Well-posedness and the energy and charge conservation for nonlinear wave equations in discrete space-time

Abstract

We consider the discretization problem for U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez (in J. Comput. Phys. 28, 271–278 (1978)) conserves the positive-definite discrete analog of the energy if the grid ratio satisfies $dt/dx \leqslant 1/\sqrt n$, where dt and dx are the mesh sizes of the time and space variables and n is the spatial dimension. We also show that, if the grid ratio is $dt/dx \leqslant 1/\sqrt n$, then there is a discrete analog of charge, and this discrete analog is conserved.

We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use energy conservation to obtain a priori bounds for finite energy solutions, thus showing that the Strauss-Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation with positive nonlinear term in the Hamiltonian is conditionally stable.