New Smoothness Indicator Within an Active Flux Framework

TL;DR

This paper introduces a new smoothness indicator for active flux methods in hyperbolic conservation laws, capable of detecting rough solution regions by measuring differences between solution sets, enhancing robustness and accuracy.

Contribution

A novel smoothness indicator based on solution differences that effectively identifies rough regions in active flux methods for hyperbolic PDEs.

Findings

The new SI accurately detects rough solution regions.

It improves the robustness of active flux methods.

Numerical examples demonstrate its effectiveness on Euler equations.

Abstract

In this work, we introduce a new smoothness indicator (SI), which is capable of detecting ``rough'' parts of the solutions computed by active flux (AF) methods for hyperbolic (systems of) conservation laws. The new SI is based on measuring the difference between the two sets of solutions (either cell averages and point values or cell averages on overlapping grids) evolved at each time step of AF methods. The key idea in the derivation of the new SI is that in the ``rough'' parts of the evolved solutions, the difference is ${\cal O}(1)$, while in the smooth areas, it is proportional to the order of the underlying AF method. The performance of the new SI, that is, its ability to automatically and robustly detect ``rough'' parts of the computed solutions, is illustrated on several numerical examples, in which the one-dimensional Euler equations of gas dynamics are numerically solved by a recently introduced semi-discrete finite-volume AF method on overlapping grids.