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High-Order Non-Oscillatory Schemes for Nonlinear Conservation Laws with Source Terms

具有源项的非线性守恒定律的高阶非振荡方案

基本信息

批准号：
15607006
负责人：
TAKAKURA Yoko
金额：
$ 2.3万
依托单位：
National University Corporation Tokyo University of Agriculture and Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15607006/
关键词：
Nonlinear Conservation Laws with Source Terms ADER schemes High-order Shock Capturing Scheme Derivative Riemann Problem (DRP)Continuous Model Discrete Model Linear Instability Nonlinear Stability 高精度数値計算法 Riemann間題 Riemann問題

项目摘要

In this study, the ADER (Arbitrary-Accuracy Derivative Riemann problem) approach for constructing non-oscillatory explicit one-step schemes with very high order of accuracy in space and time has been extended to scalar nonlinear conservation laws with source terms, and further applied to the fluid dynamics problems including source terms so as to develop shock capturing schemes having both high accuracy and high stability.From the view point of keeping the high accuracy, two methods based on the state-series expansion and the direct expansion were shown in the extension of ADER approach from a linear equation to a nonlinear equation. The ADER schemes thus constituted were verified with the Burgers' equation (convex-flux equation) for two test problems of rapidly growing waves and very long-time propagating waves. As results, it was confirmed that the ADER schemes achieve the designed order of accuracy and capture shock and expansion waves clearly Furthermore, for extensive problems wit … More h a linear flux, nonlinear convex fluxes, nonlinear non-convex fluxes numerical verification showed that the ADER schemes achieve the high accuracy, high stability, and non-oscillatory property Then the ADER schemes have been applied to the fluid dynamics problems : first, to the one-dimensional Euler equation system, and next to the shallow water equation system as a source-term problem.We have conducted theoretical analysis on stability as well, observing the difference between properties of the continuous and discrete models. It was discovered that very small errors from the discrete computation may be amplified by some property of discrete model that does not come from the continuous model i.e. the original PDE. It seems that this phenomenon may happen especially in the case of nonlinear hyperbolic conservation laws including linearly degenerate fields. Such conservation laws generally occur in describing the wave propagation phenomena over continuum. This kind of numerical behavior deteriorates the precise estimate of effect from source terms and is important from the viewpoint of reliability of numerical computation. Less

在这项研究中，本文将构造在空间和时间上具有很高精度的无振荡显式一步格式的方法推广到含源项的标量非线性守恒律问题，并进一步应用于含源项的流体动力学问题，以发展既有高精度又有高稳定性的激波捕捉格式将ADER方法从线性方程推广到非线性方程，分别采用了基于状态级数展开和直接展开的两种方法。用Burgers方程（凸通量方程）对快速增长波和长时间传播波两个试验问题进行了验证。结果表明，ADER格式达到了设计的精度要求，能清晰地捕捉到激波和膨胀波。 ...更多信息 h线性通量、非线性凸通量、非线性非凸通量的数值验证表明，ADER格式具有高精度、高稳定性和无振荡性。首先，对于一维欧拉方程组，其次将浅水方程组作为源项问题，并对稳定性进行了理论分析，观察连续模型和离散模型的属性之间的差异。有人发现，非常小的误差，从离散计算可能会被放大的离散模型的一些属性，不来自连续模型，即原来的PDE。这一现象似乎在含有线性退化场的非线性双曲守恒律方程中尤其可能发生。这种守恒定律通常出现在描述连续介质上的波传播现象中。这种数值行为恶化了对源项影响的精确估计，从数值计算的可靠性角度来看是重要的。少