In order to eliminate Courant-Friedrich-Levy(CFL) condition restraint and improvecomputational efficiency,a new finite-difference time-domain(FDTD)method based on the alternating-direction implicit(ADI) technique is i...In order to eliminate Courant-Friedrich-Levy(CFL) condition restraint and improvecomputational efficiency,a new finite-difference time-domain(FDTD)method based on the alternating-direction implicit(ADI) technique is introduced recently.In this paper,a theoretical proof of the stabilityof the three-dimensional(3-D)ADI-FDTD method is presented.It is shown that the 3-D ADI-FDTDmethod is unconditionally stable and free from the CFL condition restraint.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
Staggered-grid finite-difference(SGFD)schemes have been widely used in acoustic wave modeling for geophysical problems.Many improved methods are proposed to enhance the accuracy of numerical modeling.However,these met...Staggered-grid finite-difference(SGFD)schemes have been widely used in acoustic wave modeling for geophysical problems.Many improved methods are proposed to enhance the accuracy of numerical modeling.However,these methods are inevitably limited by the maximum Courant-Friedrichs-Lewy(CFL)numbers,making them unstable when modeling with large time sampling intervals or small grid spacings.To solve this problem,we extend a stable SGFD scheme by controlling SGFD dispersion relations and maximizing the maximum CFL numbers.First,to improve modeling stability,we minimize the error between the FD dispersion relation and the exact relation in the given wave-number region,and make the FD dispersion approach a given function outside the given wave-number area,thus breaking the conventional limits of the maximum CFL number.Second,to obtain high modeling accuracy,we use the SGFD scheme based on the Remez algorithm to compute the FD coefficients.In addition,the hybrid absorbing boundary condition is adopted to suppress boundary reflections and we find a suitable weighting coefficient for the proposed scheme.Theoretical derivation and numerical modeling demonstrate that the proposed scheme can maintain high accuracy in the modeling process and the value of the maximum CFL number of the proposed scheme can exceed that of the conventional SGFD scheme when adopting a small maximum effective wavenumber,indicating that the proposed scheme improves stability during the modeling.展开更多
The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin(DG)solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit...The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin(DG)solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping.A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy(CFL)conditions established for fixed and uniform meshes.In this work,we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations.A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws.Based on the analysis,a new selection strategy of the time step is proposed,which takes into consideration the coupling of theα-function(that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity)and the heights of the mesh elements.The analysis also suggests several stable combinations of the choices of theα-function in the numerical scheme and in the time step selection.Numerical results obtained with a moving mesh DG method for Burgers’and Euler equations are presented.For comparison purpose,numerical results obtained with an error-based time step-size selection strategy are also given。展开更多
Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectiv...Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectively,thereby limiting time step selection.Based on the definition of temporal and spatial FD operators,we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy.First,to simulate wave propagation beyond the conventional CFL stability limit,the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones.Second,to preserve the modeling accuracy,the velocity-dependent lengths of the temporal and spatial operators are adaptively varied.The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples,respectively.We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.展开更多
In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for ...In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.展开更多
The Courant-Friedrichs-Lewy condition(The CFL condition)is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations.We give a remark on the CFL condition fro...The Courant-Friedrichs-Lewy condition(The CFL condition)is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations.We give a remark on the CFL condition from a view point of stability,and we give some numerical experiments which show instability of numerical solutions even under the CFL condition.We give a mathematical model for rounding errors in order to explain the instability。展开更多
A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is k...A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional(2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.展开更多
The linear form of the error propagation of SPH,which was obtained through perturbation method,has been employed to analyze the tensile instability in SPH.The sufficient condition for tensile instability,which was eve...The linear form of the error propagation of SPH,which was obtained through perturbation method,has been employed to analyze the tensile instability in SPH.The sufficient condition for tensile instability,which was ever presented by Swegle,could also be derived from the eigenvalues of the linear form.Hence,the eigenvalues correspondingly yielded a tensile stability criterion.The criterion confirmed the Swegle's statement that the tensile instability is induced by imaginary sound speed,and revealed the origins of imaginary sound speed and some details of CFL conditions.Moreover,a reasonable numerical sound speed,which accords with the one given by Monaghan through dimensional analysis,was also derived from the criterion.The kernel's spatial derivatives,which are only with respect to the distance between particles,were found it was not accurate if the spatial derivatives of smoothing lengths were not trifle.展开更多
The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sw...The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.展开更多
基金Supported by the Specialized Research Fund for the Doctoral Program of Higher Education(No.20010614003)
文摘In order to eliminate Courant-Friedrich-Levy(CFL) condition restraint and improvecomputational efficiency,a new finite-difference time-domain(FDTD)method based on the alternating-direction implicit(ADI) technique is introduced recently.In this paper,a theoretical proof of the stabilityof the three-dimensional(3-D)ADI-FDTD method is presented.It is shown that the 3-D ADI-FDTDmethod is unconditionally stable and free from the CFL condition restraint.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金This research is supported by the National Natural Science Foundation of China(NSFC)under contract no.42274147.
文摘Staggered-grid finite-difference(SGFD)schemes have been widely used in acoustic wave modeling for geophysical problems.Many improved methods are proposed to enhance the accuracy of numerical modeling.However,these methods are inevitably limited by the maximum Courant-Friedrichs-Lewy(CFL)numbers,making them unstable when modeling with large time sampling intervals or small grid spacings.To solve this problem,we extend a stable SGFD scheme by controlling SGFD dispersion relations and maximizing the maximum CFL numbers.First,to improve modeling stability,we minimize the error between the FD dispersion relation and the exact relation in the given wave-number region,and make the FD dispersion approach a given function outside the given wave-number area,thus breaking the conventional limits of the maximum CFL number.Second,to obtain high modeling accuracy,we use the SGFD scheme based on the Remez algorithm to compute the FD coefficients.In addition,the hybrid absorbing boundary condition is adopted to suppress boundary reflections and we find a suitable weighting coefficient for the proposed scheme.Theoretical derivation and numerical modeling demonstrate that the proposed scheme can maintain high accuracy in the modeling process and the value of the maximum CFL number of the proposed scheme can exceed that of the conventional SGFD scheme when adopting a small maximum effective wavenumber,indicating that the proposed scheme improves stability during the modeling.
基金M.Zhang was partially supported by the Postdoctoral Science Foundation of China(Grant 2022M710229)J.Qiu was partially supported by National Natural Science Foundation of China(Grant 12071392).
文摘The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin(DG)solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping.A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy(CFL)conditions established for fixed and uniform meshes.In this work,we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations.A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws.Based on the analysis,a new selection strategy of the time step is proposed,which takes into consideration the coupling of theα-function(that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity)and the heights of the mesh elements.The analysis also suggests several stable combinations of the choices of theα-function in the numerical scheme and in the time step selection.Numerical results obtained with a moving mesh DG method for Burgers’and Euler equations are presented.For comparison purpose,numerical results obtained with an error-based time step-size selection strategy are also given。
基金the National Natural Science Foundation of China(No.41874144)the Research Foundation of China University of PetroleumBeijing at Karamay(RCYJ2018A-01-001).
文摘Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectively,thereby limiting time step selection.Based on the definition of temporal and spatial FD operators,we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy.First,to simulate wave propagation beyond the conventional CFL stability limit,the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones.Second,to preserve the modeling accuracy,the velocity-dependent lengths of the temporal and spatial operators are adaptively varied.The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples,respectively.We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.
文摘In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.
基金Grant-in-Aid for Scientific Research(B)22340018 from Japan Society for the Promotion of Science.
文摘The Courant-Friedrichs-Lewy condition(The CFL condition)is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations.We give a remark on the CFL condition from a view point of stability,and we give some numerical experiments which show instability of numerical solutions even under the CFL condition.We give a mathematical model for rounding errors in order to explain the instability。
基金Project supported by the National Natural Science Foundation of China(No.11302017)
文摘A stable high-order Runge-Kutta discontinuous Galerkin(RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional(2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.
基金Sponsored by the National Natural Science Foundation of China (Grant No. 10832007)
文摘The linear form of the error propagation of SPH,which was obtained through perturbation method,has been employed to analyze the tensile instability in SPH.The sufficient condition for tensile instability,which was ever presented by Swegle,could also be derived from the eigenvalues of the linear form.Hence,the eigenvalues correspondingly yielded a tensile stability criterion.The criterion confirmed the Swegle's statement that the tensile instability is induced by imaginary sound speed,and revealed the origins of imaginary sound speed and some details of CFL conditions.Moreover,a reasonable numerical sound speed,which accords with the one given by Monaghan through dimensional analysis,was also derived from the criterion.The kernel's spatial derivatives,which are only with respect to the distance between particles,were found it was not accurate if the spatial derivatives of smoothing lengths were not trifle.
基金This work is partially supported by Sloan FoundationNSF DMS0513073+1 种基金ONR grant N00014-02-1-0090DARPA grant N00014-02-1-0603
文摘The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.
