Consider the following neutral delay-differential equations with multiple delays (NMDDE)where γ> 0, L, Mj and Nj are constant complex- value d×d matrices. A sufficient condition for the asymptotic stability o...Consider the following neutral delay-differential equations with multiple delays (NMDDE)where γ> 0, L, Mj and Nj are constant complex- value d×d matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE are studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods(NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methods. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).展开更多
Central discontinuous Galerkin（CDG）method is used to solve the Navier-Stokes equations for viscous flow in this paper.The CDG method involves two pieces of approximate solutions defined on overlapping meshes.Taking ...Central discontinuous Galerkin（CDG）method is used to solve the Navier-Stokes equations for viscous flow in this paper.The CDG method involves two pieces of approximate solutions defined on overlapping meshes.Taking advantages of the redundant representation of the solution on the overlapping meshes,the cell interface of one computational mesh is right inside the staggered mesh,hence approximate Riemann solvers are not needed at cell interfaces.Third order total variation diminishing（TVD）Runge-Kutta（RK）methods are applied in time discretization.Numerical examples for 1D and2 D viscous flow simulations are presented to validate the accuracy and robustness of the CDG method.展开更多
A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the ...A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the iteration parameters of the method are tuned in such a way that fast convergence to the value of corrector is achieved.展开更多
A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)method...A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.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
We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the...We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.展开更多
Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Progra...Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Program simulations of Euler method,Heun method,lassic fourth-order Runge Kutta（RK4）method,ABM method and Hamming method are achieved based on Matlab.In addtion,the approximate solutions,local truncation errors and calculation time of the dynamic differential equations are obtained.By analyzing the simultaion results,the advantages and disadvantages of these methods are compared,which provides a basis for choice of ballistic calculation methods.展开更多
A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equation...A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.展开更多
文摘Consider the following neutral delay-differential equations with multiple delays (NMDDE)where γ> 0, L, Mj and Nj are constant complex- value d×d matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE are studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods(NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methods. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).
基金Supported by the National Natural Science Foundation of China(11602262)
文摘Central discontinuous Galerkin（CDG）method is used to solve the Navier-Stokes equations for viscous flow in this paper.The CDG method involves two pieces of approximate solutions defined on overlapping meshes.Taking advantages of the redundant representation of the solution on the overlapping meshes,the cell interface of one computational mesh is right inside the staggered mesh,hence approximate Riemann solvers are not needed at cell interfaces.Third order total variation diminishing（TVD）Runge-Kutta（RK）methods are applied in time discretization.Numerical examples for 1D and2 D viscous flow simulations are presented to validate the accuracy and robustness of the CDG method.
文摘A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the iteration parameters of the method are tuned in such a way that fast convergence to the value of corrector is achieved.
基金supported by NSF Grants DMS-1719410 and DMS-2010107by AFOSR Grant FA9550-20-1-0055.
文摘A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.
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Open Access funding enabled and organized by Projekt DEAL.Andrew Winters was funded through Vetenskapsrådet,Sweden Grant Agreement 2020-03642 VR.Some computations were enabled by resources provided by the Swedish National Infrastructure for Computing(SNIC)at Tetralith,par-tially funded by the Swedish Research Council under Grant Agreement No.2018-05973Hugo Guillermo Castro was funded through the award P2021-0004 of King Abdullah University of Science and Technol-ogy.Some of the simulations were enabled by the Supercomputing Laboratory and the Extreme Comput-ing Research Center at King Abdullah University of Science and Technology.Gregor Gassner acknowl-edges funding through the Klaus-Tschira Stiftung via the project“HiFiLab”.Gregor Gassner and Michael Schlottke-Lakemper acknowledge funding from the Deutsche Forschungsgemeinschaft through the research unit“SNuBIC”(DFG-FOR5409).
文摘We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.
文摘Several numerical methods of differential equations and their applications in ballistic calculation are discussed for the purpose of simplification of the dynamic differential equations of projectile trajectory.Program simulations of Euler method,Heun method,lassic fourth-order Runge Kutta（RK4）method,ABM method and Hamming method are achieved based on Matlab.In addtion,the approximate solutions,local truncation errors and calculation time of the dynamic differential equations are obtained.By analyzing the simultaion results,the advantages and disadvantages of these methods are compared,which provides a basis for choice of ballistic calculation methods.
文摘A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.
