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

TL;DR

This paper analyzes the discretization of nonlinear wave equations with U(1) symmetry, demonstrating energy and charge conservation under specific grid ratios, and establishing stability and well-posedness of the finite-difference scheme.

Contribution

It proves energy and charge conservation for a classical finite-difference scheme under certain grid ratios and establishes existence, uniqueness, and stability of solutions.

Findings

Energy is conserved if grid ratio dt/dx ≤ 1/√n.

Charge conservation occurs at grid ratio dt/dx = 1/√n.

The scheme is conditionally stable with proven existence and uniqueness.

Abstract

We consider the problem of discretization for the U(1)-invariant nonlinear wave equations in any dimension. We show that the classical finite-difference scheme used by Strauss and Vazquez \cite{MR0503140} conserves the positive-definite discrete analog of the energy if the grid ratio is $dt/dx\le 1/\sqrt{n}$, 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 $dt/dx=1/\sqrt{n}$, then there is the discrete analog of the charge which is conserved. We prove the existence and uniqueness of solutions to the discrete Cauchy problem. We use the energy conservation to obtain the 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.