A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System

doi:10.1007/s42967-021-00178-7

摘要/Abstract

摘要： The purpose of this paper is to develop a hybridized discontinuous Galerkin (HDG) method for solving the Ito-type coupled KdV system. In fact, we use the HDG method for discretizing the space variable and the backward Euler explicit method for the time variable. To linearize the system, the time-lagging approach is also applied. The numerical stability of the method in the sense of the L₂ norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet boundary conditions. Numerical experiments confirm that the HDG method is capable of solving the system efficiently. It is observed that the best possible rate of convergence is achieved by the HDG method. Also, it is being illustrated numerically that the corresponding conservation laws are satisfied for the approximate solutions of the Ito-type coupled KdV system. Thanks to the numerical experiments, it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.

关键词: Hybridized discontinuous Galerkin (HDG) method, Stability analysis, Ito-type coupled KdV system, Conservation laws

Abstract: The purpose of this paper is to develop a hybridized discontinuous Galerkin (HDG) method for solving the Ito-type coupled KdV system. In fact, we use the HDG method for discretizing the space variable and the backward Euler explicit method for the time variable. To linearize the system, the time-lagging approach is also applied. The numerical stability of the method in the sense of the L₂ norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet boundary conditions. Numerical experiments confirm that the HDG method is capable of solving the system efficiently. It is observed that the best possible rate of convergence is achieved by the HDG method. Also, it is being illustrated numerically that the corresponding conservation laws are satisfied for the approximate solutions of the Ito-type coupled KdV system. Thanks to the numerical experiments, it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.

Key words: Hybridized discontinuous Galerkin (HDG) method, Stability analysis, Ito-type coupled KdV system, Conservation laws

中图分类号:

Shima Baharlouei, Reza Mokhtari, Nabi Chegini. A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1351-1373.

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