高阶精度有限差分方法几何守恒律研究进展

针对高阶精度有限差分格式的几何守恒律问题,系统梳理了国内外离散几何守恒律问题方面的研究工作,以有限差分格式离散后的自由流保持问题为切入点,综述了近年来课题组在有限差分离散几何守恒律方面的研究工作,包括守恒网格导数算法(CMM)、对称守恒网格导数算法(SCMM)等,并通过若干典型算例验证了几何守恒律的满足对高阶精度有限差分方法数值模拟能力的提升。通过对有限差分离散几何守恒律问题研究工作的系统梳理,总结相关认识如下:1)直接基于网格变换导数的传统计算形式采用有限差分离散不能满足几何守恒律,需采用网格变换导数的守恒计算形式同时还需满足CMM条件;2)SCMM条件是满足几何守恒律的充分条件,且网格变换导数和雅克比均需采用其对称守恒计算形式,具有唯一性;3)自由流保持只是满足几何守恒律的一种表现形式;4)几何守恒律的满足能够有效提升高阶精度有限差分格式的数值模拟能力。

A survey of geometry conservation law for high-order finite difference method

Focusing on the geometry conservation law(GCL)for high-order finite difference schemes,the domestic and overseas studies regarding discretized geometric conservation law are reviewed systematically.Beginning with the freestream preservation phenomenon simulated by finite difference schemes,our recent studies about the geometric conservation law are summarized,including the conservative metric method(CMM)and the symmetrical conservative metric method(SCMM).Moreover,it is confirmed by several typical tests that the numerical simulations of high-order finite difference methods can be improved by satisfying the geometric conservation law.The systematic review about the geometric conservation law leads to the following conclusions:1)The geometric conservation law cannot be satisfied by finite difference methods with the discretized metrics based on the traditional computational form.The conservative computational form of metrics together with the CMM condition should be adopted to satisfy the geometric conservation law;2)The SCMM is a sufficient condition to satisfy the geometric conservation law,with an additional condition that the metrics and Jacobian must be discretized on the basis of the symmetrical conservative form,which are both unique;3)The freestream preservation behavior is merely a representation of the satisfaction of the geometric conservation law;4)The satisfaction of the geometric conservation law can effectively enhance the ability of numerical simulations with high-order finite difference schemes.