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

TL;DR

This work transfers a troubled-cell indicator from DG methods to finite-volume MUSCL schemes solving the two-dimensional Euler equations, introducing a monotonicity metric $\mu$ to assess solution quality near shocks. It demonstrates a trade-off between convergence speed and oscillation-free accuracy, identifying optimal troubled-cell configurations: $33$ for aligned shocks and $44$ for non-aligned shocks, with a practical threshold $K=0.05$ enabling near-optimal results. The approach yields convergence improvements over limiting everywhere while preserving accuracy, though unsteady cases may require initial adjustment (e.g., an initial limiting-everywhere step). The results provide actionable guidance for applying localized limiting in FVM to oblique shocks across varied grid alignments and Mach numbers.

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 $β$, and the deflection angle $θ$. 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.

Figures (24)

• Figure 1: Schematic for various test cases, illustrating the varying geometric and flow conditions, and boundary conditions. (not drawn to scale).
• Figure 2: Isentropic vortex. Density profiles along the line $y = 0$ at time $t = 10$ and $t = 20$.
• Figure 3: Stencil, $S = \{C_0, C_1, C_2, C_3, C_4\}$, used (in the troubled cell indicator) to determine whether the cell $C_0$ is a troubled cell.
• Figure 4: NonAligned oblique shock. Zoomed-in view of troubled-cells identified by the indicator for two threshold constants. Red line represents the exact shock.
• Figure 5: NonAligned oblique shock. (a) The convergence history of the residual norm as a function of number of iterations. (b) Density profiles along the line $y = 0.5$.