publication . Conference object . Other literature type . Preprint . 2017

Implicit and implicit-explicit strong stability preserving Runge–Kutta methods with high linear order

Sigal Gottlieb; Zachary Grant;
Open Access
  • Published: 15 Feb 2017
  • Publisher: Author(s)
Abstract
When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability co...
Subjects
free text keywords: Mathematics - Numerical Analysis

[1] U. Ascher, S. Ruuth, and B. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995), p. 797Ð823. [OpenAIRE]

[2] U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), pp. 151 - 167. Special Issue on Time Integration. [OpenAIRE]

[3] S. Boscarino, L. Pareschi, and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing, 35 (2013), pp. A22-A51.

[4] M. Calvo, J. de Frutos, and J. Novo, Linearly implicit runge-kutta methods for advection-reaction-diffusion equations., Applied Numerical Mathematics, 37 (2001), pp. 535- 549.

[6] [11] S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, Strong Stability Preserving RungeKutta and Multistep Time Discretizations, World Scientific Press, 2011.

[26] J. F. B. M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT, 31 (1991), pp. 482-528.

[27] E. J. Kubatko, B. A. Yeager, and D. I. Ketcheson, Optimal strong-stabilitypreserving runge-kutta time discretizations for discontinuous galerkin methods, J. Sci. Comput., 60 (2014), pp. 313-344.

[28] F. Kupka, N. Happenhofer, I. Higueras, and O. Koch, Total-variation-diminishing implicit-explicit runge-kutta methods for the simulation of double-diffusive convection in astrophysics, J. Comput. Phys., 231 (2012), pp. 3561-3586.

[29] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, Journal of Scientific Computing, 25 (2005), p. 129Ð155. [OpenAIRE]

[30] S. J. Ruuth and R. J. Spiteri, Two barriers on strong-stability-preserving time discretization methods, Journal of Scientific Computation, 17 (2002), pp. 211-220. [OpenAIRE]

[31] C.-W. Shu, Total-variation diminishing time discretizations, SIAM J. Sci. Stat. Comp., 9 (1988), pp. 1073-1084.

[32] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, Journal of Computational Physics, 77 (1988), pp. 439-471. [OpenAIRE]

Abstract
When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability co...
Subjects
free text keywords: Mathematics - Numerical Analysis

[1] U. Ascher, S. Ruuth, and B. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995), p. 797Ð823. [OpenAIRE]

[2] U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), pp. 151 - 167. Special Issue on Time Integration. [OpenAIRE]

[3] S. Boscarino, L. Pareschi, and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing, 35 (2013), pp. A22-A51.

[4] M. Calvo, J. de Frutos, and J. Novo, Linearly implicit runge-kutta methods for advection-reaction-diffusion equations., Applied Numerical Mathematics, 37 (2001), pp. 535- 549.

[6] [11] S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, Strong Stability Preserving RungeKutta and Multistep Time Discretizations, World Scientific Press, 2011.

[26] J. F. B. M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT, 31 (1991), pp. 482-528.

[27] E. J. Kubatko, B. A. Yeager, and D. I. Ketcheson, Optimal strong-stabilitypreserving runge-kutta time discretizations for discontinuous galerkin methods, J. Sci. Comput., 60 (2014), pp. 313-344.

[28] F. Kupka, N. Happenhofer, I. Higueras, and O. Koch, Total-variation-diminishing implicit-explicit runge-kutta methods for the simulation of double-diffusive convection in astrophysics, J. Comput. Phys., 231 (2012), pp. 3561-3586.

[29] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, Journal of Scientific Computing, 25 (2005), p. 129Ð155. [OpenAIRE]

[30] S. J. Ruuth and R. J. Spiteri, Two barriers on strong-stability-preserving time discretization methods, Journal of Scientific Computation, 17 (2002), pp. 211-220. [OpenAIRE]

[31] C.-W. Shu, Total-variation diminishing time discretizations, SIAM J. Sci. Stat. Comp., 9 (1988), pp. 1073-1084.

[32] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes, Journal of Computational Physics, 77 (1988), pp. 439-471. [OpenAIRE]

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