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

Abstract

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.

Keywords

• finite difference scheme
• accuracy test
• structured mesh
• weno
• muscl