The Runge--Kutta discontinuous Galerkin method with stage-dependent polynomial spaces for hyperbolic conservation laws

TL;DR

This work introduces sdRKDG, a high-order Runge--Kutta discontinuous Galerkin framework that blends two spatial operators defined on $\mathcal{P}^k$ and $\mathcal{P}^{k-1}$ within a single time step to surpass traditional MOL designs. By organizing the RK stages with stage-dependent polynomial spaces, the method achieves reduced floating-point operations and can permit larger CFL numbers, while preserving local conservation and enabling TVD/TVBM behavior with appropriate limiters. Von Neumann analysis shows improved stability margins for several sdRKDG schemes, and numerical tests (including 1D Burgers, 1D/2D Euler, and discontinuous problems) demonstrate optimal or near-optimal convergence on smooth problems and robust shock-capturing on discontinuities, with some order degeneration occurring for sonic-point problems in Class B. The results indicate a practical and flexible framework for advancing RKDG methods beyond the MOL paradigm, offering potential efficiency gains and applicability to complex systems while highlighting the need to manage sonic-point effects in certain configurations.

Abstract

In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent polynomial spaces for the spatial discretization operators. To be more specific, two different DG operators, associated with $\mathcal{P}^k$ and $\mathcal{P}^{k-1}$ piecewise polynomial spaces, are used at different RK stages. The resulting method is referred to as the sdRKDG method. It features fewer floating-point operations and may achieve larger time step sizes. For problems without sonic points, we observe optimal convergence for all the sdRKDG schemes; and for problems with sonic points, we observe that a subset of the sdRKDG schemes remains optimal. We have also conducted von Neumann analysis for the stability and error of the sdRKDG schemes for the linear advection equation in one dimension. Numerical tests, for problems including two-dimensional Euler equations for gas dynamics, are provided to demonstrate the performance of the new method.

Figures (8)

• Figure 3.1: CFL number of second-order (left) and third-order (right) sdRKDG schemes.
• Figure 3.2: The leading coefficient of the predicted error \ref{['eq:error_rk2']} at different final times.
• Figure 4.1: Solution profiles for the Sod problem in \ref{['ex:sod']}. $N=100$ and $M=10$.
• Figure 4.2: Solution profiles for the blast wave problem in \ref{['ex:blast']}. $M=200$. In the first row, $N = 200$; in the second row, $N = 400$.
• Figure 4.3: Solution profiles for the Shu--Osher problem in \ref{['ex:shuosher']} at $t=1.8$. $M=300$ and $N=400$. Here “sdRKDG” solutions are computed with the CFL numbers $0.56$ for $k = 1$ and $0.27$ for $k = 2$; the “sdRKDG small CFL” and “RKDG” solutions are computed with the CFL number $0.3$ for $k = 1$ and $0.18$ for $k = 2$.

Theorems & Definitions (23)

• Remark 2.1: RK methods in Shu--Osher form
• Remark 2.2: Implementation
• Remark 2.3: Efficiency improvement
• Remark 2.4: Sonic points
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6