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Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. (English) Zbl 1078.65080
Authors’ abstract: 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 \(\mathbb{P}^1\) (piecewise linear) elements are obtained under the usual Courant-Friedrichs-Levy (CFL) condition \(\tau\leq \gamma h\) for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where \(h\) and \(\tau\) are the maximum element lengths and time steps, respectively, and the positive constant \(\gamma\) is independent of \(h\) and \(\tau\).
However, error estimates for higher order \(\mathbb{P}^k(k\geq 2)\) elements need a more restrictive time step \(\tau\leq \gamma h^{4/3}\). We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition \(\tau\leq\gamma h\) for the \(\mathbb{P}^k\) elements of degree \(k\geq 2\). Error estimates of \(O(h^{k+1/2}+\tau^2)\) are obtained for general monotone numerical fluxes, and optimal error estimates of \(O(h^{k+1}+\tau^2)\) are obtained for upwind numerical fluxes.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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