Heuristic Modified Equation Analysis on Oscillations in Numerical Solutions of Conservation Laws
Heuristic Modified Equation Analysis on Oscillations in Numerical Solutions of Conservation Laws
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DOI：
10.1137/110822591
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发表时间：
2011-11
期刊：
影响因子：
--
通讯作者：
Jiequan Li;Zhicheng Yang
中科院分区：
文献类型：
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作者：
Jiequan Li;Zhicheng Yang
振荡现象普遍存在于双曲问题的高阶甚至一阶格式的数值解中，通常被认为是底层数值格式低耗散效应的结果。早期的分析主要是通过对线性问题的有效离散傅立叶分析或在光滑解区域的修正方程方法来完成的。本文导出了一个所谓的启发式修正方程，当应用于非线性问题时，特别是对于在线性问题中对应于高频模式解的解的振荡模式，耗散效应被区分为数值阻尼和数值扩散。前者通过启发式修正方程的零阶项反映，后者通过二阶微分项反映。结果表明，耗散效应可分为阻尼、中性和放大，数值阻尼在抵消振荡模式中起着主导作用。当考虑放大效应时，数值格式往往变得不稳定。
Oscillations are ubiquitous in numerical solutions obtained by high order or even first order schemes for hyperbolic problems and are conventionally understood as the consequence of low dissipation effects of underlying numerical schemes. Earlier analysis was done mainly through the effective discrete Fourier analysis for linear problems or the modified equation approach in smooth solution regions. In this paper, a so-called heuristic modified equation is derived when applied to nonlinear problems, particularly for oscillatory modes of solutions whose counterpart in linear problems are high frequency mode solutions, and the dissipation effect is distinguished as a numerical damping and a numerical diffusion. The former is reflected through the zero order term of the heuristic modified equation and the latter through the second order differential term. It turns out that the effect of dissipation is categorized as a damping, a neutrality, and an amplification, and that the numerical damping plays a dominant role in offsetting the oscillatory modes. When the amplification effect is taken, the numerical scheme often comes unstable.