A priori error estimates of Runge-Kutta discontinuous Galerkin schemes to smooth solutions of fractional conservation laws

Abstract

We give a priori error estimates of second order in time fully explicit Runge-Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions $\tau\leq c h$ for piecewise linear and $\tau\lesssim h^{4/3}$ for higher order finite elements, we prove a convergence rate for the energy norm $\|\cdot\|_{L^\infty_tL^2_x}+|\cdot|_{L^2_tH^{\lambda/2}_x}$ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.