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

• 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

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

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基于状态序列扩展和直接扩展的方法在ADER方法的扩展中从线性方程式到非线性方程。这样构成的ADER方案已用汉堡方程（凸 - 流体方程）进行了验证，该方程的两个测试问题是快速生长的波和很长时间的传播波。结果，证实ADER计划达到了精确的准确性顺序，并捕获了震动和扩展波，此外，对于广泛的问题而言，智慧的广泛问题……更多的线性通量，非线性凸出通量，非线性非convex通量，非线性磁通量的数值验证，表明了ADER的高精度，然后将ADER的范围用于高度稳定性，并具有高度稳定性，并且是不稳定性的。 ：首先，对于一维欧拉方程系统，在浅水方程系统旁边作为源期问题，我们也对稳定性进行了理论分析，观察到连续模型和离散模型的性质之间的差异。已经发现，离散计算中的非常小的错误可能会被某些离散模型的某些属性放大，而这些模型的某些属性不是来自连续模型，即原始PDE。看来这种现象可能发生，尤其是在非线性双曲保护定律（包括线性退化场）的情况下。这种保护定律通常在描述连续体的波传播现象时发生。这种数值行为决定了源术语的效果的精确估计，从数值计算的可靠性角度来看很重要。较少的

项目成果

ADER法に基づく生成項付非線形保存則のための高精度スキームの様々な構築手法

基于ADER方法的带有生成项的非线性守恒定律高精度格式的多种构造方法

• 发表时间: 2003
• 期刊: 2003年応用数学合同研究集会報告集 (学会発表)
• 作者: 畠山彰文;中山剛志;阿部和厚;田村至;森寿子;高倉葉子
• 通讯作者: 高倉葉子

Machinery of Numerical Instability in Conservative Difference Approximations for Compressible Euler Equations

可压缩欧拉方程保守差分近似中数值不稳定性的机制

• 发表时间: 2003
• 期刊: 京都大学数理科学研究所講究録 1353
• 作者: AISO;H.;ABOUZIAROV;M.;TAKAHASHI;T.
• 通讯作者: T.

Various Forms of ADER Schemes for Nonlinear Conservation Laws with Source Terms and Their Verification

具有源项的非线性守恒定律的各种形式的ADER方案及其验证

• 发表时间: 2005
• 期刊: Proceedings of the Tenth International Conference on Hyperbolic Problems (TO appear)
• 作者: TAKAKURA;Y.
• 通讯作者: Y.

Construction of High-Accuracy Schemes for Nonlinear Conservation Laws with Source Terms Based on the ADER Approach (in Japanese)

基于 ADER 方法的带有源项的非线性守恒定律高精度方案的构建（日语）

• 发表时间: 2003
• 期刊: Proceedings of Ouyo-Sugau-Godo--Kenkyu-Shukai (United Meeting for Applied Mathematics) 2003
• 作者: TAKAKURA;Y.
• 通讯作者: Y.

Some Applications of ADER Schemes for Nonlinear Conservation Laws (in Japanese)

非线性守恒律 ADER 方案的一些应用（日语）

• 发表时间: 2004
• 期刊: Proceedings of the 18th Computational Fluid Dynamics Symposium D7-1(CD-ROM)(Domestic)
• 作者: TAKAKURA;Y.
• 通讯作者: Y.