysics Data System (ADS)

    Pound, Adam

    2017-05-01

    When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass, it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012), 10.1103/PhysRevLett.109.051101], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass, it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighborhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations.

  2. A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics

    SciTech Connect

    Jiang, G.S.; Wu, C.

    1999-04-10

    The authors present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio-Wu Riemann problems, the two-dimensional Kelvin-Helmholtz instability problems, and the two-dimensional Orszag-Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.

  3. 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.

  4. An equation of state for the financial markets: connecting order flow to price formation.

    NASA Astrophysics Data System (ADS)

    Gerig, Austin; Mike, Szabolcs; Doyne Farmer, J.

    2006-03-01

    Many of the peculiarities of price formation in the financial marketplace can be understood as the result of a few regularities in the placement and removal of trading orders. Based on a large data set from the London Stock Exchange we show that the distribution of prices where people place orders to buy or sell follows a surprisingly simple functional form that depends on the current best prices. In addition, whether or not an order is to buy or sell is described by a long-memory process, and the cancellation of orders can be described by a few simple rules. When these results are combined, simply by following the rules of the continuous double auction, the resulting simulation model produces good predictions for the distribution of price changes and transaction costs without any adjustment of parameters. We use the model to empirically derive equations of state relating order flow and the statistical properties of prices. In contrast to previous conjectures, our results demonstrate that these distributions are not universal, but rather depend on parameters of individual markets. They also show that factors other than supply and demand play an important role in price formation.

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

    NASA Astrophysics Data System (ADS)

    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).

  6. (N+1)-dimensional fractional reduced differential transform method for fractional order partial differential equations

    NASA Astrophysics Data System (ADS)

    Arshad, Muhammad; Lu, Dianchen; Wang, Jun

    2017-07-01

    In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order partial differential equations (PDEs) can be resolved effectively. The most distinct aspect of this method is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. We utilize the proposed scheme on some initial value problems and approximate numerical solutions of linear and nonlinear time fractional PDEs are obtained, which shows that the method is highly accurate and simple to apply. The proposed technique is thus an influential technique for solving the fractional PDEs and fractional order problems occurring in the field of engineering, physics etc. Numerical results are obtained for verification and demonstration purpose by using Mathematica software.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  10. Cosmology in generalized Horndeski theories with second-order equations of motion

    NASA Astrophysics Data System (ADS)

    Kase, Ryotaro; Tsujikawa, Shinji

    2014-08-01

    We study the cosmology of an extended version of Horndeski theories with second-order equations of motion on the flat Friedmann-Lemaître-Robertson-Walker background. In addition to a dark energy field χ associated with the gravitational sector, we take into account multiple scalar fields ϕI (I =1,2,…,N-1) characterized by the Lagrangians P(I)(XI) with XI=∂μϕI∂μϕI. These additional scalar fields can model the perfect fluids of radiation and nonrelativistic matter. We derive propagation speeds of scalar and tensor perturbations as well as conditions for the absence of ghosts. The theories beyond Horndeski induce nontrivial modifications to all the propagation speeds of N scalar fields, but the modifications to those for the matter fields ϕI are generally suppressed relative to that for the dark energy field χ. We apply our results to the covariantized Galileon with an Einstein-Hilbert term in which partial derivatives of the Minkowski Galileon are replaced by covariant derivatives. Unlike the covariant Galileon with second-order equations of motion in general space-time, the scalar propagation speed square cs12 associated with the field χ becomes negative during the matter era for late-time tracking solutions, so the two Galileon theories can be clearly distinguished at the level of linear cosmological perturbations.

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

    NASA Astrophysics Data System (ADS)

    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.

  12. 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.

  13. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  15. 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.

  16. 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.

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

    NASA Astrophysics Data System (ADS)

    Owolabi, Kolade M.

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

  20. A bimodular theory for finite deformations: Comparison of orthotropic second-order and exponential stress constitutive equations for articular cartilage.

    PubMed

    Klisch, Stephen M

    2006-06-01

    Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile-compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compression. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equation, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

  1. 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

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

    PubMed

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

    2015-08-01

    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..

  3. Second order method for solving 3D elasticity equations with complex interfaces

    NASA Astrophysics Data System (ADS)

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

    2015-08-01

    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 equations. 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 have 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 discontinuities 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.

  4. 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.

  5. 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

  6. An arbitrary order diffusion algorithm for solving Schrödinger equations

    NASA Astrophysics Data System (ADS)

    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

  7. 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

  8. 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.

  9. 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

  10. 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.

  11. 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.

  12. A simple finite-difference scheme for handling topography with the first-order wave equation

    NASA Astrophysics Data System (ADS)

    Mulder, W. A.; Huiskes, M. J.

    2017-07-01

    One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results.

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

    NASA Astrophysics Data System (ADS)

    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.

  14. A hierarchical uniformly high order DG-IMEX scheme for the 1D BGK equation

    NASA Astrophysics Data System (ADS)

    Xiong, Tao; Qiu, Jing-Mei

    2017-05-01

    A class of high order nodal discontinuous Galerkin implicit-explicit (DG-IMEX) schemes with asymptotic preserving (AP) property has been developed for the one-dimensional (1D) BGK equation in Xiong et al. (2015) [40], based on a micro-macro reformulation. The schemes are globally stiffly accurate and asymptotically consistent, and as the Knudsen number becomes small or goes to zero, they recover first the compressible Navier-Stokes (CNS) and then the Euler limit. Motivated by the recent work of Filbet and Rey (2015) [27] and the references therein, in this paper, we propose a hierarchical high order AP method, namely kinetic, CNS and Euler solvers are automatically applied in regions where their corresponding models are appropriate. The numerical solvers for different regimes are coupled naturally by interface conditions. To the best of our knowledge, the resulting scheme is the very first hierarchical one being proposed in the literature, that enjoys AP property as well as uniform high order accuracy. Numerical experiments demonstrate the efficiency and effectiveness of the proposed approach. As time evolves, three different regimes are dynamically identified and naturally coupled, leading to significant CPU time savings (more than 80% for some of our test problems).

  15. 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.

  16. Rethinking Pedagogy for Second-Order Differential Equations: A Simplified Approach to Understanding Well-Posed Problems

    ERIC Educational Resources Information Center

    Tisdell, Christopher C.

    2017-01-01

    Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching "well posedness" of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a…

  17. 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.

  18. 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.

  19. 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.

  20. 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.

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

    NASA Astrophysics Data System (ADS)

    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

  2. Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity

    SciTech Connect

    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.

  3. Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

    PubMed

    Shah, A A; Xing, W W; Triantafyllidis, V

    2017-04-01

    In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

  4. 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

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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

  11. Second-order non-iterative ADI solution of non-linear partial differential equations. [Alternating Direction Implicit scheme

    NASA Technical Reports Server (NTRS)

    Wolfshtein, M.; Hirsh, R. S.; Pitts, B. H.

    1975-01-01

    A new method for the solution of non-linear partial differential equations by an ADI procedure is described. Although the method is second order accurate in time, it does not require either iterations or predictor corrector methods to overcome the nonlinearity of the equations. Thus the computational effort required for the solution of the non-linear problem becomes similar to that required for the linear case. The method is applied to a two-dimensional 'extended Burgers equation'. Linear stability is studied, and some numerical solutions obtained. The improved accuracy obtained by the 2nd order truncation error is clearly manifested.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions

    NASA Astrophysics Data System (ADS)

    Gilson, C.; Hietarinta, J.; Nimmo, J.; Ohta, Y.

    2003-07-01

    Higher-order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately, the construction of multisoliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petviashvili hierarchy. In the process, we also get bilinearizations and multisoliton formulas for a two-component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional generalization.

  17. Dissipative optical bullets modeled by the cubic-quintic-septic complex Ginzburg-Landau equation with higher-order dispersions

    NASA Astrophysics Data System (ADS)

    Djoko, Martin; Kofane, T. C.

    2017-07-01

    We investigate the propagation of dissipative optical bullets under the combined influence of dispersion, diffraction, gain, loss, spectral filtering, Raman effect and cubic-quintic-septic nonlinearities. Using the Maxwell equations, we derive a basic equation modeling the propagation of ultrashort optical solitons in optical fiber, named the higher-order (3+1)D cubic-quintic-septic complex Ginzburg-Landau [(3+1)D CQS-CGL] equation. Considering this higher-order (3+1)D CQS-CGL equation, we use a variational approach to obtain a set of differential equations characterizing the variation of the pulse parameters in fiber optic-links. The variational equations that we obtained are investigated numerically in order to observe the behavior of pulse parameters along the optical fiber. A fully direct numerical simulation of the higher-order (3+1)D CQS-CGL equation finally tests the results of the variational approach. A good agreement between analytical and numerical methods is observed. Among different behaviors, bell-shaped dissipative light bullets, double, triple and quadruple bullet complexes are obtained under certain parameter values for anomalous, zero and normal chromatic dispersion regimes.

  18. 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.

  19. Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations

    NASA Astrophysics Data System (ADS)

    Geiser, Jürgen

    2008-07-01

    In this paper we design higher-order time integrators for systems of stiff ordinary differential equations. We combine implicit Runge-Kutta and BDF methods with iterative operator-splitting methods to obtain higher-order methods. The idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently. We compare our new methods with the higher-order fractional-stepping Runge-Kutta methods, developed for stiff ordinary differential equations. The benefit is the individual handling of each operator with adapted standard higher-order time integrators. The methods are applied to equations for convection-diffusion reactions and we obtain higher-order results. Finally we discuss the applications of the iterative operator-splitting methods to multi-dimensional and multi-physical problems.

  20. 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

  1. 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.

  2. 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.

  3. Fourth-order wave equation in Bhabha-Madhavarao spin-3 2 theory

    NASA Astrophysics Data System (ADS)

    Markov, Yu. A.; Markova, M. A.; Bondarenko, A. I.

    2017-09-01

    Within the framework of the Bhabha-Madhavarao formalism, a consistent approach to the derivation of a system of the fourth-order wave equations for the description of a spin-3 2 particle is suggested. For this purpose an additional algebraic object, the so-called q-commutator (q is a primitive fourth root of unity) and a new set of matrices ημ, instead of the original matrices βμ of the Bhabha-Madhavarao algebra, are introduced. It is shown that in terms of the ημ matrices we have succeeded in reducing a procedure of the construction of fourth root of the fourth-order wave operator to a few simple algebraic transformations and to some operation of the passage to the limit z → q, where z is some (complex) deformation parameter entering into the definition of the η-matrices. In addition, a set of the matrices 𝒫1/2 and 𝒫3/2(±)(q) possessing the properties of projectors is introduced. These operators project the matrices ημ onto the spins 1/2- and 3/2-sectors in the theory under consideration. A corresponding generalization of the obtained results to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out. The application to the problem of construction of the path integral representation in para-superspace for the propagator of a massive spin-3 2 particle in a background gauge field within the Bhabha-Madhavarao approach is discussed.

  4. 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

  5. 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.

  6. 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.

  7. 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.

  8. 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…

  9. 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…

  10. Spontaneous soliton generation in the higher order Korteweg-de Vries equations on the half-line.

    PubMed

    Burde, G I

    2012-03-01

    Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.

  11. Higher-Order Equation-of-Motion Coupled-Cluster Methods for Ionization Processes

    SciTech Connect

    Kamiya, Muneaki; Hirata, So

    2006-08-21

    Compact algebraic equations defining the equation-of-motion coupled-cluster (EOM-CC) methods for ionization potentials (IP-EOM-CC) have been derived and computer implemented by virtue of a symbolic algebra system largely automating these processes. Models with connected cluster excitation operators truncated after double, triple, or quadruple level and with linear ionization operators truncated after two-hole-one-particle (2h1p), three-hole-two-particle (3h2p), or four-hole-three-particle (4h3p) level (abbreviated as IP-EOM-CCSD, CCSDT, and CCSDTQ, respectively) have been realized into parallel algorithms taking advantage of spin, spatial, and permutation symmetries with optimal size dependence of the computational costs. They are based on spin-orbital formalisms and can describe both {alpha} and {beta} and ionizations from open-shell (doublet, triplet, etc.) reference states into ionized states with various spin magnetic quantum numbers. The application of these methods to Koopmans and satellite ionizations of N{sub 2} and CO (with the ambiguity due to finite basis sets eliminated by extrapolation) has shown that IP-EOM-CCSD frequently accounts for orbital relaxation inadequately and displays errors exceeding a couple of eV. However, these errors can be systematically reduced to tenths or even hundredths of an eV by IP-EOM-CCSDT or CCSDTQ. Comparison of spectroscopic parameters of the FH{sup +} and NH{sup +} radicals between IP-EOM-CC and experiments has also underscored the importance of higher-order IP-EOM-CC treatments. For instance, the harmonic frequencies of the {tilde A} {sup 2}{Sigma}{sup -} state of NH{sup +}+ are predicted to be 1285, 1723, and 1705 cm{sup -1} by IP-EOM-CCSD, CCSDT, and CCSDTQ, respectively, as compared to the observed value of 1707 cm{sup -1}. The small adiabatic energy separation (observed 0.04 eV) between the {tilde X} {sup 2}II and {tilde a} {sup 4}{sigma}{sup -} states of NH{sup +} also requires IP-EOM-CCSDTQ for a quantitative

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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

  18. 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

  19. Validation and clinical application of a first order step response equation for nitrogen clearance during FRC measurement.

    PubMed

    Choncholas, Gary; Sondergaard, Soren; Heinonen, Erkki

    2008-02-01

    To derive a difference equation based on mass conservation and on alveolar tidal volumes for the calculation of Functional Residual Capacity. Derive an equation for the FRC from the difference equation. Furthermore, to derive and validate a step response equation as a solution of the difference equation within the framework of digital signal processing where the FRC is known a priori. A difference equation for the calculation of Functional Residual Capacity is derived and solved as step response of a first order system. The step response equation calculates endtidal fractions of nitrogen during multiple breath nitrogen clearance. The step response equation contains the eigenvalue defined as the ratio of FRC to the sum of FRC and alveolar tidal ventilation. Agreement of calculated nitrogen fractions with measured fractions is demonstrated with data from a metabolic lung model, measurements from patients in positive pressure ventilation and volunteers breathing spontaneously. Examples of eigenvalue are given and compared between diseased and healthy lungs and between ventilatory settings. Comparison of calculated and measured fractions of endtidal nitrogen demonstrates a high degree of agreement in terms of regression and bias and limits of agreement (precision) in Bland & Altman analysis. Examples illustrate the use of the eigenvalue as a possible discriminator between disease states. The first order step response equation reliably calculates endtidal fractions of nitrogen during washout based on a Functional Residual Capacity. The eigenvalue may be a clinically valuable index alone or in conjunction with other indices in the analysis of respiratory states and may aid in the setting of the ventilator.

  20. 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.

  1. Compacton solutions in a class of generalized fifth-order Korteweg--de Vries equations

    SciTech Connect

    Cooper, Fred; Hyman, James M.; Khare, Avinash

    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 Schroedinger, 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.

  2. 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.

  3. 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.

  4. 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.

  5. Structural interactions in ionic liquids linked to higher-order Poisson-Boltzmann equations.

    PubMed

    Blossey, R; Maggs, A C; Podgornik, R

    2017-06-01

    We present a derivation of generalized Poisson-Boltzmann equations starting from classical theories of binary fluid mixtures, employing an approach based on the Legendre transform as recently applied to the case of local descriptions of the fluid free energy. Under specific symmetry assumptions, and in the linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological equation introduced by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)]PRLTAO0031-900710.1103/PhysRevLett.106.046102, whereby the structuring near the surface is determined by bulk coefficients.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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}

  11. Semi-Implicit High-Order Methods for the Euler Equations used in Nonhydrostatic Mesoscale Atmospheric Modeling

    NASA Astrophysics Data System (ADS)

    Giraldo, F.; Restelli, M.; Lauter, M.

    2008-12-01

    In this work, we present the semi-implicit time-integration of the Euler equations in their various forms; in this work we compare four variations of the Euler equations used in mesoscale atmospheric modeling. The forms of the Euler equations studied are: the Exner-potential temperature, density-potential temperature, density- total energy, and density-pressure sets. All of these forms (except for the density-energy form) have been used in mesoscale atmospheric modeling; currently only one global nonhydrostatic model has proposed to use the density-energy form. We compare and contrast the vices and virtues of using these different equations in conservation and non-conservation forms. We show that exact conservation of mass and energy can be achieved as long as the density and total energy are prognostic variables. This means that to conserve both mass and energy requires the use of the density-total energy form. With the density-potential temperature or density-pressure, only mass can be conserved exactly. The Exner-potential temperature form is not capable of conserving either mass or energy. These results are irrespective of the spatial discretization used; for example, the same results hold for either spectral elements or discontinuous Galerkin methods. The semi-implicit methods that we discuss are based on the backward difference formulas (BDF). The generalized semi-implicit BDF methods we present consist of first through sixth order methods in time. While the results we show are for the Euler equations only, this generalized semi-implicit high-order method is applicable to any hyperbolic system of equations including the shallow water equations and further to the hydrostatic primitive atmospheric equations.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  16. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  1. 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.

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

    NASA Astrophysics Data System (ADS)

    Yefet, Amir; Petropoulos, Peter G.

    2001-04-01

    We consider a model explicit fourth-order staggered finite-difference method for the hyperbolic Maxwell's equations. Appropriate fourth-order accurate extrapolation and one-sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces. An eigenvalue analysis of the overall scheme provides a necessary, but not sufficient, stability condition and indicates long-time stability. Numerical results verify both the stability analysis, and the scheme's fourth-order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces. For a fixed error level, we find the fourth-order scheme is computationally cheaper in comparison to the Yee scheme by more than an order of magnitude. Some open problems encountered in the application of such high-order schemes are also discussed.

  3. Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

    PubMed Central

    Garić-Demirović, M.; Kulenović, M. R. S.; Nurkanović, M.

    2013-01-01

    We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x n+1 = x n−1 2/(ax n 2 + bx n x n−1 + cx n−1 2), n = 0,1, 2,…, where the parameters a,  b, and  c are positive numbers and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable. PMID:24369451

  4. Global dynamics of certain homogeneous second-order quadratic fractional difference equation.

    PubMed

    Garić-Demirović, M; Kulenović, M R S; Nurkanović, M

    2013-01-01

    We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x(n+1) = x²(n-1)/(ax²(n) + bx(n)x(n-1) + cx²(n-1)), n = 0,1, 2,…, where the parameters a,  b, and  c are positive numbers and the initial conditions x₋₁ and x₀ are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

  5. 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.

  6. A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation

    NASA Astrophysics Data System (ADS)

    Tayebi, A.; Shekari, Y.; Heydari, M. H.

    2017-07-01

    Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

  7. 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.

  8. 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…

  9. 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…

  10. Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation.

    PubMed

    Alexandrescu, Adrian; Montesinos, Gaspar D; Pérez-García, Víctor M

    2007-04-01

    In this paper we consider the stabilization of nonfundamental unstable stationary solutions of the cubic nonlinear Schrödinger equation. Specifically, we study the stabilization of radially symmetric solutions with nodes and asymmetric complex stationary solutions. For the first ones, we find partial stabilization similar to that recently found for vortex solutions while for the later ones stabilization does not seem possible.

  11. Second-Order Characteristic Methods for Advection-Difusion Equations and Comparison to Other Schemes

    DTIC Science & Technology

    1997-01-01

    and for nonlinear equations as well as nite volume formulations However because the characteristics for variablecoecient...there are more improved versions of SDM method with shock capturing capacity which produce better approximations they usually have nonlinear ... optimally ecient and reasonable choice of space and time steps to produce a qualitatively comparable solution to that of the BRKC and FRKC schemes in

  12. The fourth-order absorbing boundary condition with optimized coefficients for the simulation of the acoustic equation

    NASA Astrophysics Data System (ADS)

    Song, Peng; Liu, Zhaolun; Zhang, Xiaobo; Tan, Jun; Xia, Dongming; Li, Jing; Zhu, Bo

    2015-12-01

    This paper introduces the fourth-order absorbing boundary condition (ABC) into staggered-grid finite difference forward modeling of the first-order stress-velocity acoustic equation, and develops a new method to optimize coefficients of the fourth-order ABC to further improve its overall absorbing effect. Theoretical analysis and the results of numerical tests demonstrate that the fourth-order ABC with optimized coefficients has much higher absorbing efficiency than both the conventional second-order and fourth-order ABCs without optimized coefficients, for waves with large incident angles. Compared with the perfectly matched layer (PML) with 40 layers, the fourth-order ABC not only has a much better absorbing effect, but also uses far less computer memory for calculation. We present the fourth-order ABC with optimized coefficients as an ideal artificial boundary for the simulation of the acoustic equation based on extensive and complex structure models. Supported by the Fundamental Research Funds for the Central Universities (201513005).

  13. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  16. Computational solutions of three-dimensional advection-diffusion equation using fourth order time efficient alternating direction implicit scheme

    NASA Astrophysics Data System (ADS)

    Saqib, Muhammad; Hasnain, Shahid; Mashat, Daoud Suleiman

    2017-08-01

    To develop an efficient numerical scheme for three-dimensional advection diffusion equation, higher order ADI method was proposed. 2nd and fourth order ADI schemes were used to handle such problem. Von Neumann stability analysis shows that Alternating Direction Implicit scheme is unconditionally stable. The accuracy and efficiency of such schemes was depicted by two test problems. Numerical results for two test problems were carried out to establish the performance of the given method and to compare it with the others Typical methods. Fourth order ADI method were found to be very efficient and stable for solving three dimensional Advection Diffusion Equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

  17. 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.

  18. Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

    PubMed Central

    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. PMID:24511303

  19. 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.

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

    PubMed

    Liang, WanZhen; Head-Gordon, Martin

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  2. Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations.

    PubMed

    Alagoz, Baris Baykant

    2017-09-01

    With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

  3. 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.

  4. 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.

  5. 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.

  6. Rate Equation Analysis of the Dynamics of First-order Exciton Mott Transition

    NASA Astrophysics Data System (ADS)

    Sekiguchi, Fumiya; Shimano, Ryo

    2017-10-01

    We perform a rate equation analysis of the dynamics of the exciton Mott transition (EMT) assuming a detailed balance between excitons and unbound electron-hole (e-h) pairs. Using the Saha equation and adopting an empirical expression for the band-gap renormalization effect caused by unbound e-h pairs, we show that the ionization ratio of excitons exhibits bistability as a function of the total e-h pair density at low temperatures. We demonstrate that an incubation time emerges in the dynamics of the EMT from the oversaturated exciton gas phase on the verge of the bistable region. The incubation time shows slowing down behavior when the pair density approaches saddle-node bifurcation of the hysteresis curve of the exciton ionization ratio.

  7. Lyapunov-type inequality for a higher order dynamic equation on time scales.

    PubMed

    Sun, Taixiang; Xi, Hongjian

    2016-01-01

    The purpose of this work is to establish a Lyapunov-type inequality for the following dynamic equation [Formula: see text]on some time scale T under the anti-periodic boundary conditions [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], p is the quotient of two odd positive integers and [Formula: see text] with [Formula: see text].

  8. 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

  9. Intervalwise block partitioning for 3-point in solving linear systems of first order ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Mahayadin, Mahfuzah; Othman, Khairil Iskandar; Ibrahim, Zarina Bibi

    2017-04-01

    Intervalwise partitioning is a strategy to solve stiff ordinary differential equations (ODEs). This strategy using on 3-point block method will initially starts solving ODE using Adams method, and switch the system to Backward Differentiation Formula (BDF) when there is an indication of stiffness. Indication of stiffness will be based on hacc > hiter and the trace of the Jacobian. The comparison with existing method reveals that this partitioning strategy can be an alternative method to solve stiff ODEs.

  10. 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

  11. High-Order Spectral Volume Method for 2D Euler Equations

    NASA Technical Reports Server (NTRS)

    Wang, Z. J.; Zhang, Laiping; Liu, Yen; Kwak, Dochan (Technical Monitor)

    2002-01-01

    The Spectral Volume (SV) method is extended to the 2D Euler equations. The focus of this paper is to study the performance of the SV method on multidimensional non-linear systems. Implementation details including total variation diminishing (TVD) and total variation bounded (TVB) limiters are presented. Solutions with both smooth features and discontinuities are utilized to demonstrate the overall capability of the SV method.

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

    NASA Astrophysics Data System (ADS)

    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.

  13. 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

  14. 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

  15. Fractional approximations for linear first-order differential equations with polynomial coefficients—application to E1(x)

    NASA Astrophysics Data System (ADS)

    Martin, Pablo; Zamudio-Cristi, Jorge

    1982-12-01

    A method is described to obtain fractional approximations for linear first-order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all of the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the higher and lower powers of the variable after the substitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Padé method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which gives three exact digits for most of the real values of the variable. Approximations of higher accuracy than those of other authors are also obtained.

  16. Solitary wave solutions of the fourth order Boussinesq equation through the exp(-Ф(η))-expansion method.

    PubMed

    Akbar, M Ali; Hj Mohd Ali, Norhashidah

    2014-01-01

    The exp(-Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(-Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(-Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering. 35C07; 35C08; 35P99.

  17. Solution of singularly perturbed Cauchy problem for ordinary differential equation of second order with constant coefficients by Fourier method

    NASA Astrophysics Data System (ADS)

    Shaldanbayev, Amir Sh.; Shomanbayeva, Manat T.

    2017-09-01

    In this paper, using the Fourier method in the Krein space, a singularly perturbed Cauchy problem for a second-order differential equation with constant coefficients is solved, and an asymptotic expansion of this solution is found using the theory of linear operators and functional analysis. The peculiarity of the method consists in the fact that the operator of the corresponding Cauchy problem does not have a spectrum, but, nevertheless, even in this case it is possible to expand its solution in a Fourier series in the Krein space and obtain an asymptotic expansion with an estimate of the remainder term. The estimate of the remainder term is obtained in the form of a convolution operator through the right-hand side of the equation and through the coefficients of the equation itself.

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

    PubMed

    Mateo, A Muñoz; Delgado, V

    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.

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

    NASA Astrophysics Data System (ADS)

    Mateo, A. Muñoz; Delgado, V.

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  2. Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards' equation on unstructured meshes

    NASA Astrophysics Data System (ADS)

    Svyatskiy, D.; Lipnikov, K.

    2017-06-01

    Richards's equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. We propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards' equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithms for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Numerical simulations show that the new preconditioner reduces computational cost up to 2-3 times in comparison with the conventional preconditioners.

  3. 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

  4. A second-order accurate finite volume scheme with the discrete maximum principle for solving Richards’ equation on unstructured meshes

    DOE PAGES

    Svyatsky, Daniil; Lipnikov, Konstantin

    2017-03-18

    Richards’s equation describes steady-state or transient flow in a variably saturated medium. For a medium having multiple layers of soils that are not aligned with coordinate axes, a mesh fitted to these layers is no longer orthogonal and the classical two-point flux approximation finite volume scheme is no longer accurate. Here, we propose new second-order accurate nonlinear finite volume (NFV) schemes for the head and pressure formulations of Richards’ equation. We prove that the discrete maximum principles hold for both formulations at steady-state which mimics similar properties of the continuum solution. The second-order accuracy is achieved using high-order upwind algorithmsmore » for the relative permeability. Numerical simulations of water infiltration into a dry soil show significant advantage of the second-order NFV schemes over the first-order NFV schemes even on coarse meshes. Since explicit calculation of the Jacobian matrix becomes prohibitively expensive for high-order schemes due to build-in reconstruction and slope limiting algorithms, we study numerically the preconditioning strategy introduced recently in Lipnikov et al. (2016) that uses a stable approximation of the continuum Jacobian. Lastly, numerical simulations show that the new preconditioner reduces computational cost up to 2–3 times in comparison with the conventional preconditioners.« less

  5. Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

    NASA Astrophysics Data System (ADS)

    Hasnain, Shahid; Saqib, Muhammad; Mashat, Daoud Suleiman

    2017-07-01

    This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit) to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

  6. 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

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  9. Solution of Fifth-order Korteweg and de Vries Equation by Homotopy perturbation Transform Method using He's Polynomial

    NASA Astrophysics Data System (ADS)

    Sharma, Dinkar; Singh, Prince; Chauhan, Shubha

    2017-06-01

    In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is applied to solve nonlinear fifth order Korteweg de Vries (KdV) equations. The method is known as homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He's polynomials. Two test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM).

  10. Synergies from using higher order symplectic decompositions both for ordinary differential equations and quantum Monte Carlo methods

    SciTech Connect

    Matuttis, Hans-Georg; Wang, Xiaoxing

    2015-03-10

    Decomposition methods of the Suzuki-Trotter type of various orders have been derived in different fields. Applying them both to classical ordinary differential equations (ODEs) and quantum systems allows to judge their effectiveness and gives new insights for many body quantum mechanics where reference data are scarce. Further, based on data for 6 × 6 system we conclude that sampling with sign (minus-sign problem) is probably detrimental to the accuracy of fermionic simulations with determinant algorithms.

  11. 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.

  12. 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 Maxwell`s equations will also be presented.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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

  15. Higher-order analytical solutions for the equation of motion of a particle on a rotating parabola

    NASA Astrophysics Data System (ADS)

    Chowdhury, M. S. H.; Hosen, Md. Alal; Ali, Mohammad Yeakub; Ismail, Ahmad Faris

    2017-04-01

    In the present paper, a novel analytical technique to obtain higher-order approximate solutions for the equation of motion of a particle on a rotating parabola has been introduced, which is based on an energy balance method (EBM). The results are valid for small as well as large oscillation of initial amplitude. It is highly remarkable that using the introduced technique a third-order approximate solution gives an excellent agreement with the exact ones. The introduced technique is applied to the motion of a particle on a rotating parabola having high nonlinearity to illustrate its novelty, reliability and wider applicability.

  16. 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

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

    NASA Astrophysics Data System (ADS)

    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.

  18. 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.

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

    NASA Technical Reports Server (NTRS)

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

    2002-01-01

    We present high-order semi-discrete central-upwind numerical schemes for approximating solutions of multi-dimensional Hamilton-Jacobi (HJ) equations. This scheme is based on the use of fifth-order central interpolants like those developed in [1], in fluxes presented in [3]. These interpolants use the weighted essentially nonoscillatory (WENO) approach to avoid spurious oscillations near singularities, and become "central-upwind" in the semi-discrete limit. This scheme provides numerical approximations whose error is as much as an order of magnitude smaller than those in previous WENO-based fifth-order methods [2, 1]. Thee results are discussed via examples in one, two and three dimensions. We also pregnant explicit N-dimensional formulas for the fluxes, discuss their monotonicity and tl!e connection between this method and that in [2].

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

    NASA Astrophysics Data System (ADS)

    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.

  1. The extrapolated explicit midpoint scheme for variable order and step size controlled integration of the Landau-Lifschitz-Gilbert equation

    NASA Astrophysics Data System (ADS)

    Exl, Lukas; Mauser, Norbert J.; Schrefl, Thomas; Suess, Dieter

    2017-10-01

    A practical and efficient scheme for the higher order integration of the Landau-Lifschitz-Gilbert (LLG) equation is presented. The method is based on extrapolation of the two-step explicit midpoint rule and incorporates adaptive time step and order selection. We make use of a piecewise time-linear stray field approximation to reduce the necessary work per time step. The approximation to the interpolated operator is embedded into the extrapolation process to keep in step with the hierarchic order structure of the scheme. We verify the approach by means of numerical experiments on a standardized NIST problem and compare with a higher order embedded Runge-Kutta formula. The efficiency of the presented approach increases when the stray field computation takes a larger portion of the costs for the effective field evaluation.

  2. Using the seventh-order numerical method to solve first-order nonlinear coupled-wave equations for degenerate two-wave and four-wave mixing

    NASA Astrophysics Data System (ADS)

    Ja, Y. H.

    1984-12-01

    Using a new seventh-order numerical method [the O(h 7) method] for solving two-point boundary value problems, numerical solutions of the first-order nonlinear coupledwave equations for degenerate two-wave and four-wave mixing in a reflection geometry have been obtained. A computer program employing the Gauss-Jordan elimination technique has also been adopted to effectively solve the resultant large, sparse and unsymmetric matrix, obtained from the O(h 7) method and the Newton-Raphson iteration method. Numerical results from the computer calculations are presented graphically. A comparison between this O(h 7) method and the shooting method, mainly from the viewpoint of computational efficiency, is also made.

  3. 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.

  4. 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.

  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. Model order reduction for the time-harmonic Maxwell equation applied to complex nanostructures

    NASA Astrophysics Data System (ADS)

    Hammerschmidt, Martin; Herrmann, Sven; Pomplun, Jan; Burger, Sven; Schmidt, Frank

    2016-03-01

    Fields such as optical metrology and computational lithography require fast and efficient methods for solving the time-harmonic Maxwell's equation. Highly accurate geometrical modelling and numerical accuracy at low computational costs are a prerequisite for any simulation study of complex nano-structured photonic devices. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scattering at optical wavelengths of periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such as the width and height of the fin and gate can vary in a large range.

  7. 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.

  8. Residual Monte Carlo high-order solver for Moment-Based Accelerated Thermal Radiative Transfer equations

    SciTech Connect

    Willert, Jeffrey Park, H.

    2014-11-01

    In this article we explore the possibility of replacing Standard Monte Carlo (SMC) transport sweeps within a Moment-Based Accelerated Thermal Radiative Transfer (TRT) algorithm with a Residual Monte Carlo (RMC) formulation. Previous Moment-Based Accelerated TRT implementations have encountered trouble when stochastic noise from SMC transport sweeps accumulates over several iterations and pollutes the low-order system. With RMC we hope to significantly lower the build-up of statistical error at a much lower cost. First, we display encouraging results for a zero-dimensional test problem. Then, we demonstrate that we can achieve a lower degree of error in two one-dimensional test problems by employing an RMC transport sweep with multiple orders of magnitude fewer particles per sweep. We find that by reformulating the high-order problem, we can compute more accurate solutions at a fraction of the cost.

  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. 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

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

    NASA Astrophysics Data System (ADS)

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

    NASA Astrophysics Data System (ADS)

    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.

  13. Ordered Rate Constitutive Theories: Development of Rate Constitutive Equations for Solids, Liquids, and Gases

    DTIC Science & Technology

    2010-08-18

    process of being submitted for journal publication, 2010). [2] Panton , R. L. Incompressible Flow, Third Edition. John Wiley and Sons, 2005. [3] White...Constitutive Theory for Ordered Thermoelastic Solids. (In the process of being submitted for journal publication, 2010). [3] Panton , R. L

  14. 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.…

  15. Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation

    NASA Astrophysics Data System (ADS)

    Gushchin, A. K.

    2015-10-01

    We consider a statement of the Dirichlet problem which generalizes the notions of classical and weak solutions, in which a solution belongs to the space of (n-1)-dimensionally continuous functions with values in the space L_p. The property of (n-1)-dimensional continuity is similar to the classical definition of uniform continuity; however, instead of the value of a function at a point, it looks at the trace of the function on measures in a special class, that is, elements of the space L_p with respect to these measures. Up to now, the problem in the statement under consideration has not been studied in sufficient detail. This relates first to the question of conditions on the right-hand side of the equation which ensure the solvability of the problem. The main results of the paper are devoted to just this question. We discuss the terms in which these conditions can be expressed. In addition, the way the behaviour of a solution near the boundary depends on the right-hand side is investigated. Bibliography: 47 titles.

  16. Robust scale-space filter using second-order partial differential equations.

    PubMed

    Ham, Bumsub; Min, Dongbo; Sohn, Kwanghoon

    2012-09-01

    This paper describes a robust scale-space filter that adaptively changes the amount of flux according to the local topology of the neighborhood. In a manner similar to modeling heat or temperature flow in physics, the robust scale-space filter is derived by coupling Fick's law with a generalized continuity equation in which the source or sink is modeled via a specific heat capacity. The filter plays an essential part in two aspects. First, an evolution step size is adaptively scaled according to the local structure, enabling the proposed filter to be numerically stable. Second, the influence of outliers is reduced by adaptively compensating for the incoming flux. We show that classical diffusion methods represent special cases of the proposed filter. By analyzing the stability condition of the proposed filter, we also verify that its evolution step size in an explicit scheme is larger than that of the diffusion methods. The proposed filter also satisfies the maximum principle in the same manner as the diffusion. Our experimental results show that the proposed filter is less sensitive to the evolution step size, as well as more robust to various outliers, such as Gaussian noise, impulsive noise, or a combination of the two.

  17. Interconnections between various analytic approaches applicable to third-order nonlinear differential equations.

    PubMed

    Mohanasubha, R; Chandrasekar, V K; Senthilvelan, M; Lakshmanan, M

    2015-04-08

    We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. Second order nonlinear equations of motion for spinning highly flexible line-elements. [for spacecraft solar sail

    NASA Technical Reports Server (NTRS)

    Salama, M.; Trubert, M.

    1979-01-01

    A formulation is given for the second order nonlinear equations of motion for spinning line-elements having little or no intrinsic structural stiffness. Such elements have been employed in recent studies of structural concepts for future large space structures such as the Heliogyro solar sailer. The derivation is based on Hamilton's variational principle and includes the effect of initial geometric imperfections (axial, curvature, and twist) on the line-element dynamics. For comparison with previous work, the nonlinear equations are reduced to a linearized form frequently found in the literature. The comparison has revealed several new spin-stiffening terms that have not been previously identified and/or retained. They combine geometric imperfections, rotary inertia, Coriolis, and gyroscopic terms.

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

    NASA Astrophysics Data System (ADS)

    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.

  1. 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.

  2. 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.

  3. Quasi-periodic wave solutions and asymptotic properties for a fifth-order Korteweg-de Vries type equation

    NASA Astrophysics Data System (ADS)

    Qin, Chun-Yan; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian

    2016-07-01

    Under investigation in this paper is a fifth-order Korteweg-de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its N-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

  4. 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.

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

    NASA Astrophysics Data System (ADS)

    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

  6. Numerical simulation of the three dimensional Allen-Cahn equation by the high-order compact ADI method

    NASA Astrophysics Data System (ADS)

    Zhai, Shuying; Feng, Xinlong; He, Yinnian

    2014-10-01

    In this paper, a new linearized high-order compact difference method is presented for numerical simulation of three dimensional (3D) Allen-Cahn equation with three kinds of boundary conditions. The method, which is based on the Crank-Nicholson/Adams-Bashforth scheme combined with the Douglas-Gunn ADI method, is second order accurate in time and fourth order accurate in space and energy degradation. The main advantages of this method is that the nonlinear penalty term f(u) is linear and an extra stabilizing term is added to alleviate the stability constraint while maintaining accuracy and simplicity. Numerical experiments are given to demonstrate the validity and applicability of the new method.

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

    NASA Astrophysics Data System (ADS)

    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.

  8. 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.

  9. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  11. 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.

  12. High order eigenvalues for the Helmholtz equation in complicated non-tensor domains through Richardson extrapolation of second order finite differences

    NASA Astrophysics Data System (ADS)

    Amore, Paolo; Boyd, John P.; Fernández, Francisco M.; Rösler, Boris

    2016-05-01

    We apply second order finite differences to calculate the lowest eigenvalues of the Helmholtz equation, for complicated non-tensor domains in the plane, using different grids which sample exactly the border of the domain. We show that the results obtained applying Richardson and Padé-Richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values. When possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions. Our empirical findings suggest an asymptotic nature of the FD series. In all the cases studied, we are able to report numerical results which are more precise than those available in the literature.

  13. 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.

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

    NASA Astrophysics Data System (ADS)

    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

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  17. Second-order Optimality Conditions for Optimal Control of the Primitive Equations of the Ocean with Periodic Inputs

    SciTech Connect

    Tachim Medjo, T.

    2011-02-15

    We investigate in this article the Pontryagin's maximum principle for control problem associated with the primitive equations (PEs) of the ocean with periodic inputs. We also derive a second-order sufficient condition for optimality. This work is closely related to Wang (SIAM J. Control Optim. 41(2):583-606, 2002) and He (Acta Math. Sci. Ser. B Engl. Ed. 26(4):729-734, 2006), in which the authors proved similar results for the three-dimensional Navier-Stokes (NS) systems.

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

    NASA Astrophysics Data System (ADS)

    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.

  19. 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.

  20. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

  2. CRE Solvability, Nonlocal Symmetry and Exact Interaction Solutions of the Fifth-Order Modified Korteweg-de Vries Equation

    NASA Astrophysics Data System (ADS)

    Cheng, Wen-Guang; Qiu, De-Qin; Yu, Bo

    2017-06-01

    This paper is concerned with the fifth-order modified Korteweg-de Vries (fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion (CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion (CTE) method, the nonlocal symmetry related to the consistent tanh expansion (CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlevé method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed. Supported by National Natural Science Foundation of China under Grant No. 11505090, and Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009

  3. Decoupling of the DGLAP evolution equations at next-to-next-to-leading order (NNLO) at low- x

    NASA Astrophysics Data System (ADS)

    Boroun, G. R.; Rezaei, B.

    2013-05-01

    We present a set of formulas to extract two second-order independent differential equations for the gluon and singlet distribution functions. Our results extend from the LO up to NNLO DGLAP evolution equations with respect to the hard-Pomeron behavior at low- x. In this approach, both singlet quarks and gluons have the same high-energy behavior at low- x. We solve the independent DGLAP evolution equations for the functions F2s(x,Q2) and G( x, Q 2) as a function of their initial parameterization at the starting scale Q02. The results not only give striking support to the hard-Pomeron description of the low- x behavior, but give a rather clean test of perturbative QCD showing an increase of the gluon distribution and singlet structure functions as x decreases. We compared our numerical results with the published BDM (Block et al. Phys. Rev. D 77:094003 (2008)) gluon and singlet distributions, starting from their initial values at Q02=1 GeV2.

  4. 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

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

    NASA Astrophysics Data System (ADS)

    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.

  6. 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.

  7. 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

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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

  10. 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.

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    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.

  13. Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system

    NASA Astrophysics Data System (ADS)

    Yajima, Takahiro; Yamasaki, Kazuhito

    2016-03-01

    Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

  14. 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.

  15. 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

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

    NASA Astrophysics Data System (ADS)

    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.

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

    NASA Astrophysics Data System (ADS)

    Seadawy, Aly R.

    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.

  18. A low-dispersive method using the high-order stereo-modelling operator for solving 2-D wave equations

    NASA Astrophysics Data System (ADS)

    Li, Jingshuang; Yang, Dinghui; Wu, Hao; Ma, Xiao

    2017-09-01

    In this paper, we propose a 12th-order stereo-modelling operator to approximate the high-order spatial derivatives using both wavefield displacements and their gradients. On base of this compact operator (seven grids in one spatial direction) and a two-step time marching scheme, we get a new finite-difference method for solving 2-D seismic wave equations, which is 12th-order in space and fourth order in time (12-STEM). Theoretical properties of the 12-STEM including stability and errors are analysed and the numerical dispersion relationship of the 12-STEM for 1-D and 2-D cases are investigated. The computational efficiency is compared among the 12-STEM, the fourth-order stereo-modelling method and other high-order Lax-Wendroff correction (LWC) methods. Among those methods, the 12-STEM has the least computational time and memory requirement to achieve the same accuracy because large spatial and time increments can be used by the 12-STEM. What's more, for different acoustic and elastic cases, numerical simulations computed by the 12-STEM and the 12th-order LWC are presented and compared. Numerical results show that the 12-STEM can effectively suppress numerical dispersion in seismic modelling from acoustic/elastic homogeneous to heterogeneous and even complex heterogeneous models when coarse grid sizes are used or the medium has strong velocity contrast. Thus, the 12-STEM can be potentially used to solve large-scale wave-propagation problems and seismic inversion such as reverse-time migration, tomography and full waveform inversion, and so on.

  19. 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.

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

    NASA Astrophysics Data System (ADS)

    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.