Error analysis for full discretizations of quasilinear wave-type equations with two variants of the implicit midpoint rule

Abstract

Abstract We study the full discretization of a general class of first- and second-order quasilinear wave-type problems with the implicit midpoint rule and a linearized variant thereof. Based on a proof by induction, we prove wellposedness and a rigorous error estimate for both schemes, combining energy techniques, inverse estimates and a linearized fixed-point iteration for the analysis of the nonlinear scheme. To confirm the relevance of the general framework, we derive novel error estimates for the full discretization of two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity.