HFVS: An Arbitrary High Order Flux Vector Splitting Method

TL;DR

This paper introduces HFVS, a high-order flux vector splitting scheme that achieves arbitrary accuracy in space and time for hyperbolic laws, using WENO reconstruction and a Lax-Wendroff approach.

Contribution

It presents a novel, easy-to-implement high-order scheme based on flux vector splitting and WENO, extending first-order methods to very high order accuracy.

Findings

Demonstrates robustness through numerical tests

Achieves high order accuracy in space and time

Applicable to linear and nonlinear hyperbolic laws

Abstract

In this paper, a new scheme of arbitrary high order accuracy in both space and time is proposed to solve hyperbolic conservative laws. Based on the idea of flux vector splitting(FVS) scheme, we split all the space and time derivatives in the Taylor expansion of the numerical flux into two parts: one part with positive eigenvalues, another part with negative eigenvalues. According to a Lax-Wendroff procedure, all the time derivatives are then replaced by space derivatives. And the space derivatives is calculated by WENO reconstruction polynomial. One of the most important advantages of this new scheme is easy to implement.In addition, it should be pointed out, the procedure of calculating the space and time derivatives in numerical flux can be used as a building block to extend the current first order schemes to very high order accuracy in both space and time. Numerous numerical tests for linear and nonlinear hyperbolic conservative laws demonstrate that new scheme is robust and can be high order accuracy in both space and time.