Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order
Cong, Y. H.; Jiang, C. X.
2014-01-01
The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods. PMID:24977178
Diagonally implicit symplectic Runge-Kutta methods with high algebraic and dispersion order.
Cong, Y H; Jiang, C X
2014-01-01
The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods. PMID:24977178
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
NASA Technical Reports Server (NTRS)
Kennedy, Christopher A.; Carpenter, Mark H.
2016-01-01
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.
Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD
NASA Astrophysics Data System (ADS)
Aloy, Miguel-Á.; Cordero-Carrión, Isabel
2016-05-01
The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative extra loop in order to recover the primitive (physical) variables, the analytical inversion of the implicit operator is trivial and the several stages can actually be viewed as stages of explicit Runge-Kutta methods with an effective time-step. We test these methods with two different one-dimensional test beds in varied conductivity regimes, and show that our second-order schemes satisfy the theoretical expectations.
On implicit Runge-Kutta methods for parallel computations
NASA Technical Reports Server (NTRS)
Keeling, Stephen L.
1987-01-01
Implicit Runge-Kutta methods which are well-suited for parallel computations are characterized. It is claimed that such methods are first of all, those for which the associated rational approximation to the exponential has distinct poles, and these are called multiply explicit (MIRK) methods. Also, because of the so-called order reduction phenomenon, there is reason to require that these poles be real. Then, it is proved that a necessary condition for a q-stage, real MIRK to be A sub 0-stable with maximal order q + 1 is that q = 1, 2, 3, or 5. Nevertheless, it is shown that for every positive integer q, there exists a q-stage, real MIRK which is I-stable with order q. Finally, some useful examples of algebraically stable MIRKs are given.
A Low-Dispersion and Low-Dissipation Implicit Runge-Kutta Scheme
Najafi-Yazdi, A.; Mongeau, L.
2012-01-01
A fourth-order, implicit, low-dispersion, and low-dissipation Runge-Kutta scheme is introduced. The scheme is optimized for minimal dissipation and dispersion errors. High order accuracy is achieved with fewer stages than standard explicit Runge-Kutta schemes. The scheme is designed to be As table for highly stiff problems. Possible applications include wall-bounded flows with solid boundaries in the computational domain, and sound generation by reacting flows. PMID:23243319
A Low-Dispersion and Low-Dissipation Implicit Runge-Kutta Scheme.
Najafi-Yazdi, A; Mongeau, L
2013-01-15
A fourth-order, implicit, low-dispersion, and low-dissipation Runge-Kutta scheme is introduced. The scheme is optimized for minimal dissipation and dispersion errors. High order accuracy is achieved with fewer stages than standard explicit Runge-Kutta schemes. The scheme is designed to be As table for highly stiff problems. Possible applications include wall-bounded flows with solid boundaries in the computational domain, and sound generation by reacting flows. PMID:23243319
NASA Astrophysics Data System (ADS)
Langer, Stefan
2014-11-01
For unstructured finite volume methods an agglomeration multigrid with an implicit multistage Runge-Kutta method as a smoother is developed for solving the compressible Reynolds averaged Navier-Stokes (RANS) equations. The implicit Runge-Kutta method is interpreted as a preconditioned explicit Runge-Kutta method. The construction of the preconditioner is based on an approximate derivative. The linear systems are solved approximately with a symmetric Gauss-Seidel method. To significantly improve this solution method grid anisotropy is treated within the Gauss-Seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling. The agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling. Turbulence effects are included by a Spalart-Allmaras model, and the additional transport-type equation is approximately solved in a loosely coupled manner with the same method. For two-dimensional and three-dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method. Finally, we exploit the GMRES method to determine approximate spectral information of the linearized RANS equations. This approximate spectral information is used to discuss and compare characteristics of multistage Runge-Kutta methods.
A stiffly-stable implicit Runge-Kutta algorithm for CFD applications
NASA Technical Reports Server (NTRS)
Baker, A. J.; Iannelli, G. S.
1988-01-01
A stiffly-stable implicit Runge-Kutta integration algorithm is derived for CFD applications spanning the range of semidiscrete theories. The algorithm family contains the one-step 'theta' algorithms, including backwards Euler and the trapezoidal rule, and provides a versatile framework to identify expressions governing algorithm stability characteristics. Parameters of a Runge-Kutta optimal implicit algorithm, second-order accurate in time and stiffly-stable, are established. This algorithm is implemented within a weak statement finite element semidiscrete formulation for one- and two-dimensional conservation law systems. Numerical results are compared to theta-algorithm solutions, for unsteady quasi-one-dimensional Euler predictions with shocks, and for a specially derived two-dimensional conservation law system modeling the Euler equations.
Generalized disks of contractivity for explicit and implicit Runge-Kutta methods
NASA Technical Reports Server (NTRS)
Dahlquist, G.; Jeltsch, R.
1979-01-01
The A-contractivity of Runge-Kutta methods with respect to an inner product norm was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is extended to contractivity in a region bounded by a circle through the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point. It is shown that an explicit Runge-Kutta method cannot be contractive in any circle of this class if it is more than fourth order accurate.
NASA Astrophysics Data System (ADS)
Cavaglieri, Daniele; Bewley, Thomas
2015-04-01
Implicit/explicit (IMEX) Runge-Kutta (RK) schemes are effective for time-marching ODE systems with both stiff and nonstiff terms on the RHS; such schemes implement an (often A-stable or better) implicit RK scheme for the stiff part of the ODE, which is often linear, and, simultaneously, a (more convenient) explicit RK scheme for the nonstiff part of the ODE, which is often nonlinear. Low-storage RK schemes are especially effective for time-marching high-dimensional ODE discretizations of PDE systems on modern (cache-based) computational hardware, in which memory management is often the most significant computational bottleneck. In this paper, we develop and characterize eight new low-storage implicit/explicit RK schemes which have higher accuracy and better stability properties than the only low-storage implicit/explicit RK scheme available previously, the venerable second-order Crank-Nicolson/Runge-Kutta-Wray (CN/RKW3) algorithm that has dominated the DNS/LES literature for the last 25 years, while requiring similar storage (two, three, or four registers of length N) and comparable floating-point operations per timestep.
NASA Astrophysics Data System (ADS)
Kulikov, G. Yu.
2015-06-01
A technique for constructing nested implicit Runge-Kutta methods in the class of mono-implicit formulas of this type is studied. These formulas are highly efficient in practice, since the dimension of the original system of differential equations is preserved, which is not possible in the case of implicit multistage Runge-Kutta formulas of the general from. On the other hand, nested implicit Runge-Kutta methods inherit all major properties of general formulas of this form, such as A-stability, symmetry, and symplecticity in a certain sense. Moreover, they can have sufficiently high stage and classical orders and, without requiring high extra costs, can ensure dense output of integration results of the same accuracy as the order of the underlying method. Thus, nested methods are efficient when applied to the numerical integration of differential equations of various sorts, including stiff and nonstiff problems, Hamiltonian systems, and invertible equations. In this paper, previously proposed nested methods based on the Gauss quadrature formulas are generalized to Lobatto-type methods. Additionally, a unified technique for constructing all such methods is proposed. Its performance is demonstrated as applied to embedded examples of nested implicit formulas of various orders. All the methods constructed are supplied with tools for local error estimation and automatic variable-stepsize mesh generation based on an optimal stepsize selection. These numerical methods are verified by solving test problems with known solutions. Additionally, a comparative analysis of these methods with Matlab built-in solvers is presented.
Elkina, N V; Fedotov, A M; Herzing, C; Ruhl, H
2014-05-01
The Landau-Lifshitz equation provides an efficient way to account for the effects of radiation reaction without acquiring the nonphysical solutions typical for the Lorentz-Abraham-Dirac equation. We solve the Landau-Lifshitz equation in its covariant four-vector form in order to control both the energy and momentum of radiating particles. Our study reveals that implicit time-symmetric collocation methods of the Runge-Kutta-Nyström type are superior in accuracy and better at maintaining the mass-shell condition than their explicit counterparts. We carry out an extensive study of numerical accuracy by comparing the analytical and numerical solutions of the Landau-Lifshitz equation. Finally, we present the results of the simulation of particle scattering by a focused laser pulse. Due to radiation reaction, particles are less capable of penetrating into the focal region compared to the case where radiation reaction is neglected. Our results are important for designing forthcoming experiments with high intensity laser fields. PMID:25353922
NASA Technical Reports Server (NTRS)
Kanevsky, Alex
2004-01-01
My goal is to develop and implement efficient, accurate, and robust Implicit-Explicit Runge-Kutta (IMEX RK) methods [9] for overcoming geometry-induced stiffness with applications to computational electromagnetics (CEM), computational fluid dynamics (CFD) and computational aeroacoustics (CAA). IMEX algorithms solve the non-stiff portions of the domain using explicit methods, and isolate and solve the more expensive stiff portions using implicit methods. Current algorithms in CEM can only simulate purely harmonic (up to lOGHz plane wave) EM scattering by fighter aircraft, which are assumed to be pure metallic shells, and cannot handle the inclusion of coatings, penetration into and radiation out of the aircraft. Efficient MEX RK methods could potentially increase current CEM capabilities by 1-2 orders of magnitude, allowing scientists and engineers to attack more challenging and realistic problems.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
NASA Technical Reports Server (NTRS)
Kennedy, Christopher A.; Carpenter, Mark H.
2002-01-01
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.
Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations
NASA Technical Reports Server (NTRS)
Kennedy, Christopher A.; Carpenter, Mark H.
2001-01-01
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.
Two-derivative Runge-Kutta methods for differential equations
NASA Astrophysics Data System (ADS)
Chan, Robert P. K.; Wang, Shixiao; Tsai, Angela Y. J.
2012-09-01
Two-derivative Runge-Kutta (TDRK) methods are a special case of multi-derivative Runge-Kutta methods first studied by Kastlunger and Wanner [1, 2]. These methods incorporate derivatives of order higher than the first in their formulation but we consider only the first and second derivatives. In this paper we first present our study of both explicit [3] and implicit TDRK methods on stiff ODE problems. We then extend the applications of these TDRK methods to various partial differential equations [4]. In particular, we show how a 2-stage implicit TDRK method of order 4 and stage order 4 can be adapted to solve diffusion equations more efficiently than the popular Crank-Nicolson method.
Construction of IMEX methods with inherent Runge-Kutta stability
NASA Astrophysics Data System (ADS)
Braś, Michał; Izzo, Giuseppe; Jackiewicz, Zdzislaw
2016-06-01
We describe construction of implicit-explicit (IMEX) general linear methods (GLMs) with inherent Runge-Kutta stability (IRKS) for differential systems with non-stiff and stiff processes. We will use the extrapolation approach to remove implicitness in the non-stiff terms to compute unknown stage values in terms of stage derivatives at the previous step and those already computed in the current step. Highly stable IMEX GLMs of stage order equal to the order were derived up to the order four. These methods do not suffer from order reduction phenomenon which is confirmed by numerical experiments.
Runge-Kutta based generalized convolution quadrature
NASA Astrophysics Data System (ADS)
Lopez-Fernandez, Maria; Sauter, Stefan
2016-06-01
We present the Runge-Kutta generalized convolution quadrature (gCQ) with variable time steps for the numerical solution of convolution equations for time and space-time problems. We present the main properties of the method and a convergence result.
A Runge-Kutta Nystrom algorithm.
NASA Technical Reports Server (NTRS)
Bettis, D. G.
1973-01-01
A Runge-Kutta algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative. The algorithm includes the Fehlberg step control procedure.
An unconditionally stable Runge-Kutta method for unsteady flows
NASA Technical Reports Server (NTRS)
Jorgenson, Philip C. E.; Chima, Rodrick V.
1989-01-01
A quasi-three-dimensional analysis was developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The Baldwin-Lomax eddy viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing was employed to accelerate the solution of the time accurate computations. The scheme is described and accuracy analyses are given. Results are shown for a supersonic through-flow fan designed for NASA Lewis. The rotor:stator blade ratio was taken as 1:1. Results are also shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Here the blade ratio is 2:3. Implicit residual smoothing was used to increase the time step limit of the unsmoothed scheme by a factor of six with negligible differences in the unsteady results. It is felt that the implicitly smoothed Runge-Kutta scheme is easily competitive with implicit schemes for unsteady flows while retaining the simplicity of an explicit scheme.
An unconditionally stable Runge-Kutta method for unsteady flows
NASA Technical Reports Server (NTRS)
Jorgenson, Philip C. E.; Chima, Rodrick V.
1988-01-01
A quasi-three dimensional analysis was developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The Baldwin-Lomax eddy viscosity model is used for turbulent flows. The equations are integrated in time using a four stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing was employed to accelerate the solution of the time accurate computations. The scheme is described and accuracy analyses are given. Results are shown for a supersonic through-flow fan designed for NASA Lewis. The rotor:stator blade ratio was taken as 1:1. Results are also shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Here the blade ratio is 2:3. Implicit residual smoothing was used to increase the time step limit of the unsmoothed scheme by a factor of six with negligible differences in the unsteady results. It is felt that the implicitly smoothed Runge-Kutta scheme is easily competitive with implicit schemes for unsteady flows while retaining the simplicity of an explicit scheme.
Amplification and Suppression of Round-Off Error in Runge-Kutta Methods
ERIC Educational Resources Information Center
Prentice, J. S. C.
2011-01-01
A simple nonstiff linear initial-value problem is used to demonstrate the amplification of round-off error in the course of using a second-order Runge-Kutta method. This amplification is understood in terms of an appropriate expression for the global error. An implicit method is then used to show how the roundoff error may actually be suppressed.…
Obtaining Runge-Kutta Solutions Between Time Steps
NASA Technical Reports Server (NTRS)
Horn, M. K.
1984-01-01
New interpolation method used with existing Runge-Kutta algorithms. Algorithm evaluates solution at intermediate point within integration step. Only few additional computations required to produce intermediate solution data. Runge-Kutta method provides accurate solution with larger time steps than allowable in other methods.
Scaled Runge-Kutta algorithms for handling dense output
NASA Technical Reports Server (NTRS)
Horn, M. K.
1981-01-01
Low order Runge-Kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step, as well as at the end of each step. The scaled Runge-Kutta methods are designed to be used with existing Runge-Kutta formulas, using the derivative evaluations of these defining algorithms as the core of the system. For a slight increase in computing time, the solution may be generated within the integration step, improving the efficiency of the Runge-Kutta algorithms, since the step length need no longer be severely reduced to coincide with the desired output point. Scaled Runge-Kutta algorithms are presented for orders 3 through 5, along with accuracy comparisons between the defining algorithms and their scaled versions for a test problem.
Composite group of explicit Runge-Kutta methods
NASA Astrophysics Data System (ADS)
Hamid, Fatin Nadiah Abd; Rabiei, Faranak; Ismail, Fudziah
2016-06-01
In this paper,the composite groups of Runge-Kutta (RK) method are proposed. The composite group of RK method of third and second order, RK3(2) and fourth and third order RK4(3) base on classical Runge-Kutta method are derived. The proposed methods are two-step in nature and have less number of function evaluations compared to the existing Runge-Kutta method. The order conditions up to order four are obtained using rooted trees and composite rule introduced by J. C Butcher. The stability regions of RK3(2) and RK4(3) methods are presented and initial value problems of first order ordinary differential equations are carried out. Numerical results are compared with existing Runge-Kutta method.
A third order Runge-Kutta algorithm on a manifold
NASA Technical Reports Server (NTRS)
Crouch, P. E.; Grossman, R. G.; Yan, Y.
1992-01-01
A third order Runge-Kutta type algorithm is described with the property that it preserves certain geometric structures. In particular, if the algorithm is initialized on a Lie group, then the resulting iterates remain on the Lie group.
Galerkin/Runge-Kutta discretizations for semilinear parabolic equations
NASA Technical Reports Server (NTRS)
Keeling, Stephen L.
1987-01-01
A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high, optimal order convergence. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.
Convergence Acceleration of Runge-Kutta Schemes for Solving the Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Swanson, Roy C., Jr.; Turkel, Eli; Rossow, C.-C.
2007-01-01
The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 can be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. This RK/implicit scheme is used as a smoother for multigrid. Fourier analysis is applied to determine damping properties. Numerical dissipation operators based on the Roe scheme, a matrix dissipation, and the CUSP scheme are considered in evaluating the RK/implicit scheme. In addition, the effect of the number of RK stages is examined. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. Turbulent flows over an airfoil and wing at subsonic and transonic conditions are computed. The effects of the cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 x 10(exp 6) and 100 x 10(exp 6). It is demonstrated that the implicit preconditioner can reduce the computational time of a well-tuned standard RK scheme by a factor between four and ten.
Functional continuous Runge-Kutta methods for special systems
NASA Astrophysics Data System (ADS)
Eremin, A. S.; Olemskoy, I. V.
2016-06-01
We consider here numerical methods for systems of retarded functional differential equations of two equations in which the right-hand sides are cross-dependent of the unknown functions, i.e. the derivatives of unknowns don't depend on the same unknowns. It is shown that using the special structure of the system one can construct functional continuous methods of Runge-Kutta type with fewer stages than it is necessary in case of general Runge-Kutta functional continuous methods. Order conditions and example methods of orders three and four are presented. Test problems are solved, demonstrating the declared convergence order of the new methods.
An unconditionally stable Runge-Kutta method for unsteady rotor-stator interaction
NASA Technical Reports Server (NTRS)
Chima, Rodrick V.; Jorgenson, Philip C. E.
1989-01-01
A quasi-three-dimensional analysis has been developed for unsteady rotor-stator interaction in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stress-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using an explicit four-stage Runge-Kutta scheme with a constant time step. Implicit residual smoothing is used to increase the stability limit of the time-accurate computations. The scheme is described, and stability and accuracy analyses are given.
Runge-Kutta Methods for Linear Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Zingg, David W.; Chisholm, Todd T.
1997-01-01
Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.
Optimized fourth-order Runge-Kutta method for solving oscillatory problems
NASA Astrophysics Data System (ADS)
Hussain, Kasim; Ismail, Fudziah; Senu, Norazak; Rabiei, Faranak
2016-06-01
In this article, we develop a Runge-Kutta method with invalidation of phase lag, phase lag's derivatives and amplification error to solve second-order initial value problem (IVP) with oscillating solutions. The new method depends on the explicit Runge-Kutta method of algebraic order four. Numerical tests from its implementation to well-known oscillatory problems illustrate the robustness and competence of the new method as compared to the well-known Runge-Kutta methods in the scientific literature.
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.
Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping
NASA Technical Reports Server (NTRS)
Suresh, A.; Huynh, H. T.
1997-01-01
A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.
An explicit Runge-Kutta method for 3D turbulent incompressible flows
NASA Technical Reports Server (NTRS)
Sung, Chao-Ho; Lin, Cheng-Wen; Hung, C. M.
1988-01-01
A computer code has been developed to solve for the steady-state solution of the 3D incompressible Reynolds-averaged Navier-Stokes equations. The approach is based on the cell-center, central-difference, finite-volume formulation and an explicit one-step, multistage Runge-Kutta time-stepping scheme. The Baldwin-Lomax turbulence model is used. Techniques to accelerate the rate of convergence to a steady-state solution include the preconditioned method, the local time stepping, and the implicit residual smoothing. Improvements in computational efficiency have been demonstrated in several areas. This numerical procedure has been used to simulate the turbulent horseshoe vortex flow around an airfoil/flat-plate juncture.
A diagonally inverted LU implicit multigrid scheme
NASA Technical Reports Server (NTRS)
Yokota, Jeffrey W.; Caughey, David A.; Chima, Rodrick V.
1988-01-01
A new Diagonally Inverted LU Implicit scheme is developed within the framework of the multigrid method for the 3-D unsteady Euler equations. The matrix systems that are to be inverted in the LU scheme are treated by local diagonalizing transformations that decouple them into systems of scalar equations. Unlike the Diagonalized ADI method, the time accuracy of the LU scheme is not reduced since the diagonalization procedure does not destroy time conservation. Even more importantly, this diagonalization significantly reduces the computational effort required to solve the LU approximation and therefore transforms it into a more efficient method of numerically solving the 3-D Euler equations.
Multirate Runge-Kutta schemes for advection equations
NASA Astrophysics Data System (ADS)
Schlegel, Martin; Knoth, Oswald; Arnold, Martin; Wolke, Ralf
2009-04-01
Explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena. The wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step. Multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps, resulting in a moderate to substantial reduction of the computational cost, depending on the scenario to simulate [S. Osher, R. Sanders, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput. 41 (1983) 321-336; H. Tang, G. Warnecke, A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and pace grids, SIAM J. Sci. Comput. 26 (4) (2005) 1415-1431; E. Constantinescu, A. Sandu, Multirate timestepping methods for hyperbolic conservation laws, SIAM J. Sci. Comput. 33 (3) (2007) 239-278]. In air pollution modeling the advection part is usually integrated explicitly in time, where the time step is constrained by a locally varying Courant-Friedrichs-Lewy (CFL) number. Multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction. Therefore it is of major interest to apply multirate schemes to the advection equation. We introduce a generic recursive multirate Runge-Kutta scheme that can be easily adapted to an arbitrary number of refinement levels. It preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit Runge-Kutta methods as base method.
On the improvement of deconvolution with digitized data using a Runge-Kutta integration scheme
NASA Technical Reports Server (NTRS)
Houghton, J. R.; Townsend, M. A.; Packman, P. F.
1977-01-01
A relatively simple change in the treatment of the input function in numerical integration of high-order differential equations by Runge-Kutta methods provides substantial improvements in accuracy, particularly when the forcing function is in digitized form. The Runge-Kutta-Gill coefficients are modified to incorporate the changes; with pulse-type excitations, improvements on the order of 2 to 50 times greater accuracy are demonstrated.
Flux-vector splitting and Runge-Kutta methods for the Euler equations
NASA Technical Reports Server (NTRS)
Turkel, E.; Vanleer, B.
1984-01-01
Runge-Kutta schemes have been used as a method of solving the Euler equations exterior to an airfoil. In the past this has been coupled with central differences and an artificial vesocity in space. In this study the Runge-Kutta time-stepping scheme is coupled with an upwinded space approximation based on flux-vector splitting. Several acceleration techniques are also considered including a local time step, residual smoothing and multigrid.
Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations
NASA Technical Reports Server (NTRS)
Yu, S. T.; Tsai, Y.-L. P.; Hsieh, K. C.
1992-01-01
An investigation of the Runge-Kutta time-stepping, combined with compact difference schemes to solve the unsteady Euler equations, is presented. Initially, a generalized form of a N-step Runge-Kutta technique is derived. By comparing this generalized form with its Taylor's series counterpart, the criteria for the three-step and four-step schemes to be of third- and fourth-order accurate are obtained.
Low-dissipation and -dispersion Runge-Kutta schemes for computational acoustics
NASA Technical Reports Server (NTRS)
Hu, F. Q.; Hussaini, M. Y.; Manthey, J.
1994-01-01
In this paper, we investigate accurate and efficient time advancing methods for computational acoustics, where non-dissipative and non-dispersive properties are of critical importance. Our analysis pertains to the application of Runge-Kutta methods to high-order finite difference discretization. In many CFD applications multi-stage Runge-Kutta schemes have often been favored for their low storage requirements and relatively large stability limits. For computing acoustic waves, however, the stability consideration alone is not sufficient, since the Runge-Kutta schemes entail both dissipation and dispersion errors. The time step is now limited by the tolerable dissipation and dispersion errors in the computation. In the present paper, it is shown that if the traditional Runge-Kutta schemes are used for time advancing in acoustic problems, time steps greatly smaller than that allowed by the stability limit are necessary. Low-Dissipation and -Dispersion Runge-Kutta (LDDRE) schemes are proposed, based on an optimization that minimizes the dissipation and dispersion errors for wave propagation. Order optimizations of both single-step and two-step alternating schemes are considered. The proposed LDDRK schemes are remarkably more efficient than the classical Runge-Kutta schemes for acoustic computations. Moreover, low storage implementations of the optimized schemes are discussed. Special issues of implementing numerical boundary conditions in the LDDRK schemes are also addressed.
NASA Technical Reports Server (NTRS)
Hu, F. Q.; Hussaini, M. Y.; Manthey, J.
1995-01-01
We investigate accurate and efficient time advancing methods for computational aeroacoustics, where non-dissipative and non-dispersive properties are of critical importance. Our analysis pertains to the application of Runge-Kutta methods to high-order finite difference discretization. In many CFD applications, multi-stage Runge-Kutta schemes have often been favored for their low storage requirements and relatively large stability limits. For computing acoustic waves, however, the stability consideration alone is not sufficient, since the Runge-Kutta schemes entail both dissipation and dispersion errors. The time step is now limited by the tolerable dissipation and dispersion errors in the computation. In the present paper, it is shown that if the traditional Runge-Kutta schemes are used for time advancing in acoustic problems, time steps greatly smaller than that allowed by the stability limit are necessary. Low Dissipation and Dispersion Runge-Kutta (LDDRK) schemes are proposed, based on an optimization that minimizes the dissipation and dispersion errors for wave propagation. Optimizations of both single-step and two-step alternating schemes are considered. The proposed LDDRK schemes are remarkably more efficient than the classical Runge-Kutta schemes for acoustic computations. Numerical results of each Category of the Benchmark Problems are presented. Moreover, low storage implementations of the optimized schemes are discussed. Special issues of implementing numerical boundary conditions in the LDDRK schemes are also addressed.
Rendering log aesthetic curves via Runge-Kutta method
NASA Astrophysics Data System (ADS)
Gobithaasan, R. U.; Meng, T. Y.; Piah, A. R. M.; Miura, K. T.
2014-07-01
Log Aesthetic Curves (LAC) are visually pleasing curves which has been developed using monotonic curvature profile. Hence, it can be easily implemented in product design environment, e.g, Rhino 3D CAD systems. LAC is generally represented in an integral form of its turning angle. Traditionally, Gaussian-Kronrod method has been used to render this curve which consumes less than one second for a given interval. Recently, Incomplete Gamma Function was proposed to represent LAC analytically which decreases the computation time up to 13 times. However, only certain value of shape parameters (denoted as α) which dictates the types of curves generated for LAC, can be used to compute LAC. In this paper, the classical Runge-Kutta (RK4) method is proposed to evaluate LAC numerically to reduce the LAC computation time for arbitrary, α. The preliminary result looks promising where the evaluation time is decreased tremendously. This paper also demonstrates the accuracy control of LAC by reducing the stepsize of RK4. The computation time and the accuracy for various α, are also illustrated in the last section of this paper.
High-order implicit time-marching methods for unsteady fluid flow simulation
NASA Astrophysics Data System (ADS)
Boom, Pieter David
Unsteady computational fluid dynamics (CFD) is increasingly becoming a critical tool in the development of emerging technologies and modern aircraft. In spite of rapid mathematical and technological advancement, these simulations remain computationally intensive and time consuming. More efficient temporal integration will promote a wider use of unsteady analysis and extend its range of applicability. This thesis presents an investigation of efficient high-order implicit time-marching methods for application in unsteady compressible CFD. A generalisation of time-marching methods based on summation-by-parts (SBP) operators is described which reduces the number of stages required to obtain a prescribed order of accuracy, thus improving their efficiency. The classical accuracy and stability theory is formally extended for these generalised SBP (GSBP) methods, including superconvergence and nonlinear stability. Dual-consistent SBP and GSBP time-marching methods are shown to form a subclass of implicit Runge-Kutta methods, which enables extensions of nonlinear accuracy and stability results. A novel family of fully-implicit GSBP Runge-Kutta schemes based on Gauss quadrature are derived which are both algebraically stable and L-stable with order 2s - 1, where s is the number of stages. In addition, a numerical tool is developed for the construction and optimisation of general linear time-marching methods. The tool is applied to the development of several low-stage-order L-stable diagonally-implicit methods, including a diagonally-implicit GSBP Runge-Kutta scheme. The most notable and efficient method developed is a six-stage fifth-order L-stable stiffly-accurate explicit-first-stage singly-diagonally-implicit Runge-Kutta (ESDIRK5) method with stage order two. The theoretical results developed in this thesis are supported by numerical simulations, and the predicted relative efficiency of the schemes is realised.
Scaled Runge-Kutta algorithms for treating the problem of dense output
NASA Technical Reports Server (NTRS)
Horn, M. K.
1982-01-01
A set of scaled Runge-Kutta algorithms for the third- through fifth-orders are developed to determine the solution at any point within the integration step at a relatively small increase in computing time. Each scaled algorithm is designed to be used with an existing Runge-Kutta formula, using the derivative evaluations of the defining algorithm along with an additional derivative evaluation (or two). Third-order, scaled algorithms are embedded within the existing formulas at no additional derivative expense. Such algorithms can easily be adopted to generate interpolating polynomials (or dependent variable stops) efficiently.
A Runge-Kutta discontinuous Galerkin approach to solve reactive flows: The hyperbolic operator
Billet, G.; Ryan, J.
2011-02-20
A Runge-Kutta discontinuous Galerkin method to solve the hyperbolic part of reactive Navier-Stokes equations written in conservation form is presented. Complex thermodynamics laws are taken into account. Particular care has been taken to solve the stiff gaseous interfaces correctly with no restrictive hypothesis. 1D and 2D test cases are presented.
Modified Runge-Kutta methods for solving ODES. M.S. Thesis
NASA Technical Reports Server (NTRS)
Vanvu, T.
1981-01-01
A class of Runge-Kutta formulas is examined which permit the calculation of an accurate solution anywhere in the interval of integration. This is used in a code which seldom has to reject a step; rather it takes a reduced step if the estimated error is too large. The absolute stability implications of this are examined.
A Family of Trigonometrically-fitted Partitioned Runge-Kutta Symplectic Methods
Monovasilis, Th.; Kalogiratou, Z.; Simos, T. E.
2007-12-26
We are presenting a family of trigonometrically fitted partitioned Runge-Kutta symplectic methods of fourth order with six stages. The solution of the one dimensional time independent Schroedinger equation is considered by trigonometrically fitted symplectic integrators. The Schroedinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the exponential potential.
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.
NASA Technical Reports Server (NTRS)
Cockrell, C. R.
1989-01-01
Numerical solutions of the differential equation which describe the electric field within an inhomogeneous layer of permittivity, upon which a perpendicularly-polarized plane wave is incident, are considered. Richmond's method and the Runge-Kutta method are compared for linear and exponential profiles of permittivities. These two approximate solutions are also compared with the exact solutions.
NASA Astrophysics Data System (ADS)
Kalogiratou, Z.; Monovasilis, Th.; Psihoyios, G.; Simos, T. E.
2014-03-01
In this work we review single step methods of the Runge-Kutta type with special properties. Among them are methods specially tuned to integrate problems that exhibit a pronounced oscillatory character and such problems arise often in celestial mechanics and quantum mechanics. Symplectic methods, exponentially and trigonometrically fitted methods, minimum phase-lag and phase-fitted methods are presented. These are Runge-Kutta, Runge-Kutta-Nyström and Partitioned Runge-Kutta methods. The theory of constructing such methods is given as well as several specific methods. In order to present the performance of the methods we have tested 58 methods from all categories. We consider the two dimensional harmonic oscillator, the two body problem, the pendulum problem and the orbital problem studied by Stiefel and Bettis. Also we have tested the methods on the computation of the eigenvalues of the one dimensional time independent Schrödinger equation with the harmonic oscillator, the doubly anharmonic oscillator and the exponential potentials.
Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients
NASA Technical Reports Server (NTRS)
Keeling, Stephen L.
1989-01-01
A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.
A Runge-Kutta discontinuous finite element method for high speed flows
NASA Technical Reports Server (NTRS)
Bey, Kim S.; Oden, J. T.
1991-01-01
A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.
Galerkin/Runge-Kutta discretizations for parabolic equations with time dependent coefficients
NASA Technical Reports Server (NTRS)
Keeling, Stephen L.
1987-01-01
A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.
Fourth-order 2N-storage Runge-Kutta schemes
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Kennedy, Christopher A.
1994-01-01
A family of five-stage fourth-order Runge-Kutta schemes is derived; these schemes required only two storage locations. A particular scheme is identified that has desirable efficiency characteristics for hyperbolic and parabolic initial (boundary) value problems. This scheme is competitive with the classical fourth-order method (high-storage) and is considerably more efficient and accurate than existing third-order low-storage schemes.
Third-order 2N-storage Runge-Kutta schemes with error control
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Kennedy, Christopher A.
1994-01-01
A family of four-stage third-order explicit Runge-Kutta schemes is derived that requires only two storage locations and has desirable stability characteristics. Error control is achieved by embedding a second-order scheme within the four-stage procedure. Certain schemes are identified that are as efficient and accurate as conventional embedded schemes of comparable order and require fewer storage locations.
NASA Technical Reports Server (NTRS)
Fehlberg, E.
1972-01-01
The formulas include a stepsize control procedure, based on a complete coverage of the leading term of the truncation error in x. The formulas require fewer evaluations per stop than other Runge-Kutta-Nystrom formulas if the latter are operated by using the standard procedure for stepsize control. An example is presented. With results being of the same accuracy, Runge-Kutta-Nystrom formulas discussed save 50 percent or more computer time compared with other Runge-Kutta-Nystrom formulas.
NASA Astrophysics Data System (ADS)
Huang, Yong; Shi, Guo-Dong; Zhu, Ke-Yong
2016-06-01
This paper adopts the Runge-Kutta ray tracing method to obtain the ray-trajectory numerical solution in a two-dimensional gradient index medium. The emitting, absorbing and scattering processes are simulated by the Monte Carlo method. The temperature field and ray trajectory in the medium are obtained by the three methods, the Runge-Kutta ray tracing method, the ray tracing method with the cell model and the discrete curved ray tracing method with the linear refractive index cell model. Comparing the results of the three methods, it is found that the results by the Monte Carlo Runge-Kutta ray tracing method are of the highest accuracy. To improve the computational speed, the variable step-size Runge-Kutta ray tracing method is proposed, and the maximum relative error between the temperature field in the nonscattering medium by this method and the benchmark solution is less than 0.5%. The results also suggest that the Runge-Kutta ray tracing method would make the radiative transfer solution in the three-dimensional graded index media much easier.
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.
Application of Runge-Kutta scheme for high-speed inviscid internal flows
NASA Technical Reports Server (NTRS)
Moitra, A.; Turkel, E.; Kumar, A.
1986-01-01
A multi-stage Runge-Kutta method is analyzed for solving the two-dimensional Euler equations for external and internal flow problems. Subsonic, supersonic and, highly supersonic flows are studied. Various techniques for accelerating the convergence to a steady state are described and analyzed. Effects of the grid aspect ratio on the rate of convergence are evaluated. An enthalpy damping technique applicable to supersonic flows is described in detail. Numerical results for supersonic flows containing both oblique and normal shocks are presented confirming the efficiency of the method.
NASA Technical Reports Server (NTRS)
Jameson, A.; Schmidt, Wolfgang; Turkel, Eli
1981-01-01
A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.
Application of a Runge-Kutta scheme for high-speed inviscid internal flows
NASA Technical Reports Server (NTRS)
Moitra, A.; Turkel, E.; Kumar, A.
1986-01-01
A multi-stage Runge-Kutta method is analyzed for solving the two-dimensional Euler equations for external and internal flow problems. Subsonic, supersonic and, highly supersonic flows are studied. Various techniques for accelerating the convergence to a steady state are described and analyzed. Effects of the grid aspect ratio on the rate of convergence are evaluated. An enthalpy damping technique applicable to supersonic flows is described in detail. Numerical results for supersonic flows containing both oblique and normal shocks are presented confirming the efficiency of the method.
NASA Technical Reports Server (NTRS)
Lear, W. M.
1974-01-01
The integration is discussed of the vector differential equation X = F(x, t) from time t sub i to t sub (i = 1) where only the values of x sub i are available for the the integration. No previous values of x or x prime are used. Using an orbit integration problem, comparisons are made between Taylor series integrators and various types and orders of Runge-Kutta integrators. A fourth order Runge-Kutta type integrator for orbital work is presented, and approximate (there may be no exact) fifth order Runge-Kutta integrators are discussed. Also discussed and compared is a self starting integrator ising delta f/delta x. A numerical method for controlling the accuracy of integration is given, and the special equations for accurately integrating accelerometer data are shown.
NASA Technical Reports Server (NTRS)
Cannizzaro, Frank E.; Ash, Robert L.
1992-01-01
A state-of-the-art computer code has been developed that incorporates a modified Runge-Kutta time integration scheme, upwind numerical techniques, multigrid acceleration, and multi-block capabilities (RUMM). A three-dimensional thin-layer formulation of the Navier-Stokes equations is employed. For turbulent flow cases, the Baldwin-Lomax algebraic turbulence model is used. Two different upwind techniques are available: van Leer's flux-vector splitting and Roe's flux-difference splitting. Full approximation multi-grid plus implicit residual and corrector smoothing were implemented to enhance the rate of convergence. Multi-block capabilities were developed to provide geometric flexibility. This feature allows the developed computer code to accommodate any grid topology or grid configuration with multiple topologies. The results shown in this dissertation were chosen to validate the computer code and display its geometric flexibility, which is provided by the multi-block structure.
Tremblay, Jean Christophe; Carrington, Tucker Jr.
2004-12-15
If the Hamiltonian is time dependent it is common to solve the time-dependent Schroedinger equation by dividing the propagation interval into slices and using an (e.g., split operator, Chebyshev, Lanczos) approximate matrix exponential within each slice. We show that a preconditioned adaptive step size Runge-Kutta method can be much more efficient. For a chirped laser pulse designed to favor the dissociation of HF the preconditioned adaptive step size Runge-Kutta method is about an order of magnitude more efficient than the time sliced method.
NASA Technical Reports Server (NTRS)
Fehlberg, E.
1973-01-01
New Runge-Kutta-Nystrom formulas of the eighth, seventh, sixth, and fifth order are derived for the special second-order (vector) differential equation x = f (t,x). In contrast to Runge-Kutta-Nystrom formulas of an earlier NASA report, these formulas provide a stepsize control procedure based on the leading term of the local truncation error in x. This new procedure is more accurate than the earlier Runge-Kutta-Nystrom procedure (with stepsize control based on the leading term of the local truncation error in x) when integrating close to singularities. Two central orbits are presented as examples. For these orbits, the accuracy and speed of the formulas of this report are compared with those of Runge-Kutta-Nystrom and Runge-Kutta formulas of earlier NASA reports.
NASA Technical Reports Server (NTRS)
Subramanian, S. V.; Bozzola, R.
1987-01-01
Numerical solutions of the unsteady Euler equations are obtained using the classical fourth order Runge Kutta time marching scheme. This method is fully explicit and is applied to the governing equations in the finite volume, conservation law form. In order to determine the efficiency of this scheme for solving turbomachinery flows, steady blade-to-blade solutions are obtained for compressor and turbine cascades under subsonic and transonic flow conditions. Computed results are compared with other numerical methods and wind tunnel measurements. The study also focuses on other important numerical aspects influencing the performance of the algorithm and the solution accuracy such as grid types, boundary conditions and artificial viscosity. For this purpose, H, O, and C type computational grids as well as characteristic and extrapolation type boundary conditions are included in solution procedures.
An explicit Runge-Kutta method for unsteady rotor/stator interaction
NASA Technical Reports Server (NTRS)
Jorgenson, Philip C. E.; Chima, Rodrick V.
1988-01-01
A quasi-three-dimensional rotor/stator analysis has been developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant timestep. Results are shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Euler and Navier-Stokes results are compared on the scaled single- and multi-passage machine. The method is relatively fast and the quasi-three-dimensional formulation is applicable to a wide range of turbomachinery geometries.
Steady and Unsteady Numerical Solution of Generalized Newtonian Fluids Flow by Runge-Kutta method
NASA Astrophysics Data System (ADS)
Keslerová, R.; Kozel, K.; Prokop, V.
2010-09-01
In this paper the laminar viscous incompressible flow for generalized Newtonian (Newtonian and non-Newtonian) fluids is considered. The governing system of equations is the system of Navier-Stokes equations and the continuity equation. The steady and unsteady numerical solution for this system is computed by finite volume method combined with an artificial compressibility method. For time discretization the explicit multistage Runge-Kutta numerical scheme is considered. Steady state solution is achieved for t→∞ using steady boundary conditions and followed by steady residual behavior. The dual time-stepping method is considered for unsteady computation. The high artificial compressibility coefficient is used in the artificial compressibility method applied in the dual time τ. The steady and unsteady numerical results of Newtonian and non-Newtonian (shear thickening and shear thinning) fluids flow in the branching channel are presented.
Qualitatively stability of nonstandard 2-stage explicit Runge-Kutta methods of order two
NASA Astrophysics Data System (ADS)
Khalsaraei, M. M.; Khodadosti, F.
2016-02-01
When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge-Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.
Recent advances in Runge-Kutta schemes for solving 3-D Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Vatsa, Veer N.; Wedan, Bruce W.; Abid, Ridha
1989-01-01
A thin-layer Navier-Stokes has been developed for solving high Reynolds number, turbulent flows past aircraft components under transonic flow conditions. The computer code has been validated through data comparisons for flow past isolated wings, wing-body configurations, prolate spheroids and wings mounted inside wind-tunnels. The basic code employs an explicit Runge-Kutta time-stepping scheme to obtain steady state solution to the unsteady governing equations. Significant gain in the efficiency of the code has been obtained by implementing a multigrid acceleration technique to achieve steady-state solutions. The improved efficiency of the code has made it feasible to conduct grid-refinement and turbulence model studies in a reasonable amount of computer time. The non-equilibrium turbulence model of Johnson and King has been extended to three-dimensional flows and excellent agreement with pressure data has been obtained for transonic separated flow over a transport type of wing.
On spurious steady-state solutions of explicit Runge-Kutta schemes
NASA Technical Reports Server (NTRS)
Sweby, P. K.; Yee, H. C.; Griffiths, D. F.
1990-01-01
The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results.
Runge-Kutta model-based nonlinear observer for synchronization and control of chaotic systems.
Beyhan, Selami
2013-07-01
This paper proposes a novel nonlinear gradient-based observer for synchronization and observer-based control of chaotic systems. The model is based on a Runge-Kutta model of the chaotic system where the evolution of the states or parameters is derived based on the error-square minimization. The stability and convergence conditions of observer and control methods are analyzed using a Lyapunov stability approach. In numerical simulations, the proposed observer and well-known sliding-mode observer are compared for the synchronization of a Lü chaotic system and observer-based stabilization of a Chen chaotic system. The noisy case for synchronization and parameter uncertainty case for stabilization are also considered for both observer-based methods. PMID:23672740
An explicit Runge-Kutta method for turbulent reacting flow calculations
NASA Technical Reports Server (NTRS)
Boretti, A. A.
1989-01-01
The paper presents a numerical method for the solution of the conservation equations governing steady, reacting, turbulent viscous flow in two-dimensional geometries, in both Cartesian and axisymmetric coordinates. These equations are written in Favre-averaged form and closed with a first order model. A two-equation K-epsilon model, where low Reynolds number and compressibility effects are included, and a modified eddy-break up model are used to simulate fluid mechanics turbulence, chemistry and turbulence-combustion interaction. The solution is obtained by using a pseudo-unsteady method with improved perturbation propagation properties. The equations are discretized in space by using a finite volume formulation. An explicit multi-stage dissipative Runge-Kutta algorithm is then used to advance the flow equations in the pseudo-time. The method is applied to the computation of both diffusion and premixed turbulent reacting flows. The computed temperature distributions compare favorably with experimental data.
A computer program for determining truncation error coefficients for Runge-Kutta methods
NASA Technical Reports Server (NTRS)
Horn, M. K.
1980-01-01
The basic structure of a program to generate the truncation error coefficients for Runge-Kutta (RK) methods is reformulated to reduce storage requirements significantly and to accommodate variable dimensioning. This FORTRAN program, SUBROUTINE RKEQ, determines truncation error coefficients for RK algorithms for orders 1 through 10 and extends the order of coefficients through 12 with the 11th- and 12th-order terms determined following the patterns used to establish the lower order coefficients. Both subroutines (the original and RKEQ) are also written to treat RK m-fold methods which utilize m known derivatives of f to increase the order of the algorithm. Setting m = 0 gives the classical RK algorithm.
NASA Technical Reports Server (NTRS)
Subramanian, S. V.; Bozzola, R.
1985-01-01
Numerical solutions of the unsteady Euler equations are obtained using the classical fourth order Runge Kutta time marching scheme. This method is fully explicit and is applied to the governing equations in the finite volume, conservation law form. In order to determine the efficiency of this scheme for solving turbomachinery flows, steady blade-to-blade solutions are obtained for compressor and turbine cascades under subsonic and transonic flow conditions. Computed results are compared with other numerical methods and wind tunnel measurements. The present study also focuses on other important numerical aspects influencing the performance of the algorithm and the solution accuracy such as grid types, boundary conditions, and artificial viscosity. For this purpose, H, O, and C type computational grids as well as characteristic and extrapolation type boundary conditions are included in the solution procedure.
GPU acceleration of Runge Kutta-Fehlberg and its comparison with Dormand-Prince method
NASA Astrophysics Data System (ADS)
Seen, Wo Mei; Gobithaasan, R. U.; Miura, Kenjiro T.
2014-07-01
There is a significant reduction of processing time and speedup of performance in computer graphics with the emergence of Graphic Processing Units (GPUs). GPUs have been developed to surpass Central Processing Unit (CPU) in terms of performance and processing speed. This evolution has opened up a new area in computing and researches where highly parallel GPU has been used for non-graphical algorithms. Physical or phenomenal simulations and modelling can be accelerated through General Purpose Graphic Processing Units (GPGPU) and Compute Unified Device Architecture (CUDA) implementations. These phenomena can be represented with mathematical models in the form of Ordinary Differential Equations (ODEs) which encompasses the gist of change rate between independent and dependent variables. ODEs are numerically integrated over time in order to simulate these behaviours. The classical Runge-Kutta (RK) scheme is the common method used to numerically solve ODEs. The Runge Kutta Fehlberg (RKF) scheme has been specially developed to provide an estimate of the principal local truncation error at each step, known as embedding estimate technique. This paper delves into the implementation of RKF scheme for GPU devices and compares its result with Dorman Prince method. A pseudo code is developed to show the implementation in detail. Hence, practitioners will be able to understand the data allocation in GPU, formation of RKF kernels and the flow of data to/from GPU-CPU upon RKF kernel evaluation. The pseudo code is then written in C Language and two ODE models are executed to show the achievable speedup as compared to CPU implementation. The accuracy and efficiency of the proposed implementation method is discussed in the final section of this paper.
NASA Astrophysics Data System (ADS)
Igumnov, Leonid; Ipatov, Aleksandr; Belov, Aleksandr; Petrov, Andrey
2015-09-01
The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on direct-approach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-by-element approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary) and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.
Exponential Runge-Kutta integrators for modelling Predator-Prey interactions
NASA Astrophysics Data System (ADS)
Diele, F.; Marangi, C.; Ragni, S.
2012-09-01
Spatially explicit models consisting of reaction-diffusion partial differential equations are considered in order to model prey-predator interactions, since it is known that the role of spatial processes reveals of great interest in the study of the effects of habitat fragmentation on biodiversity. As almost all of the realistic models in biology, these models are nonlinear and their solution is not known in closed form. Our aim is approximating the solution itself by means of exponential Runge-Kutta integrators. Moreover, we apply the shift-and-invert Krylov approach in order to evaluate the entire functions needed for implementing the exponential method. This numerical procedure reveals to be very eff cient in avoiding numerical instability during the simulation, since it allows us to adopt high order in the accuracy. This work has received funding from the European Union's Seventh Framework Programme FP7/2007-2013, SPA.2010.1.1-04: "Stimulating the development of GMES services in specif c are", under grant agreement 263435, project title: Biodiversity Multi-Source Monitoring System:from Space To Species (BIOSOS) coordinated by CNR-ISSIA, Bari-Italy (http://www.biosos.eu).
NASA Astrophysics Data System (ADS)
Diele, F.; Marangi, C.; Ragni, S.
2009-08-01
Direct numerical approximation of a continuous-time infinite horizon control problem, requires to recast the model as a discrete-time, finite-horizon control model. The quality of the optimization results can be heavily degraded if the discretization process does not take into account features of the original model to be preserved. Restricting their attention to optimal growh problems with a steady state, Mercenier and Michel in [1] and [2], studied the conditions to be imposed for ensuring that discrete first-order approximation models have the same steady states as the infinite-horizon continuous-times counterpart. Here we show that Mercenier and Michel scheme is a first order partitioned Runge-Kutta method applied to the state-costate differential system which arises from the Pontryagin maximum principle. The main consequence is that it is possible to consider high order schemes which generalize that algorithm by preserving the steady-growth invariance of the solutions with respect to the discretization process. Numerical examples show the efficiency and accuracy of the proposed methods when applied to the classical Ramsey growth model.
Low-Storage, Explicit Runge-Kutta Schemes for the Compressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Kennedy, Chistopher A.; Carpenter, Mark H.; Lewis, R. Michael
1999-01-01
The derivation of storage explicit Runge-Kutta (ERK) schemes has been performed in the context of integrating the compressible Navier-Stokes equations via direct numerical simulation. Optimization of ERK methods is done across the broad range of properties, such as stability and accuracy efficiency, linear and nonlinear stability, error control reliability, step change stability, and dissipation/dispersion accuracy, subject to varying degrees of memory economization. Following van der Houwen and Wray, 16 ERK pairs are presented using from two to five registers of memory per equation, per grid point and having accuracies from third- to fifth-order. Methods have been assessed using the differential equation testing code DETEST, and with the 1D wave equation. Two of the methods have been applied to the DNS of a compressible jet as well as methane-air and hydrogen-air flames. Derived 3(2) and 4(3) pairs are competitive with existing full-storage methods. Although a substantial efficiency penalty accompanies use of two- and three-register, fifth-order methods, the best contemporary full-storage methods can be pearl), matched while still saving two to three registers of memory.
Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes
NASA Astrophysics Data System (ADS)
Zhu, Jun; Zhong, Xinghui; Shu, Chi-Wang; Qiu, Jianxian
2013-09-01
In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [32] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure.
Navier-Stokes calculations for DFVLR F5-wing in wind tunnel using Runge-Kutta time-stepping scheme
NASA Technical Reports Server (NTRS)
Vatsa, V. N.; Wedan, B. W.
1988-01-01
A three-dimensional Navier-Stokes code using an explicit multistage Runge-Kutta type of time-stepping scheme is used for solving the transonic flow past a finite wing mounted inside a wind tunnel. Flow past the same wing in free air was also computed to assess the effect of wind-tunnel walls on such flows. Numerical efficiency is enhanced through vectorization of the computer code. A Cyber 205 computer with 32 million words of internal memory was used for these computations.
Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations
NASA Technical Reports Server (NTRS)
Yu, Sheng-Tao
1992-01-01
Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples
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.
Cui, Hengfei; Wang, Desheng; Wan, Min; Zhang, Jun-Mei; Zhao, Xiaodan; Tan, Ru San; Huang, Weimin; Xiong, Wei; Duan, Yuping; Zhou, Jiayin; Luo, Tong; Kassab, Ghassan S; Zhong, Liang
2016-06-01
The CT angiography (CTA) is a clinically indicated test for the assessment of coronary luminal stenosis that requires centerline extractions. There is currently no centerline extraction algorithm that is automatic, real-time and very accurate. Therefore, we sought to (i) develop a hybrid approach by incorporating fast marching and Runge-Kutta based methods for the extraction of coronary artery centerlines from CTA; (ii) evaluate the accuracy of the present method compared to Van's method by using ground truth centerline as a reference; (iii) evaluate the coronary lumen area of our centerline method in comparison with the intravascular ultrasound (IVUS) as the standard of reference. The proposed method was found to be more computationally efficient, and performed better than the Van's method in terms of overlap measures (i.e., OV: [Formula: see text] vs. [Formula: see text]; OF: [Formula: see text] vs. [Formula: see text]; and OT: [Formula: see text] vs. [Formula: see text], all [Formula: see text]). In comparison with IVUS derived coronary lumen area, the proposed approach was more accurate than the Van's method. This hybrid approach by incorporating fast marching and Runge-Kutta based methods could offer fast and accurate extraction of centerline as well as the lumen area. This method may garner wider clinical potential as a real-time coronary stenosis assessment tool. PMID:27140197
Evolutionary generation of 7th order Runge - Kutta - Nyström type methods for solving y(4) = f(x,y)
NASA Astrophysics Data System (ADS)
Papakostas, S. N.; Tsitmidelis, S.; Tsitouras, Ch.
2015-12-01
We present a 7th algebraic order Runge - Kutta - Nyström method for the solution of a special fourth order initial value problem. To achieve this, a set of non - linear equations is solved using differential evolution technique. Various numerical tests justify our efforts.
Evolutionary generation of high order Runge - Kutta - Nyström type pairs for solving y(4) = f (x,y)
NASA Astrophysics Data System (ADS)
Famelis, I. Th.; Tsitmidelis, S.; Tsitouras, Ch.
2016-06-01
We present a new Runge - Kutta - Nyström type pair of orders 8(6) for the solution of a special fourth order initial value problem. To achieve this, a set of non - linear equations is solved using differential evolution technique.
NASA Astrophysics Data System (ADS)
Wang, Xiaoqiang; Ju, Lili; Du, Qiang
2016-07-01
The Willmore flow formulated by phase field dynamics based on the elastic bending energy model has been widely used to describe the shape transformation of biological lipid vesicles. In this paper, we develop and investigate some efficient and stable numerical methods for simulating the unconstrained phase field Willmore dynamics and the phase field Willmore dynamics with fixed volume and surface area constraints. The proposed methods can be high-order accurate and are completely explicit in nature, by combining exponential time differencing Runge-Kutta approximations for time integration with spectral discretizations for spatial operators on regular meshes. We also incorporate novel linear operator splitting techniques into the numerical schemes to improve the discrete energy stability. In order to avoid extra numerical instability brought by use of large penalty parameters in solving the constrained phase field Willmore dynamics problem, a modified augmented Lagrange multiplier approach is proposed and adopted. Various numerical experiments are performed to demonstrate accuracy and stability of the proposed methods.
NASA Astrophysics Data System (ADS)
Korneev, B. A.; Levchenko, V. D.
2016-03-01
In this paper we present the Runge-Kutta discontinuous Galerkin method (RKDG method) for the numerical solution of the Euler equations of gas dynamics. The method is being tested on a series of Riemann problems in the one-dimensional case. For the implementation of the method in the three-dimensional case, a DiamondTorre algorithm is proposed. It belongs to the class of the locally recursive non-locally asynchronous algorithms (LRnLA). With the help of this algorithm a significant increase of speed of calculations is achieved. As an example of the three-dimensional computing, a problem of the interaction of a bubble with a shock wave is considered.
NASA Technical Reports Server (NTRS)
Shih, C. C.
1973-01-01
A theoretical investigation of gas flow inside a multilayer insulation system has been made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas mixture was obtained by considering the diffusion mechanism of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was investigated. It has been shown that when the relaxation time is small compared with the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given, and comparisons with experimental data were made.
Xu, Zhiliang; Chen, Xu-Yan; Liu, Yingjie
2014-12-01
We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method. PMID:25414520
Starting methods for two-step Runge-Kutta methods of stage-order 3 and order 6
NASA Astrophysics Data System (ADS)
Verner, J. H.
2006-01-01
Jackiewicz and Tracogna [SIAM J. Numer. Anal. 32 (1995) 1390-1427] proposed a general formulation of two step Runge-Kutta (TSRK) methods. Using formulas for two-step pairs of TSRK methods constructed in [Japan JIAM 19 (2002) 227-248], Jackiewicz and Verner obtain results for order 8 pairs that fail to show this designated order. Hairer and Wanner [SIAM J. Numer. Anal. 34 (1997) 2087-2089] identify the problem by using B-series to formulate a complete set of order conditions for TSRK methods, and emphasize that special starting methods are necessary for the first step of implementation. They observe that for methods with stage order at least p-1, and design order p, starting methods of order at least p are sufficient. In this paper, the more general challenge to provide correct starting values for methods of low stage-order is met by showing how perturbed starting values should be selected for methods of order 6 and stage-order 3. The approach is sufficiently general that it may (and later will) be provided for such methods of higher orders. Evidence of the accompanying improvement in the implementation of TSRK methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage-order.
Xu, Zhiliang; Chen, Xu-Yan; Liu, Yingjie
2014-01-01
We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method. PMID:25414520
NASA Astrophysics Data System (ADS)
Romeo, A.; Finocchio, G.; Carpentieri, M.; Torres, L.; Consolo, G.; Azzerboni, B.
2008-02-01
The Landau-Lifshitz-Gilbert (LLG) equation is the fundamental equation to describe magnetization dynamics in microscale and nanoscale magnetic systems. In this paper we present a brief overview of a time-domain numerical method related to the fifth order Runge-Kutta formula, which has been applied to the solution of the LLG equation successfully. We discuss advantages of the method, describing the results of a numerical experiment based on the standard problem #4. The results are in good agreement with the ones present in literature. By including thermal effects in our framework, our simulations show magnetization dynamics slightly dependent on the spatial discretization.
A diagonal implicit scheme for computing flows with finite-rate chemistry
NASA Technical Reports Server (NTRS)
Eberhardt, Scott; Imlay, Scott
1990-01-01
A new algorithm for solving steady, finite-rate chemistry, flow problems is presented. The new scheme eliminates the expense of inverting large block matrices that arise when species conservation equations are introduced. The source Jacobian matrix is replaced by a diagonal matrix which is tailored to account for the fastest reactions in the chemical system. A point-implicit procedure is discussed and then the algorithm is included into the LU-SGS scheme. Solutions are presented for hypervelocity reentry and Hydrogen-Oxygen combustion. For the LU-SGS scheme a CFL number in excess of 10,000 has been achieved.
NASA Astrophysics Data System (ADS)
Im, Dong-Kyun; Choi, Seongim; Hyuck Kwon, Jang
2015-01-01
The diagonally implicit harmonic balance method is developed in an overset mesh topology and applied to unsteady rotor flows analysis. Its efficiency is by reducing the complexity of a fully implicit harmonic balance method which becomes more flexible in handling the higher harmonics of the flow solutions. Applied to the overset mesh topology, the efficiency of the method becomes greater by reducing the number of solution interpolations required during the entire solution procedure as the method reduces the unsteady computation into periodic steady state. To verify the accuracy and efficiency of the method, both hovering and unsteady forward flight of Caradonna and Tung and AH-1G rotors are solved. Compared with wind-tunnel experiments, the numerical results demonstrate good agreements at computational cost an order of magnitude more efficient than the conventional time-accurate computation method. The proposed method has great potential in other engineering applications, including flapping wing vehicles, turbo-machinery, wind-turbines, etc.
NASA Technical Reports Server (NTRS)
Swanson, R. C.; Rossow, C.-C.
2008-01-01
A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. This scheme has been applied to both the compressible and essentially incompressible Reynolds-averaged Navier-Stokes (RANS) equations using the algebraic turbulence model of Baldwin and Lomax (BL). In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulence are represented by either the Spalart-Allmaras model or the Wilcox k-! model, which are frequently used models in practical fluid dynamic applications. Convergence behavior of the scheme with these turbulence models and the BL model are directly compared. For this initial investigation we solve the flow equations and the partial differential equations of the turbulence models indirectly coupled. With this approach we examine the convergence behavior of each system. Both point and line symmetric Gauss-Seidel are considered for approximating the inverse of the implicit operator of the flow solver. To solve the turbulence equations we use a diagonally dominant alternating direction implicit (DDADI) scheme. Computational results are presented for three airfoil flow cases and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and transport-type equations for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses the RK/implicit scheme for the flow equations.
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul; Don, Wai-Sun
1993-01-01
The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: (1) impose the exact boundary condition only at the end of the complete RK cycle, (2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases, results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.
NASA Technical Reports Server (NTRS)
Chen, Shu-Cheng; Liu, Nan-Suey; Kim, Hyun Dae
1991-01-01
Presented here is an algorithm for solving the multidimensional unsteady Navier-Stokes equations for compressible flows. It is based on a diagonally-dominant approximate factorization procedure. The factorization error and the timewise linearization error associated with this procedure are reduced by performing Newton-type inner iterations at each time step. The inviscid fluxes are evaluated by the fourth-order central differencing scheme amended with a numerical dissipation directly proportional to the entire dissipative part of the truncation error intrinsic to the third order biased upwind scheme. The important features of the proposed solution are elucidated by the numerical results of the convection of a vortex and the backward-facing step flows.
Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation
NASA Astrophysics Data System (ADS)
Otero, Evelyn; Eliasson, Peter
2015-02-01
The implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solver is combined with the line-implicit technique to improve convergence on the very anisotropic grids necessary for resolving the boundary layers. The computational fluid dynamics code used is Edge, a Navier-Stokes flow solver for unstructured grids based on a dual grid and edge-based formulation. Multigrid acceleration is applied with the intention to accelerate the convergence to steady state. LU-SGS works in parallel and gives better linear scaling with respect to the number of processors, than the explicit scheme. The ordering techniques investigated have shown that node numbering does influence the convergence and that the orderings from Delaunay and advancing front generation were among the best tested. 2D Reynolds-averaged Navier-Stokes computations have clearly shown the strong efficiency of our novel approach line-implicit LU-SGS which is four times faster than implicit LU-SGS and line-implicit Runge-Kutta. Implicit LU-SGS for Euler and line-implicit LU-SGS for Reynolds-averaged Navier-Stokes are at least twice faster than explicit and line-implicit Runge-Kutta, respectively, for 2D and 3D cases. For 3D Reynolds-averaged Navier-Stokes, multigrid did not accelerate the convergence and therefore may not be needed.
NASA Technical Reports Server (NTRS)
Yokota, Jeffrey W.
1988-01-01
An LU implicit multigrid algorithm is developed to calculate 3-D compressible viscous flows. This scheme solves the full 3-D Reynolds-Averaged Navier-Stokes equation with a two-equation kappa-epsilon model of turbulence. The flow equations are integrated by an efficient, diagonally inverted, LU implicit multigrid scheme while the kappa-epsilon equations are solved, uncoupled from the flow equations, by a block LU implicit algorithm. The flow equations are solved within the framework of the multigrid method using a four-grid level W-cycle, while the kappa-epsilon equations are iterated only on the finest grid. This treatment of the Reynolds-Averaged Navier-Stokes equations proves to be an efficient method for calculating 3-D compressible viscous flows.
Satellite attitude dynamics and estimation with the implicit midpoint method
NASA Astrophysics Data System (ADS)
Hellström, Christian; Mikkola, Seppo
2009-07-01
We describe the application of the implicit midpoint integrator to the problem of attitude dynamics for low-altitude satellites without the use of quaternions. Initially, we consider the satellite to rotate without external torques applied to it. We compare the numerical solution with the exact solution in terms of Jacobi's elliptic functions. Then, we include the gravity-gradient torque, where the implicit midpoint integrator proves to be a fast, simple and accurate method. Higher-order versions of the implicit midpoint scheme are compared to Gauss-Legendre Runge-Kutta methods in terms of accuracy and processing time. Finally, we investigate the performance of a parameter-adaptive Kalman filter based on the implicit midpoint integrator for the determination of the principal moments of inertia through observations.
Explicit and implicit solution of the Navier-Stokes equations on a massively parallel computer
NASA Technical Reports Server (NTRS)
Levit, Creon; Jespersen, Dennis
1988-01-01
The design, implementation, and performance of a two-dimensional time-accurate Navier-Stokes solver for the CM2 supercomputer are described. The program uses a single processor for each grid point. Two different time-stepping methods have so far been implemented: an explicit third-order Runge-Kutta method and an implicit approximation-factorization method. The CM2 results are checked against those of a mature well-vectorized Cray 2 program, both for correctness and performance. The code is found to be correct, and the performance in some cases is up to several times that of the Cray 2.
A new alternating bi-diagonal compact scheme for non-uniform grids
NASA Astrophysics Data System (ADS)
Sengupta, Tapan K.; Sengupta, Aditi
2016-04-01
A new compact scheme has been developed for any non-uniform grid. The compact scheme has been developed for spatial discretization and is analyzed here in conjunction with four-stage, fourth order Runge-Kutta (RK4) scheme for time integration while solving the one-dimensional convection equation. The space-time discretization combination is calibrated by subjecting the system to global spectral analysis (GSA) which was developed by the authors' group. Here, the compact scheme has been obtained by using a combination of two bi-diagonal schemes. The novel aspect of this scheme is its application in the physical plane directly without the necessity of mapping or transformations. Some typical cases for problems in acoustics, as well as fluid mechanics, have been studied here and potential use in large eddy simulations (LES) has been demonstrated by solving Navier-Stokes equation for lid driven cavity.
Semi-implicit spectral deferred correction methods for ordinary differential equations
Minion, Michael L.
2002-10-06
A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples of the SISDC method applied to advection-diffusion type equations are included. The results suggest that higher-order SISDC methods are more efficient than semi-implicit Runge-Kutta methods for moderately stiff problems in terms of accuracy per function evaluation.
Global Asymptotic Behavior of Iterative Implicit Schemes
NASA Technical Reports Server (NTRS)
Yee, H. C.; Sweby, P. K.
1994-01-01
The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.
Parallel implicit unstructured grid Euler solvers
NASA Technical Reports Server (NTRS)
Venkatakrishnan, V.
1994-01-01
A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on a multiple-instruction/multiple-data stream parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed: one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All of the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the Intel iPSC/860.
Implicit integration methods for dislocation dynamics
Gardner, D. J.; Woodward, C. S.; Reynolds, D. R.; Hommes, G.; Aubry, S.; Arsenlis, A.
2015-01-20
In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events, and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. Here, this paper investigates the viability of high order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.
Implicit integration methods for dislocation dynamics
Gardner, D. J.; Woodward, C. S.; Reynolds, D. R.; Hommes, G.; Aubry, S.; Arsenlis, A.
2015-01-20
In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events, and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. Here, this paper investigates the viability of high order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a waymore » of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.« less
Implicit integration methods for dislocation dynamics
NASA Astrophysics Data System (ADS)
Gardner, D. J.; Woodward, C. S.; Reynolds, D. R.; Hommes, G.; Aubry, S.; Arsenlis, A.
2015-03-01
In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. This paper investigates the viability of high-order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.
Parallel implicit unstructured grid Euler solvers
NASA Technical Reports Server (NTRS)
Venkatakrishnan, V.
1994-01-01
A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on an MIMD (multiple instruction/multiple data stream) parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed, one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the iPSC/860.
High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs
NASA Technical Reports Server (NTRS)
Nielsen, Tanner B.; Carpenter, Mark H.; Fisher, Travis C.; Frankel, Steven H.
2014-01-01
This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.
NASA Astrophysics Data System (ADS)
Borazjani, Iman; Asgharzadeh, Hafez
2015-11-01
Flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates with explicit and semi-implicit schemes. Implicit schemes can be used to overcome these restrictions. However, implementing implicit solver for nonlinear equations including Navier-Stokes is not straightforward. Newton-Krylov subspace methods (NKMs) are one of the most advanced iterative methods to solve non-linear equations such as implicit descritization of the Navier-Stokes equation. The efficiency of NKMs massively depends on the Jacobian formation method, e.g., automatic differentiation is very expensive, and matrix-free methods slow down as the mesh is refined. Analytical Jacobian is inexpensive method, but derivation of analytical Jacobian for Navier-Stokes equation on staggered grid is challenging. The NKM with a novel analytical Jacobian was developed and validated against Taylor-Green vortex and pulsatile flow in a 90 degree bend. The developed method successfully handled the complex geometries such as an intracranial aneurysm with multiple overset grids, and immersed boundaries. It is shown that the NKM with an analytical Jacobian is 3 to 25 times faster than the fixed-point implicit Runge-Kutta method, and more than 100 times faster than automatic differentiation depending on the grid (size) and the flow problem. The developed methods are fully parallelized with parallel efficiency of 80-90% on the problems tested.
A non-linearly stable implicit finite element algorithm for hypersonic aerodynamics
NASA Technical Reports Server (NTRS)
Iannelli, G. S.; Baker, A. J.
1992-01-01
A generalized curvilinear coordinate Taylor weak statement implicit finite element algorithm is developed for the two-dimensional and axisymmetric compressible Navier-Stokes equations for ideal and reacting gases. For accurate hypersonic simulation, air is modeled as a mixture of five perfect gases, i.e., molecular and atomic oxygen and nitrogen as well as nitric oxide. The associated pressure is then determined via Newton solution of the classical chemical equilibrium equation system. The directional semidiscretization is achieved using an optimal metric data Galerkin finite element weak statement, on a developed 'companion conservation law system', permitting classical test and trial space definitions. Utilizing an implicit Runge-Kutta scheme, the terminal algorithm is then nonlinearly stable, and second-order accurate in space and time on arbitrary curvilinear coordinates. Subsequently, a matrix tensor product factorization procedure permits an efficient numerical linear algebra handling for large Courant numbers. For ideal- and real-gas hypersonic flows, the algorithm generates essentially nonoscillatory numerical solutions in the presence of strong detached shocks and boundary layer-inviscid flow interactions.
Solving the Bateman equations in CASMO5 using implicit ode numerical methods for stiff systems
Hykes, J. M.; Ferrer, R. M.
2013-07-01
The Bateman equations, which describe the transmutation of nuclides over time as a result of radioactive decay, absorption, and fission, are often numerically stiff. This is especially true if short-lived nuclides are included in the system. This paper describes the use of implicit numerical methods for o D Es applied to the stiff Bateman equations, specifically employing the Backward Differentiation Formulas (BDF) form of the linear multistep method. As is true in other domains, using an implicit method removes or lessens the (sometimes severe) step-length constraints by which explicit methods must abide. To gauge its accuracy and speed, the BDF method is compared to a variety of other solution methods, including Runge-Kutta explicit methods and matrix exponential methods such as the Chebyshev Rational Approximation Method (CRAM). A preliminary test case was chosen as representative of a PWR lattice depletion step and was solved with numerical libraries called from a Python front-end. The Figure of Merit (a combined measure of accuracy and efficiency) for the BDF method was nearly identical to that for CRAM, while explicit methods and other matrix exponential approximations trailed behind. The test case includes 319 nuclides, in which the shortest-lived nuclide is {sup 98}Nb with a half-life of 2.86 seconds. Finally, the BDF and CRAM methods were compared within CASMO5, where CRAM had a FOM about four times better than BDF, although the BDF implementation was not fully optimized. (authors)
NASA Astrophysics Data System (ADS)
Asgharzadeh, Hafez; Borazjani, Iman
2014-11-01
Time step-size restrictions and low convergence rates are major bottle necks for implicit solution of the Navier-Stokes in simulations involving complex geometries with moving boundaries. Newton-Krylov method (NKM) is a combination of a Newton-type method for super-linearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations, which can theoretically address both bottle necks. The efficiency of this method vastly depends on the Jacobian forming scheme e.g. automatic differentiation is very expensive and Jacobian-free methods slow down as the mesh is refined. A novel, computationally efficient analytical Jacobian for NKM was developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered curvilinear grids with immersed boundaries. The NKM was validated and verified against Taylor-Green vortex and pulsatile flow in a 90 degree bend and efficiently handles complex geometries such as an intracranial aneurysm with multiple overset grids, pulsatile inlet flow and immersed boundaries. The NKM method is shown to be more efficient than the semi-implicit Runge-Kutta methods and Jabobian-free Newton-Krylov methods. We believe NKM can be applied to many CFD techniques to decrease the computational cost. This work was supported partly by the NIH Grant R03EB014860, and the computational resources were partly provided by Center for Computational Research (CCR) at University at Buffalo.
A variational implementation of the implicit particle filter for the shallow water equations
NASA Astrophysics Data System (ADS)
Souopgui, I.; Morzfeld, M.; Hussaini, M.; Chorin, A. J.
2013-12-01
The estimation of initial conditions for shallow water equations is a well known test problem for operational data assimilation techniques. The state-of-the-art approach to this problem is the variational method (4D-Var), i.e. the computation of the mode of the posterior probability density function (pdf) via the adjoint technique. We add a sampling step to the variational method, thus turning a computation of the conditional mode (a biased estimator) into a computation of the conditional mean (the minimum least square error estimator). Our implementation relies on implicit sampling, which is a Monte Carlo (MC) sampling scheme. The idea in implicit sampling is to first search for the high-probability region of the posterior pdf and then to find samples in this region. Because the samples are concentrated in the high-probability region, fewer samples are required than with competing MC schemes and, thus, implicit sampling can be more efficient than other MC schemes. The search for the high-probability region can be done via a numerical minimization that is very similar to the minimization in 4D-Var. Here, we use existing 4D-Var code to implement the implicit sampling scheme. Once the minimization problem is solved, we obtain samples by solving algebraic equations with a random right-hand-side. These equations can be solved efficiently, so that the additional cost of our approach, compared to 4D-Var, is small. We present numerical experiments to demonstrate the applicability and efficiency of our approach. These numerical experiments mimic physical experiments done with the CORIOLIS turntable in Grenoble (France), which are used to study the drift of a vortex. In particular we consider shallow water equations on a square domain (2.5m x 2.5m) with open boundary conditions and discretize the equations with finite differences on a staggered grid of size 256 x 256 and a fourth order Runge-Kutta time integrator. Our goal is to estimate the initial state (velocities and
IMEX-a : an adaptive, fifth order implicit-explicit integration scheme.
Brake, Matthew Robert
2013-05-01
This report presents an efficient and accurate method for integrating a system of ordinary differential equations, particularly those arising from a spatial discretization of partially differential equations. The algorithm developed, termed the IMEX a algorithm, belongs to a class of algorithms known as implicit-explicit (IMEX) methods. The explicit step is based on a fifth order Runge-Kutta explicit step known as the Dormand-Prince algorithm, which adaptively modifies the time step by calculating the error relative to a fourth order estimation. The implicit step, which follows the explicit step, is based on a backward Euler method, a special case of the generalized trapezoidal method. Reasons for choosing both of these methods, along with the algorithm development are presented. In applications that have less stringent accuracy requirements, several other methods are available through the IMEX a toolbox, each of which simplify the fifth order Dormand-Prince explicit step: the third order Bogacki-Shampine method, the second order Midpoint method, and the first order Euler method. The performance of the algorithm is evaluated on to examples. First, a two pawl system with contact is modeled. Results predicted by the IMEX a algorithm are compared to those predicted by six widely used integration schemes. The IMEX a algorithm is demonstrated to be significantly faster (by up to an order of magnitude) and at least as accurate as all of the other methods considered. A second example, an acoustic standing wave, is presented in order to assess the accuracy of the IMEX a algorithm. Finally, sample code is given in order to demonstrate the implementation of the proposed algorithm.
Chen, Zhaoxia; Li, Juan; Zhang, Ruqiang; You, Xiong
2015-01-01
Oscillation is one of the most important phenomena in the chemical reaction systems in living cells. The general purpose simulation algorithms fail to take into account this special character and produce unsatisfying results. In order to enhance the accuracy of the integrator, the second-order derivative is incorporated in the scheme. The oscillatory feature of the solution is captured by the integrators with an exponential fitting property. Three practical exponentially fitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRK methods, the two-gene system with cross-regulation and the circadian oscillation of the period protein in Drosophila are simulated. Each EFTDRK method has the best fitting frequency which minimizes the global error. The numerical results show that the new EFTDRK methods are more accurate and more efficient than their prototype TDRK methods or RK methods of the same order and the traditional exponentially fitted RK method in the literature. PMID:26633991
IMEX Runge-Kutta Schemes and Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation
NASA Astrophysics Data System (ADS)
Boscarino, S.; Pareschi, L.; Russo, G.
2009-09-01
Hyperbolic system of conservation laws often have relaxation terms that, under a suitable scaling, lead to a reduced system of parabolic or hyperbolic type. The development of numerical methods to solve systems of this form his an active area of research. These systems in addition to the stiff relaxation term have the convection term stiff too. In this paper we will mainly concentrate on the study of the stiff regime. In fact in this stiff regime most of the popular methods for the solution of these system fail to capture the correct behavior of the relaxation limit unless the small relaxation rate is numericaly resolved. We will show how to overcome this difficulties and how to construct numerical schemes with the correct asymnptotic limit, i.e., the correct zero-relaxation limit should be preserved at a discrete level.
Explicit Runge-Kutta method with trigonometrically-fitted for solving first order ODEs
NASA Astrophysics Data System (ADS)
Fawzi, Firas Adel; Senu, N.; Ismail, F.; Majid, Z. A.
2016-06-01
In this note, an explicit trigonometrically-fitted (RK) method is developed to determine the approximate solution of the first-order IVPs with oscillatory solution. The proposed method solves first order ODEs by first converting the second order ODEs to an equivalent first order; which is based on the RK method of order four. The numerical experiment performed shows the efficacy of our newly developed method.
Analysis of numerical stability and amplification matrices: Fourth-order Runge-Kutta methods
NASA Technical Reports Server (NTRS)
Kennedy, E. W.
1979-01-01
Amplification matrices, numerical kernels, stable, and exponentially stable numerical solutions are examined. The various techniques involved in these concepts are applied to certain systems that have Jordan forms, which are nondiagonal, with particular interest in the case of imaginary or zero eigenvalues.
Chen, Zhaoxia; Li, Juan; Zhang, Ruqiang; You, Xiong
2015-01-01
Oscillation is one of the most important phenomena in the chemical reaction systems in living cells. The general purpose simulation algorithms fail to take into account this special character and produce unsatisfying results. In order to enhance the accuracy of the integrator, the second-order derivative is incorporated in the scheme. The oscillatory feature of the solution is captured by the integrators with an exponential fitting property. Three practical exponentially fitted TDRK (EFTDRK) methods are derived. To test the effectiveness of the new EFTDRK methods, the two-gene system with cross-regulation and the circadian oscillation of the period protein in Drosophila are simulated. Each EFTDRK method has the best fitting frequency which minimizes the global error. The numerical results show that the new EFTDRK methods are more accurate and more efficient than their prototype TDRK methods or RK methods of the same order and the traditional exponentially fitted RK method in the literature. PMID:26633991
NASA Astrophysics Data System (ADS)
Baird, Henry S.; Bentley, Jon L.
2004-12-01
We propose a design methodology for "implicit" CAPTCHAs to relieve drawbacks of present technology. CAPTCHAs are tests administered automatically over networks that can distinguish between people and machines and thus protect web services from abuse by programs masquerading as human users. All existing CAPTCHAs' challenges require a significant conscious effort by the person answering them -- e.g. reading and typing a nonsense word -- whereas implicit CAPTCHAs may require as little as a single click. Many CAPTCHAs distract and interrupt users, since the challenge is perceived as an irrelevant intrusion; implicit CAPTCHAs can be woven into the expected sequence of browsing using cues tailored to the site. Most existing CAPTCHAs are vulnerable to "farming-out" attacks in which challenges are passed to a networked community of human readers; by contrast, implicit CAPTCHAs are not "fungible" (in the sense of easily answerable in isolation) since they are meaningful only in the specific context of the website that is protected. Many existing CAPTCHAs irritate or threaten users since they are obviously tests of skill: implicit CAPTCHAs appear to be elementary and inevitable acts of browsing. It can often be difficult to detect when CAPTCHAs are under attack: implicit CAPTCHAs can be designed so that certain failure modes are correlated with failed bot attacks. We illustrate these design principles with examples.
NASA Astrophysics Data System (ADS)
Baird, Henry S.; Bentley, Jon L.
2005-01-01
We propose a design methodology for "implicit" CAPTCHAs to relieve drawbacks of present technology. CAPTCHAs are tests administered automatically over networks that can distinguish between people and machines and thus protect web services from abuse by programs masquerading as human users. All existing CAPTCHAs' challenges require a significant conscious effort by the person answering them -- e.g. reading and typing a nonsense word -- whereas implicit CAPTCHAs may require as little as a single click. Many CAPTCHAs distract and interrupt users, since the challenge is perceived as an irrelevant intrusion; implicit CAPTCHAs can be woven into the expected sequence of browsing using cues tailored to the site. Most existing CAPTCHAs are vulnerable to "farming-out" attacks in which challenges are passed to a networked community of human readers; by contrast, implicit CAPTCHAs are not "fungible" (in the sense of easily answerable in isolation) since they are meaningful only in the specific context of the website that is protected. Many existing CAPTCHAs irritate or threaten users since they are obviously tests of skill: implicit CAPTCHAs appear to be elementary and inevitable acts of browsing. It can often be difficult to detect when CAPTCHAs are under attack: implicit CAPTCHAs can be designed so that certain failure modes are correlated with failed bot attacks. We illustrate these design principles with examples.
Quan, Quan; Zhu, Huangjun; Liu, Si-Yuan; Fei, Shao-Ming; Fan, Heng; Yang, Wen-Li
2016-01-01
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality. PMID:26911250
NASA Astrophysics Data System (ADS)
Quan, Quan; Zhu, Huangjun; Liu, Si-Yuan; Fei, Shao-Ming; Fan, Heng; Yang, Wen-Li
2016-02-01
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality.
Steering Bell-diagonal states.
Quan, Quan; Zhu, Huangjun; Liu, Si-Yuan; Fei, Shao-Ming; Fan, Heng; Yang, Wen-Li
2016-01-01
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality. PMID:26911250
Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)
Robert Nourgaliev; Theo Theofanous; HyeongKae Park; Vincent Mousseau; Dana Knoll
2008-01-01
In recent work (Nourgaliev, Liou, Theofanous, JCP in press) we demonstrated that numerical simulations of interfacial flows in the presence of strong shear must be cast in dynamically sharp terms (sharp interface treatment or SIM), and that moreover they must meet stringent resolution requirements (i.e., resolving the critical layer). The present work is an outgrowth of that work aiming to overcome consequent limitations on the temporal treatment, which become still more severe in the presence of phase change. The key is to avoid operator splitting between interface motion, fluid convection, viscous/heat diffusion and reactions; instead treating all these non-linear operators fully-coupled within a Newton iteration scheme. To this end, the SIM’s cut-cell meshing is combined with the high-orderaccurate implicit Runge-Kutta and the “recovery” Discontinuous Galerkin methods along with a Jacobian-free, Krylov subspace iteration algorithm and its physics-based preconditioning. In particular, the interfacial geometry (i.e., marker’s positions and volumes of cut cells) is a part of the Newton-Krylov solution vector, so that the interface dynamics and fluid motions are fully-(non-linearly)-coupled. We show that our method is: (a) robust (L-stable) and efficient, allowing to step over stability time steps at will while maintaining high-(up to the 5th)-order temporal accuracy; (b) fully conservative, even near multimaterial contacts, without any adverse consequences (pressure/velocity oscillations); and (c) highorder-accurate in spatial discretization (demonstrated here up to the 12th-order for smoothin-the-bulk-fluid flows), capturing interfacial jumps sharply, within one cell. Performance is illustrated with a variety of test problems, including low-Mach-number “manufactured” solutions, shock dynamics/tracking with slow dynamic time scales, and multi-fluid, highspeed shock-tube problems. We briefly discuss preconditioning, and we introduce two physics
Numerical experiments with an implicit particle filter for the shallow water equations
NASA Astrophysics Data System (ADS)
Souopgui, I.; Chorin, A. J.; Hussaini, M.
2012-12-01
The estimation of initial conditions for the shallow water equations for a given set of later data is a well known test problem for data assimilation codes. A popular approach to this problem is the variational method (4D-Var), i.e. the computation of the mode of the posterior probability density function (pdf) via the adjoint technique. Here, we improve on 4D-Var by computing the conditional mean (the minimum least square error estimator) rather than the mode (a biased estimator) and we do so with implicit sampling, a Monte Carlo (MC) importance sampling method. The idea in implicit sampling is to first search for the high-probability region of the posterior pdf and then to find samples in this region. Because the samples are concentrated in the high-probability region, fewer samples are required than with competing MC schemes. The search for the high-probability region can be implemented by a minimization that is very similar to the minimization in 4D-Var, and we make use of a 4D-Var code in our implementation. The samples are obtained by solving algebraic equations with a random right-hand-side. These equations can be solved efficiently, so that the additional cost of our approach, compared to traditional 4D-Var, is small. The long-term goal is to assimilate experimental data, obtained with the CORIOLIS turntable in Grenoble (France), to study the drift of a vortex. We present results from numerical twin experiments as a first step towards our long-term goal. We discretize the shallow water equations on a square domain (2.5m× 2.5m) using finite differences on a staggered grid of size 28× 28 and a fourth order Runge-Kutta. We assume open boundary conditions and estimate the initial state (velocities and surface height) given noisy observations of the state. We solve the optimization problem using a 4D-Var code that relies on a L-BFGS method; the random algebraic equations are solved with random maps, i.e. we look for solutions in given, but random, directions
Nondestructive identification of the Bell diagonal state
Jin Jiasen; Yu Changshui; Song Heshan
2011-03-15
We propose a scheme for identifying an unknown Bell diagonal state. In our scheme the measurements are performed on the probe qubits instead of the Bell diagonal state. The distinct advantage is that the quantum state of the evolved Bell diagonal state ensemble plus probe states will still collapse on the original Bell diagonal state ensemble after the measurement on probe states; i.e., our identification is quantum state nondestructive. How to realize our scheme in the framework of cavity electrodynamics is also shown.
The delayed coupling method: An algorithm for solving banded diagonal matrix problems in parallel
Mattor, N.; Williams, T.J.; Hewett, D.W.; Dimits, A.M.
1997-09-01
We present a new algorithm for solving banded diagonal matrix problems efficiently on distributed-memory parallel computers, designed originally for use in dynamic alternating-direction implicit partial differential equation solvers. The algorithm optimizes efficiency with respect to the number of numerical operations and to the amount of interprocessor communication. This is called the ``delayed coupling method`` because the communication is deferred until needed. We focus here on tridiagonal and periodic tridiagonal systems.
NASA Technical Reports Server (NTRS)
Abarbanel, Saul; Gottlieb, David; Carpenter, Mark H.
1994-01-01
It has been previously shown that the temporal integration of hyperbolic partial differential equations (PDE's) may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic (PDE's) (linear and non-linear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.
NASA Technical Reports Server (NTRS)
Rogers, S. E.; Kwak, D.; Chang, J. L. C.
1986-01-01
The method of pseudocompressibility has been shown to be an efficient method for obtaining a steady-state solution to the incompressible Navier-Stokes equations. Recent improvements to this method include the use of a diagonal scheme for the inversion of the equations at each iteration. The necessary transformations have been derived for the pseudocompressibility equations in generalized coordinates. The diagonal algorithm reduces the computing time necessary to obtain a steady-state solution by a factor of nearly three. Implicit viscous terms are maintained in the equations, and it has become possible to use fourth-order implicit dissipation. The steady-state solution is unchanged by the approximations resulting from the diagonalization of the equations. Computed results for flow over a two-dimensional backward-facing step and a three-dimensional cylinder mounted normal to a flat plate are presented for both the old and new algorithms. The accuracy and computing efficiency of these algorithms are compared.
Parallelization of implicit finite difference schemes in computational fluid dynamics
NASA Technical Reports Server (NTRS)
Decker, Naomi H.; Naik, Vijay K.; Nicoules, Michel
1990-01-01
Implicit finite difference schemes are often the preferred numerical schemes in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such iterative methods are considerably more difficult and non-intuitive. The parallelization of the implicit schemes that are used for solving the Euler and the thin layer Navier-Stokes equations and that require inversions of large linear systems in the form of block tri-diagonal and/or block penta-diagonal matrices is discussed. Three-dimensional cases are emphasized and schemes that minimize the total execution time are presented. Partitioning and scheduling schemes for alleviating the effects of the global data dependencies are described. An analysis of the communication and the computation aspects of these methods is presented. The effect of the boundary conditions on the parallel schemes is also discussed.
Implicit solvers for unstructured meshes
NASA Technical Reports Server (NTRS)
Venkatakrishnan, V.; Mavriplis, Dimitri J.
1991-01-01
Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.
23. INCLINED END POST / VERTICAL / DIAGONAL / PORTAL ...
23. INCLINED END POST / VERTICAL / DIAGONAL / PORTAL BRACING DETAIL. VIEW TO SOUTHEAST. - Abraham Lincoln Memorial Bridge, Spanning Missouri River on Highway 30 between Nebraska & Iowa, Blair, Washington County, NE
30. BEARING SHOE / VERTICAL / DIAGONAL / UPPER AND ...
30. BEARING SHOE / VERTICAL / DIAGONAL / UPPER AND LOWER CHORD DETAIL OF DECK TRUSS. VIEW TO NORTHEAST. - Abraham Lincoln Memorial Bridge, Spanning Missouri River on Highway 30 between Nebraska & Iowa, Blair, Washington County, NE
Energy patterns in coupled α-helix protein chains with diagonal and off-diagonal couplings
NASA Astrophysics Data System (ADS)
Tabi, C. B.; Ondoua, R. Y.; Ekobena Fouda, H. P.; Kofané, T. C.
2016-07-01
We introduce off-diagonal effects in the three-stranded model of α-helix chains, which bring about additional nonlinear terms to enhance the way energy spreads among the coupled spines. This is analyzed through the modulational instability theory. The linear stability analysis of plane wave solutions is performed and the competitive effects of diagonal and off-diagonal interactions are studied, followed by direct numerical simulations. Some features of the obtained solitonic structures are discussed.
Diagonal chromatography to study plant protein modifications.
Walton, Alan; Tsiatsiani, Liana; Jacques, Silke; Stes, Elisabeth; Messens, Joris; Van Breusegem, Frank; Goormachtig, Sofie; Gevaert, Kris
2016-08-01
An interesting asset of diagonal chromatography, which we have introduced for contemporary proteome research, is its high versatility concerning proteomic applications. Indeed, the peptide modification or sorting step that is required between consecutive peptide separations can easily be altered and thereby allows for the enrichment of specific, though different types of peptides. Here, we focus on the application of diagonal chromatography for the study of modifications of plant proteins. In particular, we show how diagonal chromatography allows for studying proteins processed by proteases, protein ubiquitination, and the oxidation of protein-bound methionines. We discuss the actual sorting steps needed for each of these applications and the obtained results. This article is part of a Special Issue entitled: Plant Proteomics- a bridge between fundamental processes and crop production, edited by Dr. Hans-Peter Mock. PMID:26772901
Dual Diagonalization of Reactive Transport Equations
NASA Astrophysics Data System (ADS)
Yeh, G.; Tsai, C.
2013-12-01
One solves a system of species transport equations in the primitive approach to reactive transport modeling. This approach is not able to decouple equilibrium reaction rates from species concentrations. This problem has been overcome with the approach to diagonalizing the reaction matrix since mid 1990's, which yields the same number of transport equations for reaction-extents. In the diagonalization approach, first, a subset of reaction- extent transport equations is solved for concentrations of components and kinetic-variables. Then, the component, kinetic-variable, and mass action equations are solved for all species concentrations. Finally, the equilibrium reaction rates are posterior computed. The difficulty in this approach is that the solution of species concentrations in the second step is a stiff problem when the concentrations of master species are small compared to those of equilibrium species. To overcome the problem of stiffness, we propose a dual diagonalization approach. Here, a second diagonalization is performed on the decomposed unit matrix to yield species concentrations, each as a linear function of reaction extents. In this dual diagonalization approach, four steps are needed to complete the modeling. First, component and kinetic-variable transport equations are solved for the concentrations of components (a subset of reaction-extents) and kinetic-variables (another subset of reaction-extents). Second, the set of mass action equations written in terms of reaction extents are solved for equilibrium-variables (yet another subset of reaction-extents). Third, species concentrations are posterior obtained by solving the set of linear equations defining reaction-extents. Fourth, equilibrium rates are posterior calculated with transport equations for equilibrium-variables. Several example problems will be used to demonstrate the efficiency of this approach. Keywords: Reactive Transport, Reaction-Extent, Component, Kinetic-Variable, Equilibrium
Multigrid calculation of three-dimensional turbomachinery flows
NASA Technical Reports Server (NTRS)
Caughey, David A.
1989-01-01
Research was performed in the general area of computational aerodynamics, with particular emphasis on the development of efficient techniques for the solution of the Euler and Navier-Stokes equations for transonic flows through the complex blade passages associated with turbomachines. In particular, multigrid methods were developed, using both explicit and implicit time-stepping schemes as smoothing algorithms. The specific accomplishments of the research have included: (1) the development of an explicit multigrid method to solve the Euler equations for three-dimensional turbomachinery flows based upon the multigrid implementation of Jameson's explicit Runge-Kutta scheme (Jameson 1983); (2) the development of an implicit multigrid scheme for the three-dimensional Euler equations based upon lower-upper factorization; (3) the development of a multigrid scheme using a diagonalized alternating direction implicit (ADI) algorithm; (4) the extension of the diagonalized ADI multigrid method to solve the Euler equations of inviscid flow for three-dimensional turbomachinery flows; and also (5) the extension of the diagonalized ADI multigrid scheme to solve the Reynolds-averaged Navier-Stokes equations for two-dimensional turbomachinery flows.
Implicit and semi-implicit schemes: Algorithms
NASA Astrophysics Data System (ADS)
Keppens, R.; Tóth, G.; Botchev, M. A.; van der Ploeg, A.
1999-06-01
This study formulates general guidelines to extend an explicit code with a great variety of implicit and semi-implicit time integration schemes. The discussion is based on their specific implementation in the Versatile Advection Code, which is a general purpose software package for solving systems of non-linear hyperbolic (and/or parabolic) partial differential equations, using standard high resolution shock capturing schemes. For all combinations of explicit high resolution schemes with implicit and semi-implicit treatments, it is shown how second-order spatial and temporal accuracy for the smooth part of the solutions can be maintained. Strategies to obtain steady state and time accurate solutions implicitly are discussed. The implicit and semi-implicit schemes require the solution of large linear systems containing the Jacobian matrix. The Jacobian matrix itself is calculated numerically to ensure the generality of this implementation. Three options are discussed in terms of applicability, storage requirements and computational efficiency. One option is the easily implemented matrix-free approach, but the Jacobian matrix can also be calculated by using a general grid masking algorithm, or by an efficient implementation for a specific Lax-Friedrich-type total variation diminishing (TVD) spatial discretization. The choice of the linear solver depends on the dimensionality of the problem. In one dimension, a direct block tridiagonal solver can be applied, while in more than one spatial dimension, a conjugate gradient (CG)-type iterative solver is used. For advection-dominated problems, preconditioning is needed to accelerate the convergence of the iterative schemes. The modified block incomplete LU-preconditioner is implemented, which performs very well. Examples from two-dimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outflow and recover the complex magnetohydrodynamic bow shock flow in the switch-on regime
Diagonal Pade approximations for initial value problems
Reusch, M.F.; Ratzan, L.; Pomphrey, N.; Park, W.
1987-06-01
Diagonal Pade approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained. 20 refs., 3 figs., 1 tab.
Bell-state diagonal-entanglement witnesses
Jafarizadeh, M. A.; Rezaee, M.; Seyed Yagoobi, S. K. A.
2005-12-15
It has been shown that finding generic Bell-state diagonal-entanglement witnesses for d{sub 1}xd{sub 2}x{center_dot}{center_dot}{center_dot}xd{sub n} systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, 2xN and 3x3 systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a 3x3 system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable 3x3 Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable 3x3 Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some 3x3 bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.
Development of a variable time-step transient NEW code: SPANDEX
Aviles, B.N. )
1993-01-01
This paper describes a three-dimensional, variable time-step transient multigroup diffusion theory code, SPANDEX (space-time nodal expansion method). SPANDEX is based on the static nodal expansion method (NEM) code, NODEX (Ref. 1), and employs a nonlinear algorithm and a fifth-order expansion of the transverse-integrated fluxes. The time integration scheme in SPANDEX is a fourth-order implicit generalized Runge-Kutta method (GRK) with on-line error control and variable time-step selection. This Runge-Kutta method has been applied previously to point kinetics and one-dimensional finite difference transient analysis. This paper describes the application of the Runge-Kutta method to three-dimensional reactor transient analysis in a multigroup NEM code.
Numerical precision of the solution to the running-coupling Balitsky-Kovchegov equation
NASA Astrophysics Data System (ADS)
Matas, Marek; Cepila, Jan; Guillermo Contreras Nuno, Jesus
2016-03-01
We use the running coupling Balitsky-Kovchegov (rcBK) equation to study the rapidity dependence of saturation in inclusive HERA data and we discuss the behaviour of its numerical solution. The rcBK equation has been solved using Runge-Kutta methods. The influence of the parameters implicit in the numerical evolution has been studied. They include, among others, the order of the Runge-Kutta evolution, the size of the different grids and the step in the numerical evolution. Some suggestions on the minimum value of these parameters are put forward.
Langdon, A.B.
1985-03-03
Implicit time integration methods have been used extensively in numerical modelling of slowly varying phenomena in systems that also support rapid variation. Examples include diffusion, hydrodynamics and reaction kinetics. This article discussed implementation of implicit time integration in plasma codes of the ''particle-in-cell'' family, and the benefits to be gained.
NASA Technical Reports Server (NTRS)
Skliar, M.; Ramirez, W. F.
1997-01-01
For an implicitly defined discrete system, a new algorithm for Kalman filtering is developed and an efficient numerical implementation scheme is proposed. Unlike the traditional explicit approach, the implicit filter can be readily applied to ill-conditioned systems and allows for generalization to descriptor systems. The implementation of the implicit filter depends on the solution of the congruence matrix equation (A1)(Px)(AT1) = Py. We develop a general iterative method for the solution of this equation, and prove necessary and sufficient conditions for convergence. It is shown that when the system matrices of an implicit system are sparse, the implicit Kalman filter requires significantly less computer time and storage to implement as compared to the traditional explicit Kalman filter. Simulation results are presented to illustrate and substantiate the theoretical developments.
Diagonal-transition quantum cascade detector
Reininger, Peter Schwarz, Benedikt; Detz, Hermann; MacFarland, Don; Zederbauer, Tobias; Andrews, Aaron Maxwell; Schrenk, Werner; Strasser, Gottfried; Baumgartner, Oskar; Kosina, Hans
2014-09-01
We demonstrate the concept of diagonal transitions for quantum cascade detectors (QCD). Different to standard, vertical QCDs, here the active transition takes place between two energy levels in adjacent wells. Such a scheme has versatile advantages. Diagonal transitions generally yield a higher extraction efficiency and a higher resistance than vertical transitions. This leads to an improved overall performance, although the absorption strength of the active transition is smaller. Since the extraction is not based on resonant tunneling, the design is more robust, with respect to deviations from the nominal structure. In a first approach, a peak responsivity of 16.9 mA/W could be achieved, which is an improvement to the highest shown responsivity of a QCD for a wavelength of 8 μm at room-temperature by almost an order of magnitude.
Diagonal multisoliton matrix elements in finite volume
NASA Astrophysics Data System (ADS)
Pálmai, T.; Takács, G.
2013-02-01
We consider diagonal matrix elements of local operators between multisoliton states in finite volume in the sine-Gordon model and formulate a conjecture regarding their finite size dependence which is valid up to corrections exponential in the volume. This conjecture extends the results of Pozsgay and Takács which were only valid for diagonal scattering. In order to test the conjecture, we implement a numerical renormalization group improved truncated conformal space approach. The numerical comparisons confirm the conjecture, which is expected to be valid for general integrable field theories. The conjectured formula can be used to evaluate finite temperature one-point and two-point functions using recently developed methods.
Time integration algorithms for the two-dimensional Euler equations on unstructured meshes
NASA Technical Reports Server (NTRS)
Slack, David C.; Whitaker, D. L.; Walters, Robert W.
1994-01-01
Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.
Awareness of Implicit Attitudes
Hahn, Adam; Judd, Charles M.; Hirsh, Holen K.; Blair, Irene V.
2013-01-01
Research on implicit attitudes has raised questions about how well people know their own attitudes. Most research on this question has focused on the correspondence between measures of implicit attitudes and measures of explicit attitudes, with low correspondence interpreted as showing that people have little awareness of their implicit attitudes. We took a different approach and directly asked participants to predict their results on upcoming IAT measures of implicit attitudes toward five different social groups. We found that participants were surprisingly accurate in their predictions. Across four studies, predictions were accurate regardless of whether implicit attitudes were described as true attitudes or culturally learned associations (Studies 1 and 2), regardless of whether predictions were made as specific response patterns (Study 1) or as conceptual responses (Studies 2–4), and regardless of how much experience or explanation participants received before making their predictions (Study 4). Study 3 further suggested that participants’ predictions reflected unique insight into their own implicit responses, beyond intuitions about how people in general might respond. Prediction accuracy occurred despite generally low correspondence between implicit and explicit measures of attitudes, as found in prior research. All together, the research findings cast doubt on the belief that attitudes or evaluations measured by the IAT necessarily reflect unconscious attitudes. PMID:24294868
Investigations on the Incompletely Developed Plane Diagonal-Tension Field
NASA Technical Reports Server (NTRS)
Kuhn, Paul
1940-01-01
This report presents the results of an investigation on the incompletely developed diagonal-tension field. Actual diagonal-tension beams work in an intermediate stage between pure shear and pure diagonal tension; the theory developed by wagner for diagonal tension is not directly applicable. The first part of the paper reviews the most essential items of the theory of pure diagonal tension as well as previous attempts to formulate a theory of incomplete diagonal tension. The second part of the paper describes strain measurement made by the N. A. C. A. to obtain the necessary coefficients for the proposed theory. The third part of the paper discusses the stress analysis of diagonal-tension beams by means of the proposed theory.
Simultaneous diagonal and off-diagonal order in the Bose-Hubbard Hamiltonian
Scalettar, R.T.; Batrouni, G.G.; Kampf, A.P.; Zimanyi, G.T.
1995-04-01
The Bose-Hubbard model exhibits a rich phase diagram consisting both of insulating regimes where diagonal long-range (solid) order dominates as well as conducting regimes where off-diagonal long-range order (superfluidity) is present. In this paper we describe the results of quantum Monte Carlo calculations of the phase diagram, both for the hard- and soft-core cases, with a particular focus on the possibility of simultaneous superfluid and solid order. We also discuss the appearance of phase separation in the model. The simulations are compared with analytic calculations of the phase diagram and spin-wave dispersion.
Electromagnetic direct implicit PIC simulation
Langdon, A.B.
1983-03-29
Interesting modelling of intense electron flow has been done with implicit particle-in-cell simulation codes. In this report, the direct implicit PIC simulation approach is applied to simulations that include full electromagnetic fields. The resulting algorithm offers advantages relative to moment implicit electromagnetic algorithms and may help in our quest for robust and simpler implicit codes.
LU decompositions of generalized diagonally dominant matrices
Funderlic, R.E.; Neumann, M.; Plemmons, R.J.
1982-02-01
Using the simple vehicle of M-matrices, the existence and stability of LU decompositions of matrices A which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination on A are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equations Ax = 0, arising in Markov queuing networks, input-output models in economics and compartmental systems, where A or -A is an irreducible, singular M-matrix.
Universal single level implicit algorithm for gasdynamics
NASA Technical Reports Server (NTRS)
Lombard, C. K.; Venkatapthy, E.
1984-01-01
A single level effectively explicit implicit algorithm for gasdynamics is presented. The method meets all the requirements for unconditionally stable global iteration over flows with mixed supersonic and supersonic zones including blunt body flow and boundary layer flows with strong interaction and streamwise separation. For hyperbolic (supersonic flow) regions the method is automatically equivalent to contemporary space marching methods. For elliptic (subsonic flow) regions, rapid convergence is facilitated by alternating direction solution sweeps which bring both sets of eigenvectors and the influence of both boundaries of a coordinate line equally into play. Point by point updating of the data with local iteration on the solution procedure at each spatial step as the sweeps progress not only renders the method single level in storage but, also, improves nonlinear accuracy to accelerate convergence by an order of magnitude over related two level linearized implicit methods. The method derives robust stability from the combination of an eigenvector split upwind difference method (CSCM) with diagonally dominant ADI(DDADI) approximate factorization and computed characteristic boundary approximations.
An Ancient Egyptian Diagonal Star Table in Mallawi, Egypt
NASA Astrophysics Data System (ADS)
Symons, Sarah; Cockcroft, Robert
2013-11-01
A coffin belonging to an Egyptian Middle Kingdom official Hor-em-hetepu, on public display in the Mallawi Monuments Museum, Egypt, contains a previously-unpublished diagonal star table (or "diagonal star clock"). This table adds to the other twenty-four examples of this type of astronomical record or calendar from around 2100 B.C. The table displays a regular diagonal pattern of decan (star or asterism) names, with some interesting points of content, epigraphy, and typology.
Implicit Attitudes in Prosopagnosia
Knutson, Kristine M.; DeTucci, Karen A.; Grafman, Jordan
2011-01-01
We studied a male with acquired prosopagnosia using a battery of implicit association tests (IATs) to investigate whether observing faces varying by social category would activate the patient’s implicit social biases. We also asked him to categorize faces explicitly by race, gender, and political party. The patient, G.B., was marginally slower to categorize black compared to white faces. He showed congruency effects in the race and celebrity IATs, but not in the gender or political IATs. These results indicate that G.B. possesses an implicit social sensitivity to certain facial stimuli despite an inability to overtly recognize familiar faces. The results demonstrate that social biases can be retrieved based on facial stimuli via pathways bypassing the fusiform gyri. Thus the IAT effect can be added to the list of covert recognition effects found in prosopagnosia. PMID:21414330
Implicit Spacecraft Gyro Calibration
NASA Technical Reports Server (NTRS)
Harman, Richard; Bar-Itzhack, Itzhack Y.
2003-01-01
This paper presents an implicit algorithm for spacecraft onboard instrument calibration, particularly to onboard gyro calibration. This work is an extension of previous work that was done where an explicit gyro calibration algorithm was applied to the AQUA spacecraft gyros. The algorithm presented in this paper was tested using simulated data and real data that were downloaded from the Microwave Anisotropy Probe (MAP) spacecraft. The calibration tests gave very good results. A comparison between the use of the implicit calibration algorithm used here with the explicit algorithm used for AQUA spacecraft indicates that both provide an excellent estimation of the gyro calibration parameters with similar accuracies.
Droit-Volet, Sylvie
2016-01-01
This study examined the effects of emotion on implicit timing. In the implicit timing task used, the participants did not receive any temporal instructions. Instead they were simply asked and trained to press a key as quickly as possible after a stimulus (response stimulus) that was separated from a preceding stimulus by a given temporal interval (reference interval duration). However, in the testing phase, the interval duration was the reference interval duration or a shorter or longer interval duration. In addition, the participants attended two sessions: a first baseline session in which no stimulus was presented during the inter-stimulus intervals, and a second emotional session in which emotional facial expressions (angry, neutral and sad facial expressions) were presented during these intervals. Results showed faster RTs for interval durations close to the reference duration in both the baseline and the emotional conditions and yielded a U-shaped curve. This suggests that implicit processing of time persists in emotional contexts. In addition, the RT was faster for the facial expressions of anger than for those of neutrality and sadness. However, the U-shaped RT curve did not peak clearly at a shorter interval duration for the angry than for the other facial expressions. This lack of time distortion in an implicit timing task in response to arousing emotional stimuli questions the idea of an automatic speeding-up of the interval clock system involved in the representation of time. PMID:27380409
Sexual Murderers' Implicit Theories
ERIC Educational Resources Information Center
Beech, Anthony; Fisher, Dawn; Ward, Tony
2005-01-01
Interviews with 28 sexual murderers were subjected to grounded theory analysis. Five implicit theories (ITs) were identified: dangerous world, male sex drive is uncontrollable, entitlement, women as sexual objects, and women as unknowable. These ITs were found to be identical to those identified in the literature as being present in rapists. The…
Covariant Perturbation Expansion of Off-Diagonal Heat Kernel
NASA Astrophysics Data System (ADS)
Gou, Yu-Zi; Li, Wen-Du; Zhang, Ping; Dai, Wu-Sheng
2016-07-01
Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.
Generalized coordinate Bethe ansatz for non-diagonal boundaries
NASA Astrophysics Data System (ADS)
Crampe, N.; Ragoucy, E.
2012-05-01
We compute the spectrum and the eigenstates of the open XXX model with non-diagonal (triangular) boundary matrices. Since the boundary matrices are not diagonal, the usual coordinate Bethe ansatz does not work anymore, and we use a generalization of it to solve the problem.
3. VIEW OF EAST PORTION OF LOWER DIAGONAL NO. 1 ...
3. VIEW OF EAST PORTION OF LOWER DIAGONAL NO. 1 DRAIN LOOKING TOWARDS THE CENTRAL BEND, LOOKING 2742 EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
7. VIEW OF WEAPONS DELIVERY ROAD CULVERT OF LOWER DIAGONAL ...
7. VIEW OF WEAPONS DELIVERY ROAD CULVERT OF LOWER DIAGONAL NO. 1 DRAIN, LOOKING 522 EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
6. VIEW OF WEST GATE ROAD CULVERT OF LOWER DIAGONAL ...
6. VIEW OF WEST GATE ROAD CULVERT OF LOWER DIAGONAL NO. 1 DRAIN, LOOKING 2502 EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
4. VIEW OF EAST PORTION OF LOWER DIAGONAL NO. 1 ...
4. VIEW OF EAST PORTION OF LOWER DIAGONAL NO. 1 DRAIN LOOKING TOWARDS THE CENTRAL BEND, LOOKING 270t EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
5. VIEW OF WEST GATE ROAD CULVERT OF LOWER DIAGONAL ...
5. VIEW OF WEST GATE ROAD CULVERT OF LOWER DIAGONAL NO. 1 DRAIN, LOOKING 323' EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
2. VIEW OF CENTRAL BEND OF LOWER DIAGONAL NO. 1 ...
2. VIEW OF CENTRAL BEND OF LOWER DIAGONAL NO. 1 DRAIN, LOOKING 2932 EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
1. VIEW OF WEST PORTION OF LOWER DIAGONAL NO. 1 ...
1. VIEW OF WEST PORTION OF LOWER DIAGONAL NO. 1 DRAIN LOOKING TOWARDS THE WEST GATE ROAD CULVERT, LOOKING 3052 EAST OF NORTH. - Truckee-Carson Irrigation District, Lower Diagonal No. 1 Drain, Bounded by West Gate Road & Weapons Delivery Road, Naval Air Station Fallon, Fallon, Churchill County, NV
Precision measurement of the off-diagonal hyperfine interaction
Gilbert, S.L.; Masterson, B.P.; Noecker, M.C.; Wieman, C.E.
1986-10-01
We have measured the hyperfine mixing of the 6S and 7S states of cesium using a new high-precision experimental technique. By comparing the diagonal and off-diagonal hyperfine interaction for these states, we find that a single-particle description of the states is accurate to better than 2%.
Panel Post & Diagonal Brace Joint Detail; Crossbracing Center Joint ...
Panel Post & Diagonal Brace Joint Detail; Crossbracing Center Joint Detail; Chord, Panel Post, Tie Bar, & Diagonal Brace Joint Detail; Chord, Tie Bar, & Crossbracing Joint Detail - Medora Bridge, Spanning East Fork of White River at State Route 235, Medora, Jackson County, IN
A Finite-Element Approach for Modeling Inviscid and Viscous Compressible Flows using Prismatic Grids
NASA Technical Reports Server (NTRS)
Pandya, S. A.; Hefez, M.
2000-01-01
The Galerkin finite-element method is used to solve the Euler and Navier-Stokes equations on prismatic meshes. It is shown that the prismatic grid is advantageous for correctly and efficiently capturing the boundary layers in high Reynolds number flows. It can be captured accurately because of the ability to cluster grid points normal to the body. The efficiency derives from the implicit treatment of the normal direction. To treat the normal direction implicitly, a semi-implicit Runge-Kutta time stepping scheme is developed. The semi-implicit algorithm is validated on simple geometries for inviscid and viscous flows and its convergence history is compared to that of the explicit Runge-Kutta scheme. The semi-implicit scheme is shown to be a factor of 3 to 4 faster in terms of CPU time to convergence.
Implicit Learning as an Ability
ERIC Educational Resources Information Center
Kaufman, Scott Barry; DeYoung, Caroline G.; Gray, Jeremy R.; Jimenez, Luis; Brown, Jamie; Mackintosh, Nicholas
2010-01-01
The ability to automatically and implicitly detect complex and noisy regularities in the environment is a fundamental aspect of human cognition. Despite considerable interest in implicit processes, few researchers have conceptualized implicit learning as an ability with meaningful individual differences. Instead, various researchers (e.g., Reber,…
Solving block linear systems with low-rank off-diagonal blocks is easily parallelizable
Menkov, V.
1996-12-31
An easily and efficiently parallelizable direct method is given for solving a block linear system Bx = y, where B = D + Q is the sum of a non-singular block diagonal matrix D and a matrix Q with low-rank blocks. This implicitly defines a new preconditioning method with an operation count close to the cost of calculating a matrix-vector product Qw for some w, plus at most twice the cost of calculating Qw for some w. When implemented on a parallel machine the processor utilization can be as good as that of those operations. Order estimates are given for the general case, and an implementation is compared to block SSOR preconditioning.
Numerical solution of 3D Navier-Stokes equations with upwind implicit schemes
NASA Technical Reports Server (NTRS)
Marx, Yves P.
1990-01-01
An upwind MUSCL type implicit scheme for the three-dimensional Navier-Stokes equations is presented. Comparison between different approximate Riemann solvers (Roe and Osher) are performed and the influence of the reconstructions schemes on the accuracy of the solution as well as on the convergence of the method is studied. A new limiter is introduced in order to remove the problems usually associated with non-linear upwind schemes. The implementation of a diagonal upwind implicit operator for the three-dimensional Navier-Stokes equations is also discussed. Finally the turbulence modeling is assessed. Good prediction of separated flows are demonstrated if a non-equilibrium turbulence model is used.
Iterative diagonalization for orbital optimization in natural orbital functional theory.
Piris, M; Ugalde, J M
2009-10-01
A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F. The off-diagonal elements of the latter are determined explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be determined from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calculated with the Hartree-Fock orbitals after the occupation numbers have been optimized. The method has been tested on the G2/97 set of molecules for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off-diagonal elements of F. The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. PMID:19219918
Resonant energy transfer assisted by off-diagonal coupling.
Wu, Ning; Sun, Ke-Wei; Chang, Zhe; Zhao, Yang
2012-03-28
Dynamics of resonant energy transfer of a single excitation in a molecular dimer system are studied in the simultaneous presence of diagonal and off-diagonal exciton-phonon coupling. It is found that, at given temperatures, the off-diagonal coupling can enhance both the coherence of the resonant energy transfer and the net quantity of energy transferred from an initially excited monomer to the other. Also studied is the dynamics of entanglement between the dimer system and the phonon bath as measured by the von Neumann entanglement entropy, and the inter-monomer entanglement dynamics for the excitonic system. PMID:22462880
Teleportation of an arbitrary mixture of diagonal states of multiqudit
NASA Astrophysics Data System (ADS)
Du, Qian-Hua; Lin, Xiu-Min; Chen, Zhi-Hua; Lin, Gong-Wei; Chen, Li-Bo; Gu, Yong-Jian
2008-03-01
This paper proposes a scheme to teleport an arbitrary mixture of diagonal states of multiqutrit via classical correlation and classical communication. To teleport an arbitrary mixture of diagonal states of N qutrits, N classically correlated pairs of two qutrits are used as channel. The sender (Alice) makes Fourier transform and conditional gate (i.e., XOR(3) gate) on her qutrits and does measurement in appropriate computation bases. Then she sends N ctrits to the receiver (Bob). Based on the received information, Bob performs the corresponding unitary transformation on his qutrits and obtains the teleported state. Teleportation of an arbitrary mixture of diagonal states of multiqudit is also discussed.
Block-diagonal similarity renormalization group and effective nucleon-nucleon interactions
NASA Astrophysics Data System (ADS)
Szpigel, S.; Timóteo, V. S.; Ruiz Arriola, E.
2016-04-01
We apply the block-diagonal similarity renormalization group to a simple toy-model for the nucleon-nucleon (NN) interaction in the 1 S 0 channel, aiming to analyze the complementarity between the explicit and the implicit renormalization approaches in nuclear physics. By explicit renormalization we mean the methods based on the wilsonian renormalization group in which high-energy modes above a given cutoff scale are integrated out while their effects are replaced by scale dependent effective interactions consistently generated in the process. We call implicit renormalization the usual procedure of cutoff effective theories in which the high-energy modes above the cutoff scale are simply removed and their effects are included through parametrized cutoff dependent counterterms whose strengths are fixed by fitting low-energy data. We compare the effective interactions obtained in both schemes and find a wide range of cutoff scales where they overlap. We further analyze the role played by the one-pion exchange (OPE) considering a δ-shell plus OPE representation for the NN interaction.
15. VIEW LOOKING NORTH AT END POSTS WITH DIAGONAL TENSION ...
15. VIEW LOOKING NORTH AT END POSTS WITH DIAGONAL TENSION MEMBERS. NOTE: MODERN VEHICULAR BRIDGE TO RIGHT. - Baltimore & Ohio Railroad, Bollman Truss Bridge, Spanning Little Patuxent River, Savage, Howard County, MD
Detail view of turnbuckle in diagonal member, with kodachrome film ...
Detail view of turnbuckle in diagonal member, with kodachrome film box on right turnbuckle for scale. - Pennsylvania Railroad, Whitford Bridge, Spanning Amtrak tracks at Whitford Road, Whitford, Chester County, PA
10. View of end portals, top chords, diagonals, verticals and ...
10. View of end portals, top chords, diagonals, verticals and strut connections. Looking from east span to east end of west span. - Boomershine Bridge, Spanning Twin Creek, Farmersville, Montgomery County, OH
3. VIEW OF CENTER PANEL SHOWING DIAGONAL TENSION MEMBERS AND ...
3. VIEW OF CENTER PANEL SHOWING DIAGONAL TENSION MEMBERS AND LOWER CHORD CONNECTIONS. EAST BANK LOOKING SOUTHWEST. - Cunningham Lane Bridge, Spanning Pine River on Cunningham Lane near Highway 80, Rockbridge, Richland County, WI
8. SOUTH VIEW, RIVETED CONNECTION POINT OF TOP CHORD, DIAGONAL ...
8. SOUTH VIEW, RIVETED CONNECTION POINT OF TOP CHORD, DIAGONAL TENSION MEMBER, INCLINED END POST AND VERTICAL MEMBER - Jordan Narrows Bridge, Crossing Jordan River at 9600 North, Lehi, Utah County, UT
13. VIEW OF TOP CHORD CONNECTION, SHOWING DIAGONAL AND VERTICAL ...
13. VIEW OF TOP CHORD CONNECTION, SHOWING DIAGONAL AND VERTICAL WEB MEMBERS AND TOP LATERAL BRACING, SOUTHEAST SPAN, LOOKING NORTHEAST - Linden Avenue Bridge, Spanning Purgatoire River on Linden Avenue, Trinidad, Las Animas County, CO
33. Coal Fuel Elevator (diagonal in foreground), Fuel Elevator (left), ...
33. Coal Fuel Elevator (diagonal in foreground), Fuel Elevator (left), Fuel Storage Bins (center), and Power Plant (right) Photographs taken by Joseph E.B. Elliot - Huber Coal Breaker, 101 South Main Street, Ashley, Luzerne County, PA
Average diagonal entropy in nonequilibrium isolated quantum systems.
Giraud, Olivier; García-Mata, Ignacio
2016-07-01
The diagonal entropy was introduced as a good entropy candidate especially for isolated quantum systems out of equilibrium. Here we present an analytical calculation of the average diagonal entropy for systems undergoing unitary evolution and an external perturbation in the form of a cyclic quench. We compare our analytical findings with numerical simulations of various quantum systems. Our calculations elucidate various heuristic relations proposed recently in the literature. PMID:27575092
Implicit measure for yoga research: Yoga implicit association test
Ilavarasu, Judu V; Rajesh, Sasidharan K; Hankey, Alex
2014-01-01
Context: The implicit association test (IAT), a new tool for yoga research is presented. Implicit measures could be used in those situations where (1) The construct is difficult to self-report, (2) there is a threat of social desirability. Clinically, we can assess cognitive dissonance by evaluating incongruence between implicit and explicit measures. Explicit preferences are self-reported. Implicit preferences are what we inherently believe, often without our conscious awareness. Aims: The primary objective of this study is to provide a bird's eye view of the field, implicit cognition, with emphasis on the IAT and the secondary objective is to illustrate through an example of our study to develop an implicit tool to assess implicit preference toward yoga. Settings and Design: A total of 5 independent samples of total 69 students undergoing short and long-term yoga courses in a Yoga University were assessed for their implicit and explicit preferences towards yoga. Materials and Methods: The yoga-IAT (Y-IAT), explicit self-rating scale was administered through computers using the Inquisit program by Millisecond Software. Experimental and scoring materials are provided. Results: A moderate preference toward yoga was detected, with a lower implicit-explicit congruence, reflecting possible confound of social desirability in the self-report of preference toward yoga. Conclusions: Implicit measures may be used in the yoga field to assess constructs, which are difficult to self-report or may have social desirability threat. Y-IAT may be used to evaluate implicit preference toward yoga. PMID:25035621
A multigrid nonoscillatory method for computing high speed flows
NASA Technical Reports Server (NTRS)
Li, C. P.; Shieh, T. H.
1993-01-01
A multigrid method using different smoothers has been developed to solve the Euler equations discretized by a nonoscillatory scheme up to fourth order accuracy. The best smoothing property is provided by a five-stage Runge-Kutta technique with optimized coefficients, yet the most efficient smoother is a backward Euler technique in factored and diagonalized form. The singlegrid solution for a hypersonic, viscous conic flow is in excellent agreement with the solution obtained by the third order MUSCL and Roe's method. Mach 8 inviscid flow computations for a complete entry probe have shown that the accuracy is at least as good as the symmetric TVD scheme of Yee and Harten. The implicit multigrid method is four times more efficient than the explicit multigrid technique and 3.5 times faster than the single-grid implicit technique. For a Mach 8.7 inviscid flow over a blunt delta wing at 30 deg incidence, the CPU reduction factor from the three-level multigrid computation is 2.2 on a grid of 37 x 41 x 73 nodes.
Auditory spatial resolution in horizontal, vertical, and diagonal planes
NASA Astrophysics Data System (ADS)
Grantham, D. Wesley; Hornsby, Benjamin W. Y.; Erpenbeck, Eric A.
2003-08-01
Minimum audible angle (MAA) and minimum audible movement angle (MAMA) thresholds were measured for stimuli in horizontal, vertical, and diagonal (60°) planes. A pseudovirtual technique was employed in which signals were recorded through KEMAR's ears and played back to subjects through insert earphones. Thresholds were obtained for wideband, high-pass, and low-pass noises. Only 6 of 20 subjects obtained wideband vertical-plane MAAs less than 10°, and only these 6 subjects were retained for the complete study. For all three filter conditions thresholds were lowest in the horizontal plane, slightly (but significantly) higher in the diagonal plane, and highest for the vertical plane. These results were similar in magnitude and pattern to those reported by Perrott and Saberi [J. Acoust. Soc. Am. 87, 1728-1731 (1990)] and Saberi and Perrott [J. Acoust. Soc. Am. 88, 2639-2644 (1990)], except that these investigators generally found that thresholds for diagonal planes were as good as those for the horizontal plane. The present results are consistent with the hypothesis that diagonal-plane performance is based on independent contributions from a horizontal-plane system (sensitive to interaural differences) and a vertical-plane system (sensitive to pinna-based spectral changes). Measurements of the stimuli recorded through KEMAR indicated that sources presented from diagonal planes can produce larger interaural level differences (ILDs) in certain frequency regions than would be expected based on the horizontal projection of the trajectory. Such frequency-specific ILD cues may underlie the very good performance reported in previous studies for diagonal spatial resolution. Subjects in the present study could apparently not take advantage of these cues in the diagonal-plane condition, possibly because they did not externalize the images to their appropriate positions in space or possibly because of the absence of a patterned visual field.
Implicit Sequence Learning in Children.
ERIC Educational Resources Information Center
Meulemans, Thierry; Van der Linden, Martial; Perruchet, Pierre
1998-01-01
Examined implicit learning ability in 6- and 10-year olds and adults as assessed by a serial reaction-time task, along with retention of knowledge after one week and explicit knowledge developed by children. Found no age-related difference in serial reaction-time performance, consistent with the idea that implicit learning abilities may be…
The Neuropharmacology of Implicit Learning
Uddén, Julia; Folia, Vasiliki; Petersson, Karl Magnus
2010-01-01
Two decades of pharmacologic research on the human capacity to implicitly acquire knowledge as well as cognitive skills and procedures have yielded surprisingly few conclusive insights. We review the empirical literature of the neuropharmacology of implicit learning. We evaluate the findings in the context of relevant computational models related to neurotransmittors such as dopamine, serotonin, acetylcholine and noradrenalin. These include models for reinforcement learning, sequence production, and categorization. We conclude, based on the reviewed literature, that one can predict improved implicit acquisition by moderately elevated dopamine levels and impaired implicit acquisition by moderately decreased dopamine levels. These effects are most prominent in the dorsal striatum. This is supported by a range of behavioral tasks in the empirical literature. Similar predictions can be made for serotonin, although there is yet a lack of support in the literature for serotonin involvement in classical implicit learning tasks. There is currently a lack of evidence for a role of the noradrenergic and cholinergic systems in implicit and related forms of learning. GABA modulators, including benzodiazepines, seem to affect implicit learning in a complex manner and further research is needed. Finally, we identify allosteric AMPA receptors modulators as a potentially interesting target for future investigation of the neuropharmacology of procedural and implicit learning. PMID:21629444
Implicit Theories of Peer Relationships
ERIC Educational Resources Information Center
Rudolph, Karen D.
2010-01-01
This research investigated the role of children's implicit theories of peer relationships in their psychological, emotional, and behavioral adjustment. Participants included 206 children (110 girls; 96 boys; M age = 10.13 years, SD = 1.16) who reported on their implicit theories of peer relationships, social goal orientation, need for approval,…
Development of Implicit Methods in CFD NASA Ames Research Center 1970's - 1980's
NASA Technical Reports Server (NTRS)
Pulliam, Thomas H.
2010-01-01
The focus here is on the early development (mid 1970's-1980's) at NASA Ames Research Center of implicit methods in Computational Fluid Dynamics (CFD). A class of implicit finite difference schemes of the Beam and Warming approximate factorization type will be addressed. The emphasis will be on the Euler equations. A review of material pertinent to the solution of the Euler equations within the framework of implicit methods will be presented. The eigensystem of the equations will be used extensively in developing a framework for various methods applied to the Euler equations. The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented.
Diagonal unitary entangling gates and contradiagonal quantum states
NASA Astrophysics Data System (ADS)
Lakshminarayan, Arul; Puchała, Zbigniew; Życzkowski, Karol
2014-09-01
Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as lnN, in contrast to the lnN2 behavior characteristic of random unitary gates. Entangling power of a diagonal gate U is related to the von Neumann entropy of an auxiliary quantum state ρ =AA†/N2, where the square matrix A is obtained by reshaping the vector of diagonal elements of U of length N2 into a square matrix of order N. This fact provides a motivation to study the ensemble of non-Hermitian unimodular matrices A, with all entries of the same modulus and random phases and the ensemble of quantum states ρ, such that all their diagonal entries are equal to 1/N. Such a state is contradiagonal with respect to the computational basis, in the sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse-graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle." This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.
Net efficiency of roller skiing with a diagonal stride.
Nakai, Akira; Ito, Akira
2011-02-01
The aims of this study were: (a) to determine net efficiency during roller skiing with a diagonal stride at various speeds; (b) to assess the development of net efficiency across speeds; and (c) to examine the characteristics of efficiency in diagonal roller skiing. Two-dimensional kinematics and oxygen uptake were determined in eight male collegiate cross-country ski athletes who roller skied with the diagonal stride at various speeds on a level track. Net efficiency was calculated from rates of internal and external work and net energy expenditure. Individual net efficiency ranged from 17.7% to 52.1%. Net efficiency in the entire group of athletes increased with increasing speed, reached a maximum value of 37.3% at 3.68 m · s(-1), before slowly decreasing. These findings indicate that roller skiing with the diagonal stride at high speed is a highly efficient movement and that an optimal speed exists at which net efficiency can be maximally enhanced in diagonal roller skiing. PMID:21184346
Diagonal free homonuclear correlation using heteronuclei at natural abundance.
Baishya, Bikash
2015-07-01
Homonuclear correlated spectroscopy such as COSY and TOCSY provides crucial structural information. In all homonuclear correlation, the most intense peaks are represented by the diagonal. As a result, the useful cross peaks close to the diagonal get obscured by the huge tails of diagonal peaks. Herein, we show that by editing the proton magnetization by a 13C nucleus in natural abundance, it is possible to eliminate the inphase coherence or untransferred magnetization that leads to the diagonal peak while retaining the antiphase coherence or transferred magnetization required for creation of cross peak. After the coherence transfer step, the untransferred magnetization directly attached to 13C evolves under one bond heteronuclear coupling while the transferred transverse magnetization directly attached to remote 12C does not. As a result, the untransferred magnetization directly attached to 13C can be converted to an unobservable heteronuclear multiple quantum coherence leading to a diagonal free correlated spectrum with a sensitivity penalty of two orders of magnitude but comparable to HSQC kind of experiments at natural abundance. The method demonstrated for COSY and TOCSY allows all proton-proton correlations to be observed except the geminal proton-proton correlations. Further, protons directly attached to heteronuclei other than 13C must be scalar coupled to protons directly attached to 13C to have a detectable cross peak. PMID:26001137
Improving DTI tractography by including diagonal tract propagation.
Taylor, Paul A; Cho, Kuan-Hung; Lin, Ching-Po; Biswal, Bharat B
2012-01-01
Tractography algorithms have been developed to reconstruct likely WM pathways in the brain from diffusion tensor imaging (DTI) data. In this study, an elegant and simple means for improving existing tractography algorithms is proposed by allowing tracts to propagate through diagonal trajectories between voxels, instead of only rectilinearly to their facewise neighbors. A series of tests (using both real and simulated data sets) are utilized to show several benefits of this new approach. First, the inclusion of diagonal tract propagation decreases the dependence of an algorithm on the arbitrary orientation of coordinate axes and therefore reduces numerical errors associated with that bias (which are also demonstrated here). Moreover, both quantitatively and qualitatively, including diagonals decreases overall noise sensitivity of results and leads to significantly greater efficiency in scanning protocols; that is, the obtained tracts converge much more quickly (i.e., in a smaller amount of scanning time) to those of data sets with high SNR and spatial resolution. Importantly, the inclusion of diagonal propagation adds essentially no appreciable time of calculation or computational costs to standard methods. This study focuses on the widely-used streamline tracking method, FACT (fiber assessment by continuous tracking), and the modified method is termed "FACTID" (FACT including diagonals). PMID:22970125
Implicit emotion perception in schizophrenia.
Trémeau, Fabien; Antonius, Daniel; Todorov, Alexander; Rebani, Yasmina; Ferrari, Kelsey; Lee, Sang Han; Calderone, Daniel; Nolan, Karen A; Butler, Pamela; Malaspina, Dolores; Javitt, Daniel C
2015-12-01
Explicit but not implicit facial emotion perception has been shown to be impaired in schizophrenia. In this study, we used newly developed technology in social neuroscience to examine implicit emotion processing. It has been shown that when people look at faces, they automatically infer social traits, and these trait judgments rely heavily on facial features and subtle emotion expressions even with neutral faces. Eighty-one individuals with schizophrenia or schizoaffective disorder and 62 control subjects completed a computer task with 30 well-characterized neutral faces. They rated each face on 10 trait judgments: attractive, mean, trustworthy, intelligent, dominant, fun, sociable, aggressive, emotionally stable and weird. The degree to which trait ratings were predicted by objectively-measured subtle emotion expressions served as a measure of implicit emotion processing. Explicit emotion recognition was also examined. Trait ratings were significantly predicted by subtle facial emotional expressions in controls and patients. However, impairment in the implicit emotion perception of fear, happiness, anger and surprise was found in patients. Moreover, these deficits were associated with poorer everyday problem-solving skills and were relatively independent of explicit emotion recognition. Implicit emotion processing is impaired in patients with schizophrenia or schizoaffective disorder. Deficits in implicit and explicit emotion perception independently contribute to the patients' poor daily life skills. More research is needed to fully understand the role of implicit and explicit processes in the functional deficits of patients, in order to develop targeted and useful remediation interventions. PMID:26473695
Spectral sharpening of color sensors: diagonal color constancy and beyond.
Vazquez-Corral, Javier; Bertalmío, Marcelo
2014-01-01
It has now been 20 years since the seminal work by Finlayson et al. on the use of spectral sharpening of sensors to achieve diagonal color constancy. Spectral sharpening is still used today by numerous researchers for different goals unrelated to the original goal of diagonal color constancy e.g., multispectral processing, shadow removal, location of unique hues. This paper reviews the idea of spectral sharpening through the lens of what is known today in color constancy, describes the different methods used for obtaining a set of sharpening sensors and presents an overview of the many different uses that have been found for spectral sharpening over the years. PMID:24577523
Neural network based diagonal decoupling control of powered wheelchair systems.
Nguyen, Tuan Nghia; Su, Steven; Nguyen, Hung T
2014-03-01
This paper proposes an advanced diagonal decoupling control method for powered wheelchair systems. This control method is based on a combination of the systematic diagonalization technique and the neural network control design. As such, this control method reduces coupling effects on a multivariable system, leading to independent control design procedures. Using an obtained dynamic model, the problem of the plant's Jacobian calculation is eliminated in a neural network control design. The effectiveness of the proposed control method is verified in a real-time implementation on a powered wheelchair system. The obtained results confirm that robustness and desired performance of the overall system are guaranteed, even under parameter uncertainty effects. PMID:23981543
Spectral Sharpening of Color Sensors: Diagonal Color Constancy and Beyond
Vazquez-Corral, Javier; Bertalmío, Marcelo
2014-01-01
It has now been 20 years since the seminal work by Finlayson et al. on the use of spectral sharpening of sensors to achieve diagonal color constancy. Spectral sharpening is still used today by numerous researchers for different goals unrelated to the original goal of diagonal color constancy e.g., multispectral processing, shadow removal, location of unique hues. This paper reviews the idea of spectral sharpening through the lens of what is known today in color constancy, describes the different methods used for obtaining a set of sharpening sensors and presents an overview of the many different uses that have been found for spectral sharpening over the years. PMID:24577523
Quantum Discord of 2 n -Dimensional Bell-Diagonal States
NASA Astrophysics Data System (ADS)
Jafarizadeh, M. A.; Karimi, N.; Amidi, D.; Zahir Olyaei, H.
2016-03-01
In this study, using the concept of relative entropy as a distance measure of correlations we investigate the important issue of evaluating quantum correlations such as entanglement, dissonance and classical correlations for 2 n -dimensional Bell-diagonal states. We provide an analytical technique, which describes how we find the closest classical states(CCS) and the closest separable states(CSS) for these states. Then analytical results are obtained for quantum discord of 2 n -dimensional Bell-diagonal states. As illustration, some special cases are examined. Finally, we investigate the additivity relation between the different correlations for the separable generalized Bloch sphere states.
The efficient calculation of chemically reacting flow
NASA Technical Reports Server (NTRS)
Eklund, D. R.; Hassan, H. A.; Drummond, J. P.
1986-01-01
A semi-implicit finite volume formulation is used to study flows with chemical reactions. In this formulation the source terms resulting from the chemical reactions are treated implicitly and the resulting system of partial differential equations is solved using two time-stepping schemes. The first is based on the Runge-Kutta method while the second is based on an Adams predictor-corrector method. Results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated. Further, analysis and computation indicate that the Runge-Kutta method is more efficient than the Adams methods. Finally, an adaptive time stepping scheme is developed to study problems involving shock ignition. Calculations for a hydrogen-air system agree well with other methods.
View, from south showing verticals, diagonals, and overhead bracing of ...
View, from south showing verticals, diagonals, and overhead bracing of south span Pratt through trusses, south portal of north span, pipe rails and posts, and concrete deck with bituminous wearing surface - Castle Garden Bridge, Township Route 343 over Bennetts Branch of Sinnemahoning Creek, Driftwood, Cameron County, PA
1. Aerial view of turnpike path running diagonally up from ...
1. Aerial view of turnpike path running diagonally up from lower left (present-day Orange Turnpike alignment) and containing on towards upper right through tree clump in center of the bare spot on the landscape, and on through the trees. View looking south. - Orange Turnpike, Parallel to new Orange Turnpike, Monroe, Orange County, NY
Improving stochastic estimates with inference methods: calculating matrix diagonals.
Selig, Marco; Oppermann, Niels; Ensslin, Torsten A
2012-02-01
Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image reconstruction and statistical inference. Here, methods of statistical inference are used to improve the accuracy or the computational costs of matrix probing methods to estimate matrix diagonals. In particular, the generalized Wiener filter methodology, as developed within information field theory, is shown to significantly improve estimates based on only a few sampling probes, in cases in which some form of continuity of the solution can be assumed. The strength, length scale, and precise functional form of the exploited autocorrelation function of the matrix diagonal is determined from the probes themselves. The developed algorithm is successfully applied to mock and real world problems. These performance tests show that, in situations where a matrix diagonal has to be calculated from only a small number of computationally expensive probes, a speedup by a factor of 2 to 10 is possible with the proposed method. PMID:22463179
Penguins and Pandas: A Note on Teaching Cantor's Diagonal Argument
ERIC Educational Resources Information Center
Rauff, James V.
2008-01-01
Cantor's diagonal proof that the set of real numbers is uncountable is one of the most famous arguments in modern mathematics. Mathematics students usually see this proof somewhere in their undergraduate experience, but it is rarely a part of the mathematical curriculum of students of the fine arts or humanities. This note describes contexts that…
34. Coal Fuel Elevator (diagonal in foreground), Fuel Elevator (left), ...
34. Coal Fuel Elevator (diagonal in foreground), Fuel Elevator (left), Fuel Storage Bins (center), and Power Plant (far center), and Retail Coal Storage Bins (right) Photograph taken by George Harven - Huber Coal Breaker, 101 South Main Street, Ashley, Luzerne County, PA
35. Coal Fuel Elevator (diagonal in center), Fuel Elevator (left), ...
35. Coal Fuel Elevator (diagonal in center), Fuel Elevator (left), Fuel Storage Bins (center), and Power Plant (far center), and Retail Coal Storage Bins (right) Photograph taken by George Harven - Huber Coal Breaker, 101 South Main Street, Ashley, Luzerne County, PA
5. DETAIL OF SOUTHEAST DOOR SHOWING DIAGONAL WOOD PATTERN OF ...
5. DETAIL OF SOUTHEAST DOOR SHOWING DIAGONAL WOOD PATTERN OF PANELS AND RINGS IN BRICK WALL USED FOR SECURING THE DOORS WITH CHAINS WHEN OPEN. - Presidio of San Francisco, Cavalry Stables, Cowles Street, between Lincoln Boulevard & McDowell Street, San Francisco, San Francisco County, CA
Semi-implicit finite difference methods for three-dimensional shallow water flow
Casulli, Vincenzo; Cheng, Ralph T.
1992-01-01
A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.
Stiff modes in spinvalve simulations with OOMMF
NASA Astrophysics Data System (ADS)
Mitropoulos, Spyridon; Tsiantos, Vassilis; Ovaliadis, Kyriakos; Kechrakos, Dimitris; Donahue, Michael
2016-04-01
Micromagnetic simulations are an important tool for the investigation of magnetic materials. Micromagnetic software uses various techniques to solve differential equations, partial or ordinary, involved in the dynamic simulations. Euler, Runge-Kutta, Adams, and BDF (Backward Differentiation Formulae) are some of the methods used for this purpose. In this paper, spinvalve simulations are investigated. Evidence is presented showing that these systems have stiff modes, and that implicit methods such as BDF are more effective than explicit methods in such cases.
Multigrid time-accurate integration of Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Arnone, Andrea; Liou, Meng-Sing; Povinelli, Louis A.
1993-01-01
Efficient acceleration techniques typical of explicit steady-state solvers are extended to time-accurate calculations. Stability restrictions are greatly reduced by means of a fully implicit time discretization. A four-stage Runge-Kutta scheme with local time stepping, residual smoothing, and multigridding is used instead of traditional time-expensive factorizations. Some applications to natural and forced unsteady viscous flows show the capability of the procedure.
Stable time integration suppresses unphysical oscillations in the bidomain model
NASA Astrophysics Data System (ADS)
Torabi Ziaratgahi, Saeed; Marsh, Megan; Sundnes, Joakim; Spiteri, Raymond
2014-07-01
The bidomain model is a popular model for simulating electrical activity in cardiac tissue. It is a continuum-based model consisting of non-linear ordinary differential equations (ODEs) describing spatially averaged cellular reactions and a system of partial differential equations (PDEs) describing electrodiffusion on tissue level. Because of this multi-scale, ODE/PDE structure of the model, operator-splitting methods that treat the ODEs and PDEs in separate steps are natural candidates as numerical solution methods. Second-order methods can generally be expected to be more effective than first-order methods under normal accuracy requirements. However, the simplest and the most commonly applied second-order method for the PDE step, the Crank--Nicolson (CN) method, may generate unphysical oscillations. In this paper, we investigate the performance of a two-stage, L-stable singly diagonally implicit Runge--Kutta method for solving the PDEs of the bidomain model. Numerical experiments show that the enhanced stability property of this method leads to more physically realistic numerical simulations compared to both the CN and backward Euler methods.
[Using the Implicit Association Test (IAT) to measure implicit shyness].
Aikawa, Atsushi; Fujii, Tsutomu
2011-04-01
Previous research has shown that implicitly measured shyness predicted spontaneous shy behavior in social situations, while explicit self-ratings of shyness predicted controlled shy behavior (Asendorpf, Banse, & Mücke, 2002). The present study examined whether these same results would be replicated in Japan. In Study 1, college students (N=47) completed a shyness Implicit Association Test (IAT for shyness) and explicit self-ratings of shyness. In Study 2, friends (N=69) of the Study 1 participants rated those participants on various personality scales. Covariance structure analysis, revealed that only implicit self-concept measured by the shyness IAT predicted other-rated high interpersonal tension (spontaneous shy behavior). Also, only explicit self-concept predicted other-rated low praise seeking (controlled shy behavior). The results of this study are similar to the findings of the previous research. PMID:21706822
Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory
Jiang Tongsong
2005-05-01
By means of complex representation and real representation of a quaternion matrix, this paper studies the problem of diagonalization of a quaternion matrix, gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory.
Posture modulates implicit hand maps.
Longo, Matthew R
2015-11-01
Several forms of somatosensation require that afferent signals be informed by stored representations of body size and shape. Recent results have revealed that position sense relies on a highly distorted body representation. Changes of internal hand posture produce plastic alterations of processing in somatosensory cortex. This study therefore investigated how such postural changes affect implicit body representations underlying position sense. Participants localised the knuckles and tips of each finger in external space in two postures: the fingers splayed (Apart posture) or pressed together (Together posture). Comparison of the relative locations of the judgments of each landmark were used to construct implicit maps of represented hand structure. Spreading the fingers apart produced increases in the implicit representation of hand size, with no apparent effect on hand shape. Thus, changes of internal hand posture produce rapid modulation of how the hand itself is represented, paralleling the known effects on somatosensory cortical processing. PMID:26117153
Off-diagonal Jacobian support for Nodal BCs
Peterson, John W.; Andrs, David; Gaston, Derek R.; Permann, Cody J.; Slaughter, Andrew E.
2015-01-01
In this brief note, we describe the implementation of o-diagonal Jacobian computations for nodal boundary conditions in the Multiphysics Object Oriented Simulation Environment (MOOSE) [1] framework. There are presently a number of applications [2{5] based on the MOOSE framework that solve complicated physical systems of partial dierential equations whose boundary conditions are often highly nonlinear. Accurately computing the on- and o-diagonal Jacobian and preconditioner entries associated to these constraints is crucial for enabling ecient numerical solvers in these applications. Two key ingredients are required for properly specifying the Jacobian contributions of nonlinear nodal boundary conditions in MOOSE and nite element codes in general: 1. The ability to zero out entire Jacobian matrix rows after \
Block-bordered diagonalization and parallel iterative solvers
Alvarado, F.; Dag, H.; Bruggencate, M. ten
1994-12-31
One of the most common techniques for enhancing parallelism in direct sparse matrix methods is the reorganization of a matrix into a blocked-bordered structure. Incomplete LDU factorization is a very good preconditioner for PCG in serial environments. However, the inherent sequential nature of the preconditioning step makes it less desirable in parallel environments. This paper explores the use of BBD (Blocked Bordered Diagonalization) in connection with ILU preconditioners. The paper shows that BBD-based ILU preconditioners are quite amenable to parallel processing. Neglecting entries from the entire border can result in a blocked diagonal matrix. The result is a great increase in parallelism at the expense of additional iterations. Experiments on the Sequent Symmetry shared memory machine using (mostly) power system that matrices indicate that the method is generally better than conventional ILU preconditioners and in many cases even better than partitioned inverse preconditioners, without the initial setup disadvantages of partitioned inverse preconditioners.
A CLT on the SNR of Diagonally Loaded MVDR Filters
NASA Astrophysics Data System (ADS)
Rubio, Francisco; Mestre, Xavier; Hachem, Walid
2012-08-01
This paper studies the fluctuations of the signal-to-noise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples. Specifically, a central limit theorem (CLT) is established for the fluctuations of the SNR of the diagonally loaded MVDR filter, under both supervised and unsupervised training settings in adaptive filtering applications. Our second-order analysis is based on the Nash-Poincar\\'e inequality and the integration by parts formula for Gaussian functionals, as well as classical tools from statistical asymptotic theory. Numerical evaluations validating the accuracy of the CLT confirm the asymptotic Gaussianity of the fluctuations of the SNR of the MVDR filter.
A Summary of Diagonal Tension Part I : Methods of Analysis
NASA Technical Reports Server (NTRS)
Kuhn, Paul; Peterson, James P; Levin, L Ross
1952-01-01
Previously published methods for stress and strength analysis of plane and curved shear webs working in diagonal tension are presented as a unified method. The treatment is sufficiently comprehensive and detailed to make the paper self-contained. Part 1 discusses the theory and methods for calculating the stresses and shear deflections of web systems as well as the strengths of the web, the stiffeners, and the riveting. Part 2, published separately, presents the experimental evidence. (author)
Covariant diagonalization of the perfect fluid stress-energy tensor
NASA Astrophysics Data System (ADS)
Garat, Alcides
2015-02-01
We introduce new tetrads that manifestly and covariantly diagonalize the stress-energy tensor for a perfect fluid with vorticity at every spacetime point. This new tetrad can be applied to introduce simplification in the analysis of astrophysical relativistic problems where vorticity is present through the Carter-Lichnerowicz equation. We also discuss the origin of inertia in this special case from the standpoint of our new local tetrads.
INTERIOR OF HOG BARN SHOWING MILKING STANCHIONS AND DIAGONAL SHEATHING, ...
INTERIOR OF HOG BARN SHOWING MILKING STANCHIONS AND DIAGONAL SHEATHING, LOOKING EAST. (In the 1940s the hog barn was converted to a calf barn to service the growing dairy. After a fire on the property took the Engles main barn in 1954, the building was converted into a milking parlor.) - Engle Farm, Barn, 89 South Ebey Road, Coupeville, Island County, WA
Diagonal dominance for the multivariable Nyquist array using function minimization
NASA Technical Reports Server (NTRS)
Leininger, G. G.
1977-01-01
A new technique for the design of multivariable control systems using the multivariable Nyquist array method was developed. A conjugate direction function minimization algorithm is utilized to achieve a diagonal dominant condition over the extended frequency range of the control system. The minimization is performed on the ratio of the moduli of the off-diagonal terms to the moduli of the diagonal terms of either the inverse or direct open loop transfer function matrix. Several new feedback design concepts were also developed, including: (1) dominance control parameters for each control loop; (2) compensator normalization to evaluate open loop conditions for alternative design configurations; and (3) an interaction index to determine the degree and type of system interaction when all feedback loops are closed simultaneously. This new design capability was implemented on an IBM 360/75 in a batch mode but can be easily adapted to an interactive computer facility. The method was applied to the Pratt and Whitney F100 turbofan engine.
Polaronic discontinuities induced by off-diagonal coupling
NASA Astrophysics Data System (ADS)
Zhang, Yuyu; Duan, Liwei; Chen, Qinghu; Zhao, Yang
2012-07-01
In this paper, we study a form of the Holstein molecular crystal model in which the influence of lattice vibrations on the transfers of electronic excitations between neighboring sites (off-diagonal coupling) is taken into account. Using the Toyozawa Ansatz and the Lanczos algorithm, the Holstein Hamiltonian with two types of off-diagonal coupling is studied focusing on a number of analyticity issues in the ground state. For finite-sized lattices and antisymmetric coupling, a sequence of discontinuities are found in the polaron energy dispersion, the size of the ground-state phonon cloud, and the linearized von Neumann entropy used to quantify the quantum entanglement between the exciton and the phonons in the ground state. Such behavior is accompanied by a shift of the ground-state crystal momentum from zero to nonzero values as the coupling strength is increased. In the thermodynamic limit, all discontinuities associated with antisymmetric coupling vanish except the one corresponding to the initial departure of the ground-state wavevector from the Brillouin zone center. For the case of symmetric off-diagonal coupling, a smooth crossover is found to exist in all parameters regimes.
Polaronic discontinuities induced by off-diagonal coupling.
Zhang, Yuyu; Duan, Liwei; Chen, Qinghu; Zhao, Yang
2012-07-21
In this paper, we study a form of the Holstein molecular crystal model in which the influence of lattice vibrations on the transfers of electronic excitations between neighboring sites (off-diagonal coupling) is taken into account. Using the Toyozawa Ansatz and the Lanczos algorithm, the Holstein Hamiltonian with two types of off-diagonal coupling is studied focusing on a number of analyticity issues in the ground state. For finite-sized lattices and antisymmetric coupling, a sequence of discontinuities are found in the polaron energy dispersion, the size of the ground-state phonon cloud, and the linearized von Neumann entropy used to quantify the quantum entanglement between the exciton and the phonons in the ground state. Such behavior is accompanied by a shift of the ground-state crystal momentum from zero to nonzero values as the coupling strength is increased. In the thermodynamic limit, all discontinuities associated with antisymmetric coupling vanish except the one corresponding to the initial departure of the ground-state wavevector from the Brillouin zone center. For the case of symmetric off-diagonal coupling, a smooth crossover is found to exist in all parameters regimes. PMID:22830684
A Partitioning Algorithm for Block-Diagonal Matrices With Overlap
Guy Antoine Atenekeng Kahou; Laura Grigori; Masha Sosonkina
2008-02-02
We present a graph partitioning algorithm that aims at partitioning a sparse matrix into a block-diagonal form, such that any two consecutive blocks overlap. We denote this form of the matrix as the overlapped block-diagonal matrix. The partitioned matrix is suitable for applying the explicit formulation of Multiplicative Schwarz preconditioner (EFMS) described in [3]. The graph partitioning algorithm partitions the graph of the input matrix into K partitions, such that every partition {Omega}{sub i} has at most two neighbors {Omega}{sub i-1} and {Omega}{sub i+1}. First, an ordering algorithm, such as the reverse Cuthill-McKee algorithm, that reduces the matrix profile is performed. An initial overlapped block-diagonal partition is obtained from the profile of the matrix. An iterative strategy is then used to further refine the partitioning by allowing nodes to be transferred between neighboring partitions. Experiments are performed on matrices arising from real-world applications to show the feasibility and usefulness of this approach.
Yao, Yao; Zhou, Nengji; Prior, Javier; Zhao, Yang
2015-01-01
It has long been a puzzle on what drives charge separation in artificial polymeric solar cells as a consensus has yet to emerge among rivaling theories based upon electronic localization and delocalization pictures. Here we propose an alternative using the two-bath spin-boson model with simultaneous diagonal and off-diagonal coupling: the critical phase, which is born out of the competition of the two coupling types, and is neither localized nor delocalized. The decoherence-free feature of the critical phase also helps explain sustained coherence of the charge-transfer state. Exploiting Hamiltonian symmetries in an enhanced algorithm of density-matrix renormalization group, we map out boundaries of the critical phase to a precision previously unattainable, and determine the bath spectral densities inducive to the existence of the charge-transfer state. PMID:26412693
Rosta, Edina; Warshel, Arieh
2012-01-01
Understanding the relationship between the adiabatic free energy profiles of chemical reactions and the underlining diabatic states is central to the description of chemical reactivity. The diabatic states form the theoretical basis of Linear Free Energy Relationships (LFERs) and thus play a major role in physical organic chemistry and related fields. However, the theoretical justification for some of the implicit LFER assumptions has not been fully established by quantum mechanical studies. This study follows our earlier works(1,2) and uses the ab initio frozen density functional theory (FDFT) method(3) to evaluate both the diabatic and adiabatic free energy surfaces and to determine the corresponding off-diagonal coupling matrix elements for a series of S(N)2 reactions. It is found that the off-diagonal coupling matrix elements are almost the same regardless of the nucleophile and the leaving group but change upon changing the central group. Furthermore, it is also found that the off diagonal elements are basically the same in gas phase and in solution, even when the solvent is explicitly included in the ab initio calculations. Furthermore, our study establishes that the FDFT diabatic profiles are parabolic to a good approximation thus providing a first principle support to the origin of LFER. These findings further support the basic approximation of the EVB treatment. PMID:23329895
Rosta, Edina; Warshel, Arieh
2012-01-01
Understanding the relationship between the adiabatic free energy profiles of chemical reactions and the underlining diabatic states is central to the description of chemical reactivity. The diabatic states form the theoretical basis of Linear Free Energy Relationships (LFERs) and thus play a major role in physical organic chemistry and related fields. However, the theoretical justification for some of the implicit LFER assumptions has not been fully established by quantum mechanical studies. This study follows our earlier works1,2 and uses the ab initio frozen density functional theory (FDFT) method3 to evaluate both the diabatic and adiabatic free energy surfaces and to determine the corresponding off-diagonal coupling matrix elements for a series of SN2 reactions. It is found that the off-diagonal coupling matrix elements are almost the same regardless of the nucleophile and the leaving group but change upon changing the central group. Furthermore, it is also found that the off diagonal elements are basically the same in gas phase and in solution, even when the solvent is explicitly included in the ab initio calculations. Furthermore, our study establishes that the FDFT diabatic profiles are parabolic to a good approximation thus providing a first principle support to the origin of LFER. These findings further support the basic approximation of the EVB treatment. PMID:23329895
NASA Technical Reports Server (NTRS)
Atkins, Harold
1991-01-01
A multiple block multigrid method for the solution of the three dimensional Euler and Navier-Stokes equations is presented. The basic flow solver is a cell vertex method which employs central difference spatial approximations and Runge-Kutta time stepping. The use of local time stepping, implicit residual smoothing, multigrid techniques and variable coefficient numerical dissipation results in an efficient and robust scheme is discussed. The multiblock strategy places the block loop within the Runge-Kutta Loop such that accuracy and convergence are not affected by block boundaries. This has been verified by comparing the results of one and two block calculations in which the two block grid is generated by splitting the one block grid. Results are presented for both Euler and Navier-Stokes computations of wing/fuselage combinations.
Navier-Stokes cascade analysis with a stiff Kappa-Epsilon turbulence solver
NASA Technical Reports Server (NTRS)
Liu, Jong-Shang; Sockol, Peter M.; Prahl, Joseph M.
1987-01-01
The two dimensional, compressible, thin layer Navier-Stokes equations with the Baldwin-Lomax turbulence model and the kinetic energy-energy dissipation (k-epsilon) model are solved numerically to simulate the flow through a cascade. The governing equations are solved for the entire flow domain, without the boundary layer assumptions. The stiffness of the k-epsilon equations is discussed. A semi-implicit, Runge-Kutta, time-marching scheme is developed to solve the k-epsilon equations. The impact of the k-epsilon solver on the explicit Runge-Kutta Navier-Stokes solver is discussed. Numerical solutions are presented for two dimensional turbulent flow over a flat plate and a double circular arc cascade and compared with experimental data.
A brief introduction to symplectic integrators and recent results
Channell, P.J.
1994-02-01
The author begins with a brief synopsis about Hamiltonian systems and symplectic maps. A symplectic integrator is a symplectic map {phi}(q,p;t) that systematically approximates the time t flow of a Hamiltonian system. Systematic means: (1) in time step, t, i.e. the error should vanish as some power of the time step, and (2) in order of approximation, i.e. one would like a hierarchy of such {phi} that have errors that vanish as successively higher powers of the time step. At present the authors known two general types of symplectic integrators: (1) implicit integrators that are derived from a generating function or from algebraic conditions on Runge-Kutta schemes, and (2) explicit integrators that are derived from integrable Hamiltonians or from algebraic conditions on Runge-Kutta schemes.
Semantic Generalization in Implicit Language Learning
ERIC Educational Resources Information Center
Paciorek, Albertyna; Williams, John N.
2015-01-01
Despite many years of investigation into implicit learning in nonlinguistic domains, the potential for implicit learning to deliver the kinds of generalizations that underlie natural language competence remains unclear. In a series of experiments, we investigated implicit learning of the semantic preferences of novel verbs, specifically, whether…
Measuring individual differences in implicit cognition: the implicit association test.
Greenwald, A G; McGhee, D E; Schwartz, J L
1998-06-01
An implicit association test (IAT) measures differential association of 2 target concepts with an attribute. The 2 concepts appear in a 2-choice task (2-choice task (e.g., flower vs. insect names), and the attribute in a 2nd task (e.g., pleasant vs. unpleasant words for an evaluation attribute). When instructions oblige highly associated categories (e.g., flower + pleasant) to share a response key, performance is faster than when less associated categories (e.g., insect & pleasant) share a key. This performance difference implicitly measures differential association of the 2 concepts with the attribute. In 3 experiments, the IAT was sensitive to (a) near-universal evaluative differences (e.g., flower vs. insect), (b) expected individual differences in evaluative associations (Japanese + pleasant vs. Korean + pleasant for Japanese vs. Korean subjects), and (c) consciously disavowed evaluative differences (Black + pleasant vs. White + pleasant for self-described unprejudiced White subjects). PMID:9654756
NASA Astrophysics Data System (ADS)
Jiang, Zhen-Hua; Yan, Chao; Yu, Jian
2013-08-01
Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously.
Performance Theory of Diagonal Conducting Wall MHD Accelerators
NASA Technical Reports Server (NTRS)
Litchford, R. J.
2003-01-01
The theoretical performance of diagonal conducting wall crossed field accelerators is examined on the basis of an infinite segmentation assumption using a cross-plane averaged generalized Ohm's law for a partially ionized gas, including ion slip. The desired accelerator performance relationships are derived from the cross-plane averaged Ohm's law by imposing appropriate configuration and loading constraints. A current dependent effective voltage drop model is also incorporated to account for cold-wall boundary layer effects including gasdynamic variations, discharge constriction, and electrode falls. Definition of dimensionless electric fields and current densities lead to the construction of graphical performance diagrams, which further illuminate the rudimentary behavior of crossed field accelerator operation.
Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition
Weinstein, Marvin; Auerbach, Assa; Chandra, V.Ravi; /Technion
2011-11-04
We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements. The lattice of size N is partitioned into two subclusters. At each iteration the Lanczos vector is projected into a set of n{sub svd} smaller subcluster vectors using singular value decomposition. For low entanglement entropy S{sub ee}, (satisfied by short range Hamiltonians), we expect the truncation error to vanish as exp(-n{sup 1/S{sub ee}}{sub svd}). Convergence is tested for the Heisenberg model on Kagome clusters of up to 36 sites, with no symmetries exploited, using less than 15GB of memory. Generalization to multiple partitioning is discussed.
Diagonal-Axes Stage for Pointing an Optical Communications Transceiver
NASA Technical Reports Server (NTRS)
Regehr, Martin W.; Garkanian, Vachik
2011-01-01
Traditional azimuth-elevation ("az-el") stages are used to point a variety of devices ranging from large optical telescopes to tank guns. Such a stage typically has an elevation stage having a horizontal axis mounted on an azimuth stage with a vertical axis. Both stages are often motorized. Optical communications transceivers often require two-axis motorized control, as when the communications link is between a ground station and an aircraft or satellite. In such applications, the traditional azimuth-elevation stage has two important drawbacks: a gimbal lock exclusion zone at zenith and susceptibility to pointing errors caused by backlash. Az-el stages in which the azimuth stage cannot rotate a full 360deg have the additional drawback of an azimuth exclusion zone. The diagonal-axes stage described here mitigates or eliminates all of these problems. Instead of one vertical axis and one horizontal axis, a diagonal-axes stage has two horizontal axes, both oriented at 45 to the trajectory of the target. For example, a ground station located on the equator tracking a satellite with an equatorial orbit would have one axis parallel to northeast and southwest, and the other axis parallel to northwest and southeast. The diagonal-axes stage is considerably less vulnerable to backlash. If it is correctly oriented, its axes rotate in only one direction during an overhead pass by a satellite. As a result, the effects of backlash may be inherently eliminated. If the gravity-induced torque on either axis changes during the pass, then backlash may become important during the part of the pass where the gravity torque, instead of opposing the motion of the stage, pushes the stage in the direction of motion. This can result in the loss of gear-to-gear contact in one or more stages of the gear reduction mechanism. In this case, a preload spring used to eliminate backlash need only be sufficiently strong to overcome the gravity torque, i.e. it need not overcome friction in the gear
Applications and accuracy of the parallel diagonal dominant algorithm
NASA Technical Reports Server (NTRS)
Sun, Xian-He
1993-01-01
The Parallel Diagonal Dominant (PDD) algorithm is a highly efficient, ideally scalable tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is introduced. Then the algorithm is extended to solve periodic tridiagonal systems. A variant, the reduced PDD algorithm, is also proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric, and anti-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the algorithm is a good candidate for the emerging massively parallel machines.
A deterministic global optimization using smooth diagonal auxiliary functions
NASA Astrophysics Data System (ADS)
Sergeyev, Yaroslav D.; Kvasov, Dmitri E.
2015-04-01
In many practical decision-making problems it happens that functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. In this paper, a global optimization problem is considered where both the goal function f (x) and its gradient f‧ (x) are black-box functions. It is supposed that f‧ (x) satisfies the Lipschitz condition over the search hyperinterval with an unknown Lipschitz constant K. A new deterministic 'Divide-the-Best' algorithm based on efficient diagonal partitions and smooth auxiliary functions is proposed in its basic version, its convergence conditions are studied and numerical experiments executed on eight hundred test functions are presented.
Supersonic flow in an expanding MHD channel of diagonal type
Likhachev, A.P.; Medin, S.A.
1983-03-01
A quasi-two-dimensional numerical analysis of the supersonic flow of a nonviscous, non-heat-conducting gas in an expanding (with respect to the insulation walls) MHD channel of diagonal type is performed. The induced magnetic fields are neglected. The influence of the conditions of channel loading and the commutation angle of the short-circuited, ideally sectioned electrodes on the flow structure, the distribution of the electrodynamic parameters, and the integral characteristics of the generator is considered. The results of quasi-two-dimensional and quasi-one-dimensional calculations are compared.
Subsonic flow in the channel of a diagonal MHD generator
Isakova, N.P.; Medin, S.A.
1981-05-01
A numerical study has been made of the local and integral characteristics of the planar subsonic flow in the channel of an MHD generator with diagonal electrode connection. It is shown that the inhomogeneity in the parameter distribution is dependent on the electrical loading, and the largest deviations from homogeneous flow occur on open circuit and short circuit. A comparison is made with a channel of Faraday type as regards the main integral characteristics. The data from two-dimensional analysis are compared with those from a one-dimensional flow model.
Revealing children's implicit spelling representations.
Critten, Sarah; Pine, Karen J; Messer, David J
2013-06-01
Conceptualizing the underlying representations and cognitive mechanisms of children's spelling development is a key challenge for literacy researchers. Using the Representational Redescription model (Karmiloff-Smith), Critten, Pine and Steffler (2007) demonstrated that the acquisition of phonological and morphological knowledge may be underpinned by increasingly explicit levels of spelling representation. However, their proposal that implicit representations may underlie early 'visually based' spelling remains unresolved. Children (N = 101, aged 4-6 years) were given a recognition task (Critten et al., 2007) and a novel production task, both involving verbal justifications of why spellings are correct/incorrect, strategy use and word pattern similarity. Results for both tasks supported an implicit level of spelling characterized by the ability to correctly recognize/produce words but the inability to explain operational strategies or generalize knowledge. Explicit levels and multiple representations were also in evidence across the two tasks. Implications for cognitive mechanisms underlying spelling development are discussed. PMID:23659891