Lax-Wendroff flux reconstruction method for hyperbolic conservation laws

TL;DR

This paper introduces a flux reconstruction Lax-Wendroff method for hyperbolic conservation laws that is single-step, quadrature-free, and achieves higher CFL numbers, demonstrated through theoretical analysis and numerical experiments.

Contribution

It develops a novel flux reconstruction Lax-Wendroff method applicable to general hyperbolic laws, with improved stability and efficiency over existing methods.

Findings

Higher CFL numbers achieved compared to existing methods

Uniform performance at all orders demonstrated by Fourier analysis

Numerical results confirm fifth-order accuracy for linear and non-linear problems

Abstract

The Lax-Wendroff method is a single step method for evolving time dependent solutions governed by partial differential equations, in contrast to Runge- Kutta methods that need multiple stages per time step. We develop a flux reconstruction version of the method in combination with a Jacobian-free Lax- Wendroff procedure that is applicable to general hyperbolic conservation laws. The method is of collocation type, is quadrature free and can be cast in terms of matrix and vector operations. Special attention is paid to the construction of numerical flux, including for non-linear problems, resulting in higher CFL numbers than existing methods, which is shown through Fourier analysis and yielding uniform performance at all orders. Numerical results up to fifth order of accuracy for linear and non-linear problems are given to demonstrate the performance and accuracy of the method.