The Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws

TL;DR

This paper introduces a new Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws, improving boundary treatment and efficiency while maintaining accuracy.

Contribution

The paper presents a hybridized RKDG method that enhances compactness and simplifies boundary handling compared to traditional approaches.

Findings

Method achieves similar accuracy to original RKDG

Improved compactness reduces computational complexity

Effective for 2D Euler equations in gas dynamics

Abstract

In this paper, we develop a new type of Runge--Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple boundary treatment. The key idea is to hybridize two different spatial operators in an explicit RK scheme, utilizing local projected derivatives for inner RK stages and the usual DG spatial discretization for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. We also explore the connections between our method and Lax--Wendroff DG schemes and ADER-DG schemes. Numerical examples are given to confirm that the new RKDG method is as accurate as the original RKDG method, while being more compact, for problems including two-dimensional Euler equations for compressible gas dynamics.