Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems

王阿霞 1; 2; 马逸尘 1

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Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems

更新时间：2025-12-08

- Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems
- Vol. 42, Issue 2, Pages: 234-237(2008)
- 作者机构：
  
  1. 西安交通大学理学院,西安,710049
  2. 长安大学理学院,西安,710064
- 作者简介：
- 基金信息：
- DOI：
  CLC： O241.82
- Online First：10 February 2008，
  
  Published：2008
- 稿件说明：
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王阿霞 1, 2, 马逸尘 1. Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems[J]. 2008, 42(2): 234-237.
DOI：

王阿霞 1, 2, 马逸尘 1. Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems[J]. 2008, 42(2): 234-237. DOI：

摘要

针对具有周期性边界条件对流占优的扩散问题中的二阶导数

引入辅助变量

构造了局部间断Galerkin(LDG)方法

并给出了方法的稳定性结果和误差估计式.局部间断Galerkin方法是Runge-Kutta间断Galerkin方法的推广

具有高阶精度

能够灵活处理复杂区域

易于处理复杂边界的边值问题

能够有效去除近似解在间断、大梯度处产生的虚假振荡.数值实验说明

当有限元空间取为一次多项式空间时

LDG方法具有二阶收敛

误差满足理论估计式.该方法可以推广到更高阶的方程

如Korteweg-de Vries方程、重调和方程等.

Abstract

Local discontinuous Galerkin method(LDGM)for convection-dominated diffusion problems with periodic boundary conditions is constructed. To deal with the derivative of second order

an auxiliary variable is introduced. The stability and error estimations are presented simultaneously. LDGM is an extension of the Runge-Kutta discontinuous Galerkin method with higher accuracy and flexibility

especially for complicated geometries and boundaries. The method enables to effectively remove the spurious oscillations around the discontinuities or strong gradients regions. The numerical experiments show that second order convergence can be obtained when polynomials of degree 1 are chosen and the numerical errors are consistent with the theoretical error estimations. It can be extended to solve partial differential equations with higher order such as KdV equations and biharmonic equations.

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references

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COCKBURN B, HOUAND S, SHU C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws: IV[J].Mathematics of Computation, 1990,54:545-581.

COCKBURN B, SHU C W. The local discontinuous Galerkin for time dependent convection-diffusion systems[J].SIAM Journal of Numerical Analysis, 1998,175:2440-2463.

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