A comprehensive numerical investigation of the application of troubled-cells to finite volume methods using a novel monotonicity parameter

In this paper, we adapt a troubled-cell indicator proposed by Fu and Shu [1] for discontinuous Galerkin (DG) methods to finite volume methods (FVM) employing third-order MUSCL reconstruction. We show that limiting only in troubled-cells is advantageous in terms of convergence but at the expense of the overall solution quality. We investigate the optimal number of troubled-cells required in the vicinity of an oblique shock to obtain a solution with minimal oscillations and enhanced convergence using a novel monotonicity parameter. 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 by employing two dimensional compressible Euler equations and find that the degree of the shock misalignment with the grid determines the optimal number of troubled-cells. We show that, 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, for a threshold constant K = 0.05 , the adapted troubled-cell indicator identifies a set of cells that is close to and contains the optimal set of cells, and consequently, produces a solution close to that obtained by limiting everywhere, but with improved convergence. We demonstrate the effectiveness of the adapted troubled-cell indicator for unsteady problems using the double Mach reflection test case.