| 自相似Euler方程的数值方法 |
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| 作者姓名： | 刘福军 董海涛 |
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| 作者单位： | 北京航空航天大学数学与系统科学学院,北京100191;北京航空航天大学航空科学与工程学院国家计算流体力学实验室,北京100191 |
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| 摘 要： | 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.
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| 关 键 词： | 自相似 准自相似 EULER方程 激波管问题 二维Riemann问题 激波反射 |
| 收稿时间： | 2019-06-21 |
Numerical Methods for Euler Equations with Self-Similar Solutions |
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| Authors: | LIU Fu-jun DONG Hai-tao |
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| Affiliation: | 1.School of Mathematic Sciences, Beihang University, Beijing 100191, China2.National Laboratory of Computational Fluid Dynamics, School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China |
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| Abstract: | 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. |
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| Keywords: | self-similar quasi self-similar Euler equations shock tube problem 2D Riemann problem shock reflection |
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