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首页> 外文期刊>Journal of Computational Physics >An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations
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An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations

机译：一般非定常扩散方程的具有局部时间步长的显式不连续Galerkin方案

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摘要

In this paper we propose a discontinuous Galerkin scheme for the numerical approximation of unsteady heat conduction and diffusion problems in multi dimensions. The scheme is based on a discrete space-time variational formulation and uses an explicit approximative solution as predictor. This predictor is obtained by a Taylor expansion about the barycenter of each grid cell at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the differential equation several times. The heat flux between adjacent grid cells is approximated by a local analytical solution. It takes into account that the approximate solution may be discontinuous at grid cell interfaces and allows the approximation of discontinuities in the heat conduction coefficient. The presented explicit scheme has to satisfy a typical parabolic stability restriction. The loss of efficiency, especially in the case of strongly varying sizes of cells in unstructured grids, is circumvented by allowing different time steps in each grid cell which are adopted to the local stability restrictions. We discuss the linear stability properties in this case of varying diffusion coefficients, varying space increments and local time steps and extent these considerations also to a modified symmetric interior penalization scheme. In numerical simulations we show the efficiency and the optimal order of convergence in space and time. (c) 2008 Elsevier Inc. All rights reserved.
机译：在本文中，我们提出了一个不连续的Galerkin方案，用于多维非稳态热传导和扩散问题的数值逼近。该方案基于离散的时空变分公式，并使用显式近似解作为预测变量。通过在旧时间级别上围绕每个网格单元的重心的泰勒展开式获得该预测因子，在该时间范围内，所有时间或混合时空导数都使用微分方程多次替换为空间导数。相邻网格单元之间的热通量通过局部解析解近似。考虑到近似解在网格单元界面处可能是不连续的，并且允许热传导系数的不连续性近似。提出的显式方案必须满足典型的抛物线稳定性限制。通过允许每个网格单元中采用不同的时间步长来避免效率损失，尤其是在非结构化网格中单元大小变化非常大的情况下，这些时间步长将采用局部稳定性限制。在变化的扩散系数，变化的空间增量和局部时间步长的情况下，我们讨论了线性稳定性，并将这些考虑也扩展到了一种改进的对称内部惩罚方案中。在数值模拟中，我们显示了时空收敛的效率和最佳顺序。 （c）2008 Elsevier Inc.保留所有权利。

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