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

TL;DR

This paper introduces a novel high-order Runge--Kutta discontinuous Galerkin scheme that employs stage-dependent polynomial spaces, leading to computational efficiency and larger time steps for hyperbolic conservation laws.

Contribution

The paper develops a new sdRKDG method using stage-dependent polynomial spaces, extending traditional frameworks and improving efficiency and stability for hyperbolic problems.

Findings

Optimal convergence for all schemes on problems without sonic points

Some schemes remain optimal with sonic points

Numerical tests demonstrate improved performance on gas dynamics problems

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.