Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws

Abstract

In this paper we present the analysis for the Runge-Kutta discontinuous Galerkin (RKDG) method to solve scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta (TVDRK3) method. We use an energy technique to present the L 2-norm stability for scalar linear conservation laws, and obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree k ≥ 1 under the standard temporal-spatial CFL condition τ ≤ γh, where h and τ , respectively, are the element length and time step, and the positive constant γ is independent of h and τ .

A discontinuous Galerkin scheme for front propagation with obstacles

“…Conversely this last equation implies also (22). In conclusion, the DG scheme in (20) is equivalent to (24), i.e.…”

“…We begin by collecting some of the properties of the operator H in (19) that was reported in [22]. Below, || · || denotes the L 2 norm on I, and || · || Γ h denotes the L 2 norm on the boundaries, i.e.…”

“…Our strategy will be mainly to use stability estimates of Zhang and Shu [22] for linear equations, and extend them to the obstacle case (under a CFL condition such as τ /h ≤ const, where τ and h are the time and space steps respectively, for the RK2 and RK3 schemes).…”

On the long-time simulation accuracy of the discontinuous Galerkin method for 1D transport equation

“…Случаи нелинейных уравнений и систем уравнений исследованы в [6][7][8]. Пост-обработка численного решения [5] была обобщена на случай линейных уравне-ний с переменными коэффициентами и нелинейных уравнений, а также задач с граничными условиями в [9][10][11].…”

Stationary Discrete Shock Profiles for Scalar Conservation Laws with a Discontinuous Galerkin Method

“…They gave estimates of the size of extent of oscillations upstream and downstream of the discontinuity based on suitable weights introduced in [15]. Recently, their work was extended to arbitrary approximation order in space and a third-order Runge-Kutta method on non-uniform meshes [34,35]. The case of δ-singularities in initial condition and source term of a scalar hyperbolic equation was then investigated in [33] where superconvergence in negative-order norms outside the pollution region was proved and sharp estimates over the whole domain were given.…”