The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

TL;DR

This paper introduces a jump filter for the discontinuous Galerkin method that reduces oscillations near discontinuities in hyperbolic conservation law simulations, maintaining high accuracy and efficiency.

Contribution

The paper presents a novel jump filter that preserves key DG properties, integrates seamlessly with hybrid limiters, and offers low computational cost without characteristic decomposition.

Findings

Effective suppression of oscillations near shocks

Maintains high-order accuracy and conservation

Low computational overhead and good parallelization

Abstract

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, $L^2$ stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.