New Smoothness Indicator Within an Active Flux Framework

TL;DR

This work introduces a new smoothness indicator (SI) for active flux (AF) methods solving hyperbolic conservation laws by measuring the difference between two evolved solution representations and applying a noise-filtered threshold. The SI is designed so that rough regions yield an O(1) signal, while smooth regions decay as O((Delta x)^r) with r the method order, demonstrated on 1-D Euler equations with a second-order semi-discrete FV AF method on overlapping grids. Numerical tests show the SI accurately flags shocks and its smooth regions shrink with mesh refinement, indicating reliable shock detection and potential for adaptive strategies. The approach offers a path toward adaptive limiting and mesh refinement in AF frameworks and can extend to multidimensional and polygonal meshes.

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.

Figures (3)

• Figure 1: Example 1: Density computed using $N=800$ (top panel) and the corresponding SI values obtained on several meshes with $N=800$ (middle left), $N=1600$ (middle right), $N=6400$ (low left), and $N=12800$ (low right). $C=0.01$, $K=1$.
• Figure 2: Example 2: Density computed using $N=800$ (top panel) and the corresponding SI values obtained on several meshes with $N=800$ (middle left), $N=1600$ (middle right), $N=6400$ (low left), and $N=12800$ (low right). $C=0.2$, $K=6$.
• Figure 3: Example 3: Density computed using $N=400$ (top panel) and the corresponding SI values obtained on several meshes with $N=400$ (middle left), $N=800$ (middle right), $N=6400$ (low left), and $N=12800$ (low right). $C=200$, $K=1.2$.