Application of troubled-cells to finite volume methods -- an optimality study using a novel monotonicity parameter

TL;DR

This study adapts a troubled-cell indicator from DG to finite volume methods, analyzing the trade-off between convergence and solution quality for shock flows using a novel monotonicity parameter.

Contribution

It introduces a new troubled-cell indicator for finite volume methods and systematically studies the optimal troubled-cell configuration for shock capturing.

Findings

Optimal troubled-cell sets depend on shock alignment with the grid.

The adapted indicator effectively identifies near-optimal troubled-cell sets.

The method improves convergence while maintaining solution quality.

Abstract

We adapt a troubled-cell indicator from discontinuous Galerkin (DG) methods to finite volume methods (FVM) with MUSCL reconstruction and using a novel monotonicity parameter show there is a trade-off between convergence and quality of the solution. Employing two dimensional compressible Euler equations for flows with oblique shocks, this trade-off is studied by varying the number of troubled-cells systematically. An oblique shock is characterized primarily by the upstream Mach number, the shock angle $\beta$, and the deflection angle $\theta$. We study these factors and their combinations and find that the degree of the shock misalignment with the grid determines the optimal number of troubled-cells. On each side of the shock, the optimal set consists of three troubled-cells for aligned shocks, and the troubled-cells identified by tracing the shock and four lines parallel to it, separated by the grid spacing, for nonaligned shocks. We show that the adapted troubled-cell indicator identifies a set of cells that is close to and contains the optimal set of cells for a threshold constant $K = 0.05$, and consequently, produces a solution close to that obtained by limiting everywhere, but with improved convergence.