The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

doi:10.1007/s42967-020-00114-1

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 271-292.doi: 10.1007/s42967-020-00114-1

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The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

Haijin Wang¹, Qiang Zhang²

1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China;
2 Department of Mathematics, Nanjing University, Nanjing 210093, China

收稿日期:2020-08-27 修回日期:2020-11-27 出版日期:2022-03-20 发布日期:2022-03-01
通讯作者: Haijin Wang, Qiang Zhang E-mail:hjwang@njupt.edu.cn;qzh@nju.edu.cn

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

Haijin Wang¹, Qiang Zhang²

1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China;
2 Department of Mathematics, Nanjing University, Nanjing 210093, China

Received:2020-08-27 Revised:2020-11-27 Online:2022-03-20 Published:2022-03-01
Contact: Haijin Wang, Qiang Zhang E-mail:hjwang@njupt.edu.cn;qzh@nju.edu.cn

摘要/Abstract

摘要： In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-difusion problems. In the spatial discretization, both the original DDG methods and the refned DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the difusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step τ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

关键词: Direct discontinuous Galerkin method, Implicit-explicit scheme, Stability analysis, Energy method, Convection-difusion problem

Abstract: In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-difusion problems. In the spatial discretization, both the original DDG methods and the refned DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the difusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step τ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

Key words: Direct discontinuous Galerkin method, Implicit-explicit scheme, Stability analysis, Energy method, Convection-difusion problem

中图分类号:

Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.

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The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems

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