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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.

Constraint Preserving Schemes for Some Gauge Invariant Wave Equations