Abstract

In this talk, we will study the property of finite difference (FD) weighted essentially non-oscillatory (WENO) schemes on curvilinear meshes. To maintain the free-stream solution of Euler equation exactly, we will look at the alternative formulation of the FD WENO schemes, in which WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Theoretical derivation and numerical results show that the FD WENO schemes based on the alternative flux formulation can preserve free-stream and vortex solutions on generalized coordinate systems, hence giving much better performance than the standard FD WENO schemes for such problems. Furthermore, we extend the method to solve the ideal magnetohydrodynamics equations and the shallow water equations, such that it can preserve the free-stream condition and well-balanced property on curvilinear meshes, respectively.