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 τ≤ c h for piecewise linear and τ h4/3 for higher order finite elements, we prove a convergence rate for the energy norm \|·\|L^∞ₜL²ₓ+|·|_L²ₜHλ/2ₓ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.