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

TL;DR

The paper addresses the stencil growth and boundary-handling challenges in high-order RKDG methods for hyperbolic conservation laws by introducing the compact RKDG (cRKDG) method. It hybridizes two spatial operators within a single time step: inner RK stages use a local projected derivative, while the final stage uses the standard DG operator, with limiting applied only at the end. A Lax–Wendroff type convergence theorem is established, and connections to LWDG and ADER–DG are explored. Numerical experiments in 1D and 2D demonstrate that cRKDG achieves the same order of accuracy as RKDG with a significantly more compact stencil, along with improved boundary behavior and practical CFL performance. The work lays groundwork for future stability analysis and extensions to implicit time stepping and structure-preserving schemes.

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.

Figures (8)

• Figure 3.1: Stencils of RKDG and cRKDG methods with a second-order RK method.
• Figure 4.1: Solution profiles for two Riemann problems of the Buckley--Leverett equation in \ref{['ex:1dbl']}. $N = 80$ and $M = 1$.
• Figure 4.2: Solution profiles for the Sod problem in \ref{['ex:sod']}. $N=100$ and $M=1$.
• Figure 4.3: Solution profiles for the blast wave problem in \ref{['ex:blast']}. $N=300$ and $M=200$.
• Figure 4.4: Solution profiles for the Shu--Osher problem in \ref{['ex:shuosher']} at $t=1.8$. $M=200$ and $N=200$.

Theorems & Definitions (34)

• Remark 2.1: RK methods in Butcher form
• Remark 2.2: Limiters
• Proposition 3.1
• Theorem 3.1
• Theorem 3.2: Grant, 2022. grant2022perturbed
• Theorem 3.3
• Proof 1
• Remark 3.4
• Remark 3.5
• Remark 3.6