A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation

TL;DR

This paper introduces a finite-difference numerical scheme for solving radially symmetric solutions of a dissipative nonlinear Klein-Gordon equation, analyzing its accuracy, stability, and physical damping effects.

Contribution

It develops a second-order consistent finite-difference method for the nonlinear Klein-Gordon equation with damping, including stability analysis and physical damping comparison.

Findings

Scheme is second-order accurate for zero nonlinearity

Necessary condition for scheme stability of order n

Physical effects of damping coefficients analyzed

Abstract

In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions in an open sphere around the origin, where the internal and external damping coefficients are constant, and the nonlinear term follows a power law. We prove that our scheme is consistent of second order when the nonlinearity is identically equal to zero, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to compare the physical effects of the damping coefficients.