A low dissipation finite difference nested multi-resolution WENO scheme for Euler/Navier-Stokes equations

Abstract

In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller L1 and L∞ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.