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

TL;DR

The paper introduces a dissipation-based stabilization for high-order RKDG methods by incorporating a WENO-based smoothness sensor to adaptively add diffusion near discontinuities. By using a reconstruction-based HWENO approach and a relative-difference sensor, the method blends low- and high-order diffusion in a noninvasive, semi-discrete formulation, achieving both high-order accuracy on smooth regions and sharp shock-capturing. It demonstrates optimal convergence for smooth problems and entropy-stable, sharply resolved solutions across a broad suite of scalar and Euler tests, outperforming conventional LO stabilization. The work offers a modular, extensible framework for stabilizing DG methods on hyperbolic problems and suggests further improvements in smoothness sensors and theoretical analysis to broaden applicability.

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.

Figures (17)

• Figure 1: One-dimensional linear advection with constant velocity, numerical solutions at $t=1.0$ obtained using $E_h=128$ and $p\in\{1,2,3\}$.
• Figure 2: One-dimensional Burgers equation, numerical solutions at $t=1.0$ obtained using $E_h=128$ and $p\in\{1,2,3\}$.
• Figure 3: Solid body rotation, numerical solutions at $t=1.0$ obtained using $E_h=128^2$ and $p=2$.
• Figure 4: Two-dimensional Burgers equation, density profiles at $t=0.5$ obtained using $E_h=128^2$ and $p=2$.
• Figure 5: Two-dimensional KPP problem, numerical solutions at $t=1.0$ obtained using $E_h=128^2$ and $p=2$.

Theorems & Definitions (4)

• remark 1
• remark 2
• remark 3
• remark 4