A new hybrid WENO scheme for hyperbolic conservation laws

Abstract

In [28], a simple fifth order weighted essentially non-oscillatory (WENO) scheme was presented in the finite difference framework for the hyperbolic conservation laws, in which the reconstruction of fluxes is a convex combination of a fourth degree polynomial with two linear polynomials. In this follow-up paper, we propose a new fifth order hybrid weighted essentially non-oscillatory (WENO) scheme based on the simple WENO. The main idea of the hybrid WENO scheme is that if all extreme points of the reconstruction polynomial for numerical flux in the big stencil are located outside of the big stencil, then we reconstruct the numerical flux by upwind linear approximation directly, otherwise use the simple WENO procedure. Compared with the simple WENO, the major advantage is its higher efficiency with less numerical errors in smooth regions and less computational costs. Likewise, the hybrid WENO scheme still keeps the simplicity and robustness of the simple WENO scheme. Extensive numerical results for both one and two dimensional equations are performed to verify these good performance of the proposed scheme.