TVD格式计算一维Euler方程间断解存在的几个理论问题

本文讨论了TVD格式按本文第二节A段意义推广到拟线性方程组的几个理论问题。指出:1.对于Riemann间断分解,Euler方程的解和Riemann不变量都没有TVD性质,因此不能将TVD格式自然推广;2.TVD格式是提高分辨率的一种途径,它不能保证得到物理解,能得到物理解的TVD格式一定是耗散守恒格式,因此将TVD格式推广时,必须保证格式是耗散的,否则仍会得到非物理解。并给出了一阶迎风格式,TVD2,UNO2得到的若干非物理解算例;3.非物理解现象只是发生在具有剧烈膨胀的单波区的计算上,而某些TVD格式若会得到非物理解,则一定发生在特征值变号的单波区的计算上。

Some Theoretical Problems on TVD Schemes for Computation of One-Dimensional Euler Equations

In this paper, some theoretical problems on TVD schemes have been discussed. We have come to the follwing conclusions: 1. For Riemann problem, the solution of Euler equation and the Riemann invariants do not have TVD property, so it is unreasonble to extend TVD scheme naturally to system of quasilinear equation in the meaning of section 2A. 2. TVD schemes are just a way to raise the resolution of discontinuity, they can not guarantee that the difference solutions converge to physical solutions.The TVD schemes giving physical solutions must be dissi-pative conservative schemes. When extending TVD scheme for scalar equation to Euler equations, we must ensure that the scheme is dissipative, otherwise the difference solution may converge to non-physical solution. The examples of non-physical solution by first-order upwind scheme, TVD2, UN02 schemes are presented. 3. We point out that non-physical solution phenomena only happen inside a simple wave with strong rarefaction. Furthermore, if some TVD schemes give non-physical solutions, they must happen inside the simple wave in which the sign of the general eigenvalue changes.