Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

doi:10.1007/s42967-023-00285-7

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (1): 658-687.doi: 10.1007/s42967-023-00285-7

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Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

Joseph Hunter¹, Zheng Sun², Yulong Xing¹

1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA;
2. Department of Mathematics, The University of Alabama, Tuscaloosa, AL, 35487, USA

收稿日期:2023-03-01 修回日期:2023-05-07 发布日期:2024-04-16
通讯作者: Yulong Xing,E-mail:xing.205@osu.edu;Joseph Hunter,E-mail:hunter.926@osu.edu;Zheng Sun,E-mail:zsun30@ua.edu E-mail:xing.205@osu.edu;hunter.926@osu.edu;zsun30@ua.edu
基金资助:
The work of Z. Sun is partially supported by the NSF under Grant DMS-2208391. The work of Y. Xing is partially sponsored by the NSF under Grant DMS-1753581.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

Joseph Hunter¹, Zheng Sun², Yulong Xing¹

1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA;
2. Department of Mathematics, The University of Alabama, Tuscaloosa, AL, 35487, USA

Received:2023-03-01 Revised:2023-05-07 Published:2024-04-16
Contact: Yulong Xing,E-mail:xing.205@osu.edu;Joseph Hunter,E-mail:hunter.926@osu.edu;Zheng Sun,E-mail:zsun30@ua.edu E-mail:xing.205@osu.edu;hunter.926@osu.edu;zsun30@ua.edu
Supported by:
The work of Z. Sun is partially supported by the NSF under Grant DMS-2208391. The work of Y. Xing is partially sponsored by the NSF under Grant DMS-1753581.

摘要/Abstract

摘要： 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) condition \(\tau \leqslant \hat{\lambda } h\). Here, \(\hat{\lambda }\) is the CFL number, \(\tau\) is the time-step size, and h is the spatial mesh size. We study several IMEX schemes and characterize their CFL number as a function of \(\theta =d/h^2\) 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.

关键词: Linearized Korteweg-de Vries (KdV) equation, Implicit-explicit (IMEX) Runge-Kutta (RK) method, Stability, Courant-Friedrichs-Lewy (CFL) condition, Finite difference (FD) method, Local discontinuous Galerkin (DG) method

Abstract: 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) condition \(\tau \leqslant \hat{\lambda } h\). Here, \(\hat{\lambda }\) is the CFL number, \(\tau\) is the time-step size, and h is the spatial mesh size. We study several IMEX schemes and characterize their CFL number as a function of \(\theta =d/h^2\) 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.

Key words: Linearized Korteweg-de Vries (KdV) equation, Implicit-explicit (IMEX) Runge-Kutta (RK) method, Stability, Courant-Friedrichs-Lewy (CFL) condition, Finite difference (FD) method, Local discontinuous Galerkin (DG) method

Joseph Hunter, Zheng Sun, Yulong Xing. Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation[J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 658-687.

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Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

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