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Analysis of Local Discontinuous Galerkin Methods with Implicit-Explicit Time Marching for Linearized KdV Equations
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-09-19 , DOI: 10.1137/24m1635818
Haijin Wang, Qi Tao, Chi-Wang Shu, Qiang Zhang
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-09-19 , DOI: 10.1137/24m1635818
Haijin Wang, Qi Tao, Chi-Wang Shu, Qiang Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2222-2248, October 2024.
Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.
中文翻译:
线性化KdV方程的隐式-显式时间推进局部间断伽辽金法分析
《SIAM 数值分析杂志》,第 62 卷,第 5 期,第 2222-2248 页,2024 年 10 月。
抽象的。在本文中,我们针对线性化一维 KdV 方程,提出了两种完全离散 IMEX-LDG 方案的稳定性和误差分析,将局部不连续 Galerkin 空间离散化与隐式显式 Runge-Kutta 时间离散化相结合。能量稳定性分析从阶段解的一系列时间差异开始。然后从时间差异出发探索稳定性机制,并构造与色散项离散化有关的半负定对称形式,并利用辅助变量与主变量之间的重要关系来控制反耗散项,得到涉及主变量和所有辅助变量的离散能量的无条件稳定性,即时间步长受独立于空间网格大小的常数限制。我们还提出了一种新的投影技术,并采用时间方向上的分部求和技术来实现精度的最佳顺序。新的投影技术可以作为一种分析工具应用于一般奇数阶波动方程。最后,通过数值实验来测试所考虑方案的稳定性和准确性。
更新日期:2024-09-20
Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.
中文翻译:
线性化KdV方程的隐式-显式时间推进局部间断伽辽金法分析
《SIAM 数值分析杂志》,第 62 卷,第 5 期,第 2222-2248 页,2024 年 10 月。
抽象的。在本文中,我们针对线性化一维 KdV 方程,提出了两种完全离散 IMEX-LDG 方案的稳定性和误差分析,将局部不连续 Galerkin 空间离散化与隐式显式 Runge-Kutta 时间离散化相结合。能量稳定性分析从阶段解的一系列时间差异开始。然后从时间差异出发探索稳定性机制,并构造与色散项离散化有关的半负定对称形式,并利用辅助变量与主变量之间的重要关系来控制反耗散项,得到涉及主变量和所有辅助变量的离散能量的无条件稳定性,即时间步长受独立于空间网格大小的常数限制。我们还提出了一种新的投影技术,并采用时间方向上的分部求和技术来实现精度的最佳顺序。新的投影技术可以作为一种分析工具应用于一般奇数阶波动方程。最后,通过数值实验来测试所考虑方案的稳定性和准确性。