In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge-Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge-Kutta method. Error estimates for the P 1 (piecewise linear) elements are obtained under the usual CFL condition τ ≤ γh for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and τ are the maximum element lengths and time steps, respectively, and the positive constant γ is independent of h and τ . However, error estimates for higher order P k (k ≥ 2) elements need a more restrictive time step τ ≤ γh 4/3 . We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition τ ≤ γh for the P k elements of degree k ≥ 2. Error estimates of O(h k+1/2 + τ 2 ) are obtained for general monotone numerical fluxes, and optimal error estimates of O(h k+1 + τ 2 ) are obtained for upwind numerical fluxes.
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 b s t r a c tThe main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving one-dimensional convection-diffusion equations with a nonlinear convection. Both Runge-Kutta and multi-step IMEX methods are considered. By the aid of the energy method, we show that the IMEX LDG schemes are unconditionally stable for the nonlinear problems, in the sense that the time-step s is only required to be upper-bounded by a positive constant which depends on the flow velocity and the diffusion coefficient, but is independent of the mesh size h. We also give optimal error estimates for the IMEX LDG schemes, under the same temporal condition, if a monotone numerical flux is adopted for the convection. Numerical experiments are given to verify our main results.
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