The L$^2$-norm Stability Analysis of Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper we propose a simple and unified framework to investigate the L$^2$-norm stability of the explicit Runge--Kutta discontinuous Galerkin (RKDG) methods when solving the linear constant-coefficient hyperbolic equations. Two key ingredients in the energy analysis are the temporal differences of numerical solutions in different Runge--Kutta stages and a matrix transferring process. Many popular schemes, including the fourth order RKDG schemes, are discussed in this paper to show that the presented technique is flexible and useful. Different performances in the L$^2$-norm stability of different RKDG schemes are carefully investigated. For some lower-degree piecewise polynomials, the monotonicity stability is proved if the stability mechanism can be provided by the upwind-biased numerical fluxes. Some numerical examples are also given.