Constraint Preserving Schemes for Some Gauge Invariant Wave Equations

Abstract

We study discretizations of the Maxwell–Klein–Gordon equation as an example of a constrained geometric nonlinear evolution partial differential equation. For the temporal gauge we propose a fully discrete scheme which preserves the nonlinear constraint thanks to a special application of Lagrange multipliers. We show that the method generalizes to Hamiltonian wave equations whose kinetic and potential energy are both invariant under a group of transformations, even though the Galerkin spaces are not invariant. We then extend the method to the Lorenz gauge. Numerical results illustrate the discussion.