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A study of troubled-cell indicators applied to finite volume methods using a novel monotonicity parameter

R. Shivananda Rao, M. Ramakrishna

TL;DR

This paper transfers a troubled-cell indicator from discontinuous Galerkin methods to finite-volume schemes for the 2D Euler equations and introduces a monotonicity parameter $μ$ to assess local solution quality near shocks. By restricting limiting to the identified troubled cells (limiting restricted region) rather than applying it everywhere, the authors demonstrate comparable accuracy with faster convergence and reduced computational cost across multiple steady-state problems, while highlighting trade-offs between the number of troubled cells and oscillations. The framework uses a 5-cell stencil to detect troubled cells and evaluates performance using residual convergence, error norms, density profiles, and the newly defined monotonicity measure. The results indicate that the restricted-region limiting approach is typically advantageous, though care must be taken to ensure stability in certain unsteady scenarios, and the introduction of surrounding troubled cells may be necessary for robust performance. Overall, the work provides a practical, efficient strategy for shock-capturing in FVM by leveraging localized limiting guided by a DG-inspired indicator and a quantitative monotonicity metric $μ$.

Abstract

We adapt a troubled-cell indicator developed for discontinuous Galerkin (DG) methods to the finite volume method (FVM) framework for solving hyperbolic conservation laws. This indicator depends solely on the cell-average data of the target cell and its immediate neighbours. Once the troubled-cells are identified, we apply the limiter only in these cells instead of applying in all computational cells. We introduce a novel technique to quantify the quality of the solution in the neighbourhood of the shock by defining a monotonicity parameter $μ$. Numerical results from various two-dimensional simulations on the hyperbolic systems of Euler equations using a finite volume solver employing MUSCL reconstruction validate the performance of the troubled-cell indicator and the approach of limiting only in the troubled-cells. These results show that limiting only in the troubled-cells is preferable to limiting everywhere as it improves convergence without compromising on the solution accuracy.

A study of troubled-cell indicators applied to finite volume methods using a novel monotonicity parameter

TL;DR

This paper transfers a troubled-cell indicator from discontinuous Galerkin methods to finite-volume schemes for the 2D Euler equations and introduces a monotonicity parameter to assess local solution quality near shocks. By restricting limiting to the identified troubled cells (limiting restricted region) rather than applying it everywhere, the authors demonstrate comparable accuracy with faster convergence and reduced computational cost across multiple steady-state problems, while highlighting trade-offs between the number of troubled cells and oscillations. The framework uses a 5-cell stencil to detect troubled cells and evaluates performance using residual convergence, error norms, density profiles, and the newly defined monotonicity measure. The results indicate that the restricted-region limiting approach is typically advantageous, though care must be taken to ensure stability in certain unsteady scenarios, and the introduction of surrounding troubled cells may be necessary for robust performance. Overall, the work provides a practical, efficient strategy for shock-capturing in FVM by leveraging localized limiting guided by a DG-inspired indicator and a quantitative monotonicity metric .

Abstract

We adapt a troubled-cell indicator developed for discontinuous Galerkin (DG) methods to the finite volume method (FVM) framework for solving hyperbolic conservation laws. This indicator depends solely on the cell-average data of the target cell and its immediate neighbours. Once the troubled-cells are identified, we apply the limiter only in these cells instead of applying in all computational cells. We introduce a novel technique to quantify the quality of the solution in the neighbourhood of the shock by defining a monotonicity parameter . Numerical results from various two-dimensional simulations on the hyperbolic systems of Euler equations using a finite volume solver employing MUSCL reconstruction validate the performance of the troubled-cell indicator and the approach of limiting only in the troubled-cells. These results show that limiting only in the troubled-cells is preferable to limiting everywhere as it improves convergence without compromising on the solution accuracy.

Paper Structure

This paper contains 7 sections, 6 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Numerical Methodology
  3. Troubled-cell indicator
  4. Limiting Strategies
  5. Comparison techniques:
  6. Results of two limiting approaches
  7. Conclusions

Figures (17)

  • Figure 1: Schematic for various steady-state test cases, illustrating the varying geometric and flow conditions, and boundary conditions. (not drawn to scale).
  • Figure 2: Solution domain and initial conditions for unsteady test cases (not drawn to scale).
  • 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: Aligned oblique shock. Zoomed-in view of troubled-cells identified by the indicator for two threshold constants. Red line represents the exact shock. Percentage of number of troubled-cells identified in whole computational domain of 200 $\times$ 200 cells: (a) 7.71% (b) 6.93% (c) 4.87% (d) 5.55%
  • Figure 5: NonAligned oblique shock. Zoomed-in view of troubled-cells identified by the indicator for two threshold constants. Red line represents the exact shock. Percentage of number of troubled-cells identified in whole computational domain of 800 $\times$ 200 cells: (a) 4.11% (b) 3.01% (c) 1.81% (d) 1.76%
  • ...and 12 more figures