HFVS: An arbitrary high order approach based on flux vector splitting

Abstract

In this paper, a new scheme of arbitrary high order accuracy in both space and time is proposed to solve hyperbolic conservative laws. The basic idea in the construction is that, based on the idea of the flux vector splitting (FVS), we split all the spatial and time derivatives in the Taylor expansion of the numerical flux into two parts: one part with positive eigenvalues, another with negative eigenvalues. According to a Lax–Wendroff procedure, all the time derivatives are then replaced by spatial derivatives, which are evaluated by using WENO reconstruction polynomials. One of the most significant advantages of the current scheme is very easy to implement. In addition, it is found that the higher spatial and time derivatives produced in the construction of the numerical flux can be regarded as a building block, in the sense that they can be coupled with any extact/approximate Riemann solvers to extend a first-order scheme to very high order accuracy in both space and time. Numerous numerical tests for linear and nonlinear hyperbolic conservative laws are carried out, and the numerical results demonstrate that the proposed scheme is robust and can be of high order accuracy in both space and time.