High Order in Space and Time Schemes Through an Approximate Lax-Wendroff Procedure

TL;DR

This paper analyzes a high-order space-time scheme based on an approximate Lax-Wendroff procedure, introducing fluctuation control via WENO interpolation to improve accuracy near discontinuities.

Contribution

It proposes a novel fluctuation control method using WENO interpolation for flux derivatives, enhancing high-order schemes' robustness near discontinuities.

Findings

WENO-based fluctuation control improves accuracy near discontinuities

The scheme achieves high-order accuracy with controlled errors

Numerical results confirm the benefits of the proposed method

Abstract

This paper deals with the scheme proposed by the authors in Zor\'io, Baeza and Mulet (J Sci Comput 71(1):246-273, 2017). This scheme is an alternative to the techniques proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) to obtain high-order accurate schemes using Weighted Essentially Non Oscillatory finite differences and approximating the flux derivatives required by the Cauchy-Kovalevskaya procedure by simple centered finite differences. We analyse how errors in first-order terms near discontinuities propagate through both versions of the Cauchy-Kovalevskaya procedure. We propose a fluctuation control, for which the approximation of the first-order derivative to be used in the Cauchy-Kovalevskaya procedure is obtained from a Weighted Essentially Non Oscillatory (WENO) interpolation of flux derivatives, instead of the usual finite difference of WENO flux reconstructions. The numerical results that we obtain confirm the benefits of this fluctuation control.