Very-low-dissipation, physics-based, and parameter-free shock capturing for discontinuous Galerkin methods

This work addresses the challenge of shock capturing in numerical simulations of hyperbolic conservation laws, focusing on the discontinuous Galerkin (DG) method. It proposes a novel modal filtering approach based on physical criteria to detect elements near discontinuities and tune modal damping in the solutions. The minimum entropy principle combined with prescribed physical criteria is used to detect discontinuities and the Ducros sensor, a contact discontinuity sensor, and the Persson modal sensor are used to appropriately adjust filter intensity. The filter function is designed to apply higher dissipation to high-frequency modes, which often contribute to spurious oscillations. Additionally, temporal blending of the filter intensity is introduced to handle non-stationary discontinuities effectively. The effectiveness of the proposed approach is demonstrated through various test cases involving Euler and Navier–Stokes equations, ranging from smooth compressible flows to complex shock-boundary layer interactions. Results show that the method successfully eliminates spurious oscillations while maintaining high-order accuracy in smooth regions. It proves to be computationally efficient and less dissipative compared to other shock-capturing techniques. The proposed modal filtering approach offers an effective, efficient, and robust solution for shock capturing in the DG framework, capable of handling a wide range of flow scenarios while preserving solution accuracy.