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

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 $\tau \le ch$ for piecewise linear and $\tau \lesssim h^{4\mathrm{/}3}$ for higher order finite elements, we prove a convergence rate for the energy norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L_t^{\mathrm{\infty }}{L}_x^2}+\mathrm{\mid }\cdot {\mathrm{\mid }}_{L_t^2{H}_x^{\lambda \mathrm{/}2}}$ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.