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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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