Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems
Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems
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DOI：
10.1137/140956750
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发表时间：
2015-01
期刊：
影响因子：
--
通讯作者：
Haijin Wang;Chi-Wang Shu;Qiang Zhang
中科院分区：
文献类型：
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作者：
Haijin Wang;Chi-Wang Shu;Qiang Zhang
本文的主要目的是分析局部间断Galerkin（LDG）方法与精心选择的隐式-显式（IMEX）Runge-Kutta时间离散相结合求解一维线性对流扩散方程的三阶精度的稳定性和误差估计。在时间离散中，对流项被显式处理，扩散项被隐式处理。这项工作有三个亮点。第一，利用LDG方法中梯度的独立数值解，建立了梯度与数值解界面跳跃之间的重要关系。第二，借助于上述关系和能量方法，我们证明了IMEX LDG格式对于线性问题是无条件稳定的，即时间步长τ只需要由一个依赖于扩散系数与平流系数平方之比的常数来确定上界，并且与时间步长τ无关。
The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge--Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step $\tau$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent...