An efficient fifth-order finite difference multi-resolution WENO scheme for inviscid and viscous flow problems

Abstract

In this paper, a more efficient fifth-order finite difference multi-resolution WENO scheme is designed to solve the compressible inviscid and viscous flow problems based on the original multi-resolution WENO scheme presented in Zhu and Shu (2018). This new improved method inherits all the following features of the original multi-resolution WENO scheme (Zhu and Shu , 2018): (1) a series of unequal-sized central stencils are used to obtain the spatial reconstruction polynomials; (2) any positive numbers whose sum is one can be set as the linear weights; (3) smaller L1 and L∞ errors could be obtained for smooth problems; (4) it is easier to extend to the unstructured finite volume framework. However, different from the original multi-resolution WENO scheme, this improved method simplifies the reconstruction process, significantly improves the computational efficiency, and has greater engineering application potential. Numerical results show that the two types of fifth-order finite difference multi-resolution WENO schemes have similar results, but the CPU time of this new multi-resolution WENO scheme is about 0.7 times that of the original one. Moreover, the new fifth-order multi-resolution WENO scheme with a small increase in the computational cost shows less dissipation error than the classical WENO scheme (Jiang and Shu, 1996), and can capture more subtle flow structures for solving inviscid and viscous flow problems on the same mesh level. Several benchmark inviscid and viscous problems are illustrated to verify the above conclusions and the improved performance of this fifth-order multi-resolution WENO scheme.