An alternative formulation of finite difference WENO schemes with Lax-Wendroff time discretization for conservation laws

We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the 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. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that it is more straightforward to construct a Lax-Wendroff time discretization procedure, with a Taylor expansion in time and with all time derivatives replaced by spatial derivatives through the partial differential equations, resulting in a narrower effective stencil compared with previous high order finite difference WENO scheme based on the reconstruction of flux functions with a Lax-Wendroff time discretization. We will describe the scheme formulation and present numerical tests for oneand two-dimensional scalar and system conservation laws demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.