Dissipative WENO stabilization of high-order discontinuous Galerkin methods for hyperbolic problems

TL;DR

This paper introduces a novel WENO-based stabilization method for high-order discontinuous Galerkin schemes that enhances accuracy and discontinuity capturing by controlling numerical dissipation through a smoothness sensor.

Contribution

The paper proposes a dissipation-based WENO stabilization approach for DG methods that improves accuracy and discontinuity resolution compared to traditional slope limiters.

Findings

Achieves high-order accuracy with optimal convergence rates.

Effectively captures discontinuities sharply.

Provides an alternative stabilization technique for DG schemes.

Abstract

We present a new approach to stabilizing high-order Runge-Kutta discontinuous Galerkin (RKDG) schemes using weighted essentially non-oscillatory (WENO) reconstructions in the context of hyperbolic conservation laws. In contrast to RKDG schemes that overwrite finite element solutions with WENO reconstructions, our approach employs the reconstruction-based smoothness sensor presented by Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) to control the amount of added numerical dissipation. Incorporating a dissipation-based WENO stabilization term into a discontinuous Galerkin (DG) discretization, the proposed methodology achieves high-order accuracy while effectively capturing discontinuities in the solution. As such, our approach offers an attractive alternative to WENO-based slope limiters for DG schemes. The reconstruction procedure that we use performs Hermite interpolation on stencils composed of a mesh cell and its neighboring cells. The amount of numerical dissipation is determined by the relative differences between the partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. The employed smoothness sensor takes all derivatives into account to properly assess the local smoothness of a high-order DG solution. Numerical experiments demonstrate the ability of our scheme to capture discontinuities sharply. Optimal convergence rates are obtained for all polynomial degrees.