自相似Euler方程的数值方法

Numerical Methods for Euler Equations with Self-Similar Solutions

Euler方程某些问题的解具有自相似特点,可以使用更准确的方法求解.提出了两种数值方法,分别称为自相似和准自相似方法,新方法可以使用现有守恒律方程的数值格式,无须设计特殊方法.对一维激波管问题、二维Riemann问题、激波反射以及激波折射问题进行了数值计算.对自相似Euler方程,一维计算结果显示数值解基本等同于精确解,二维结果也比现有文献计算的结果有更高的分辨率.对准自相似Euler方程,新方法可以求解不具有自相似性但接近自相似的问题,并在计算时间足够长时可以取得自相似Euler方程的效果.数值求解自相似Euler方程对自相似问题的研究,高分辨率、高精度格式的设计乃至Euler方程的精确解都有重要启示.

Some problems of Euler equations have self-similar solutions which can be solved by more accurate method.The current paper proposes two new numerical methods for Euler equations with self-similar and quasi self-similar solutions respectively,which can use existing difference schemes for conservation laws and do not need to redesign specified schemes.Numerical simulations were implemented on one-dimensional shock tube problems,two-dimensional Riemann problems,shock reflection from a solid wedge,and shock refraction at a gaseous interface.For self-similar equations,one-dimensional results are almost equal to the exact solutions,and two-dimensional results also exhibit considerable high resolution.For quasi self-similar equations,the method can solve solutions that are not but close to self-similar,i.e.,quasi self-similar,and this method can also achieve very high resolution when computing time is long enough.Numerical simulations to self-similar and quasi self-similar Euler equations have important implications on the study of self-similar problems,development of high resolution schemes,and even the research on exact solutions of Euler equations.