非均匀结构网格上MUSCL和WENO格式的精度

Accuracy of MUSCL and WENO Schemes on Non-Uniform Structured Meshes

基于一维均匀网格条件下构造的差分格式,在实际应用中须推广到非均匀或者曲线网格上,坐标变换过程引入几何诱导误差。目前常用收敛解误差随着网格细化变化的精度测试方法评估差分格式的精度。在二维柱坐标均匀网格上,采用1阶迎风、2阶MUSCL和5阶WENO计算流场参数为常数的自由流问题,按照精度测试方法比较收敛曲线斜率,发现1阶迎风的网格收敛精度是2阶的,5阶WENO的网格收敛精度不到1阶。理论分析表明,这种精度测试方法与差分格式精度定义不等价,而且所采用的数据无法反映差分格式的固有缺陷,因此,不能用来作为差分格式精度评价指标。很多研究WENO的文献经常模拟双Mach反射问题、二维Riemann问题等经典算例,把接触间断是否演变成不稳定涡结构作为特征,理论上可以证明涡结构是非物理现象,因此用是否出现涡结构作为算法高精度的论据并不合适。

The difference schemes constructed on the basis of one-dimensional uniform grids must be extended to non-uniform or curvilinear grids in practical applications,and the coordinate transformation process introduces geometry-induced errors.The accuracy of the difference schemes is evaluated by the accuracy test,in which the convergence solution error varies with the grid refinement.In this paper,the first-order upwind scheme,the second-order MUSCL scheme and the fifth-order WENO scheme were used to calculate the uniform free flow problem with constant flow parameters on a two-dimensional cylindrical coordinate uniform grid system,and the slope of the convergence curve was compared according to the accuracy test method,and it was found that the grid convergence accuracy of the first-order upwind scheme was second order,and the grid convergence accuracy of the fifth-order WENO scheme was less than first order.Theoretical analysis shows that this accuracy test method is not equivalent to the definition of difference scheme accuracy,and the data used cannot reflect the inherent defects of the difference scheme.Therefore,it cannot be used as a criterion for evaluating the accuracy of the difference scheme.Many studies of WENO schemes often simulate benchmarks such as the double Mach reflection problem and the two-dimensional Riemann problem,and use whether the contact discontinuity develops into an unstable vortex structure as a characteristic of the algorithm with high accuracy,which can be theoretically proved to be a non-physical phenomenon,so it is not appropriate to use whether a vortex structure appears as an argument for the algorithm′s high accuracy.