Convergence of Explicit Runge–Kutta Discontinuous Galerkin Approximations of the First-Order Form of Maxwell’s Equations with Low Regularity Solutions

Abstract.

We establish a convergence result for the approximation of low-regularity solutions to time-dependent PDE systems that have an involution structure similar to Maxwell’s equations and the linear wave equations. The approximation is based on an explicit Runge–Kutta (ERK) time-stepping and the discontinuous Galerkin (dG) method with stabilization (so-called upwind fluxes) in space. The regularity setting only assumes that the exact solution and its first time derivative are in [Formula: see text] with a Sobolev regularity index [Formula: see text] in [Formula: see text] (here, [Formula: see text] is the time interval and [Formula: see text] the space domain), and that its second time derivative is in [Formula: see text]. The two main tools for the convergence analysis are a Ritz projection in space that leverages recent convergence results in operator norm for the dG approximation of the steady form of the PDE, and the [Formula: see text]-stability under a standard CFL condition of three-stage, third-order and four-stage, fourth-order ERK schemes. These latter results are known in the literature, but we provide here a somewhat simpler argument to prove the [Formula: see text]-stability.