Rescaling method for blow-up solutions of nonlinear wave equations

TL;DR

This paper introduces a hybrid numerical scheme combining finite differences and a rescaling technique to accurately simulate blow-up solutions in nonlinear wave equations, inspired by methods used for parabolic equations.

Contribution

It presents a novel rescaling approach for nonlinear wave equations and analyzes its convergence, extending techniques from parabolic equations to wave phenomena.

Findings

Successfully reproduces blow-up behavior numerically

Demonstrates convergence of the proposed scheme

Provides numerical examples illustrating the method

Abstract

We develop a hybrid scheme based on a finite difference scheme and a rescaling technique to approximate the solution of nonlinear wave equation. In order to numerically reproduce the blow-up phenomena, we propose a rule of scaling transformation, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A careful study of the convergence of the proposed scheme is carried out and several numerical examples are performed in illustration.