Stability Analysis and Error Estimate of the Explicit Single-Step Time-Marching Discontinuous Galerkin Methods with Stage-Dependent Numerical Flux Parameters for a Linear Hyperbolic Equation in One Dimension

Abstract

In this paper, we present the \(\hbox {L}^2\)-norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper.