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 τ ≤ ch for piecewise linear and τ ≲ h4/3 for higher order finite elements, we prove a convergence rate for the energy norm ‖⋅‖Lt∞Lx2+|⋅|Lx2Hxλ/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.