A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws

Luo, Juan; Shu, Chi-Wang; Zhang, Qiang

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 991-1018.

In this paper we present an a priori error estimate of the Runge–Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge–Kutta method and the finite element space is made up of piecewise polynomials of degree $k\ge 2$. Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition $\tau \le \gamma h$, where $h$ is the element length and $\tau$ is time step, and $\gamma$ is a positive constant independent of $h$ and $\tau$. Optimal estimates are also considered when the upwind numerical flux is used.

Classification : 65M60, 65M12
Mots clés : Discontinuous Galerkin method, Runge–Kutta method, error estimates, symmetrizable system of conservation laws, energy analysis

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