Convergence of explicit Runge-Kutta discontinuous Galerkin approximations of the first-order form of Maxwell's equations with low regularity solutions
Résumé
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 $L^∞(J; H^s(D))$ with a Sobolev regularity index $s \in (0, \frac12$) (here, $J$ is the time interval and $D$ the space domain), and that its second time-derivative is in $L^∞(J; L^2(D))$. 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 $L^2$-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 $L^2$-stability.
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