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 L^∞(0,T;H^s(Ω)) with a Sobolev regularity index s in ]0,1/2[ (here, T is the time horizon and Ω the space domain), and that its second time-derivative is in L^∞(0,T;L²(Ω)).
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 L²-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 argumentation to prove the $L²$-stability. This is joint work with J.-L. Guermond (Texas A&M).