Shock Capturing via Limiting for High-Order Methods Including Discontinuous Galerkin and Flux Reconstruction

High-order methods, such as discontinuous Galerkin (DG), spectral, and flux reconstruction (FR), are prone to generating unwanted oscillations near shocks and discontinuities. Conventional limiting techniques, while effective in suppressing oscillations near shocks, often compromise accuracy near extrema, where the solution is only first-order accurate. In this presentation, we introduce a novel limiting technique for these high-order schemes, aimed at effectively managing shocks while preserving accuracy. Unlike typical approaches, which rely on successful detection to activate limiting, our limiter depends continuously on the data, thereby avoiding potential issues if detection fails. To reduce computing time, efficient criteria for detecting smooth regions where limiting is unnecessary are presented. Numerical tests for advection and Euler equations are conducted to demonstrate the effectiveness of the proposed method.