The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique
The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique
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两个有趣的双曲问题的解析解作为采用新网格细化技术的有限体积方法的测试用例
DOI：
10.1016/j.cam.2007.03.008
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发表时间：
2008
影响因子：
2.4
通讯作者：
Matthias Kunik
中科院分区：
文献类型：
--
作者：
W. Heineken;Matthias Kunik
将网格自适应有限体积法应用于两个双曲型问题：超相对论欧拉方程和标量守恒律。这两个问题都是在两个空间维中考虑的，并具有运动激波的共同特征。与经典欧拉方程不同，超相对论欧拉方程的适当初始条件的推导是一个利用一维激波条件和系统的洛伦兹不变性来求解的非平凡问题。这两个问题的离散都是基于三角形网格上的空间和时间上的二阶有限体积法。我们介绍了最小模限幅器的一种变体，它避免了欧拉系统的非物理状态。网格在集成过程中进行调整。网格自适应的频率是自动控制的，以保证运动激波阵面的高分辨率。我们引入了“宽度细化”的概念，扩大了激波阵面周围强细化区域的宽度；通过数值研究找到了最佳宽度。因此，我们能够通过减少适应步骤的数量来提高效率。将有限体积格式与几种低阶方法的性能进行了比较。
A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.