Application of troubled-cells to finite volume methods -- an optimality study using a novel monotonicity parameter
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915301015158784 |
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| author | Rao, R Shivananda Ramakrishna, M |
| author_facet | Rao, R Shivananda Ramakrishna, M |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Application of troubled-cells to finite volume methods -- an optimality study using a novel monotonicity parameter Rao, R Shivananda Ramakrishna, M Numerical Analysis 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. |
| title | Application of troubled-cells to finite volume methods -- an optimality study using a novel monotonicity parameter |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.17753 |