New Fully Discrete Active Flux Methods with Truly Multi-Dimensional Evolution Operators and WENO Reconstruction

TL;DR

This paper introduces fully discrete third-order Active Flux and WENO methods for 2D acoustic equations, utilizing multidimensional evolution operators and bicharacteristics to enhance stability and accuracy on coarse grids.

Contribution

It presents novel fully discrete multidimensional Active Flux and WENO schemes with improved stability and accuracy for 2D acoustic equations, based on new evolution operators.

Findings

Enhanced stability with higher CFL numbers.

Robustness on coarse grids for both continuous and discontinuous problems.

Achieved third-order accuracy in two-dimensional simulations.

Abstract

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.