A multi-resolution limiter for the Runge-Kutta discontinuous Galerkin method

Abstract: We propose a novel multi-resolution (MR) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws on a general unstructured mesh. Unlike classical limiters, which detects only solution discontinuities to dichotomize cells into good or troubled, the proposed MR limiter also takes into account the derivative discontinuities to divide cells into several groups. The method operates by performing a successive comparison of the local DG polynomial's derivatives, from high-order to low-order, against a baseline constructed from neighboring cell averages. If a $k$th-order derivative of the DG polynomial is larger than the baseline, then we reduce the order to $k-1$ and set the corresponding $k$th-order terms to be 0; Otherwise, the remaining $k$th-order DG polynomial is used to represent the final solution. Only if all the derivatives are larger than the baseline, a TVD slope limiter is used to reconstruct the solution. In this manner, the limiter dynamically selects an optimal polynomial suited to the local solution smoothness without problem-dependent parameter to tune. Notably, it also possesses a scale-invariance property that is absent in most classical limiters. A series of numerical examples demonstrate the accuracy and robustness of the proposed MR limiter.