Convergence Analysis of the Finite Difference Solution for the Nonlinear Klein-Gordon Equation

Ekhlass Al-Rawi

doi:10.33899/csmj.2008.163952

Abstract

This paper is devoted to drive the matrix algebraic equation for the one-dimensional nonlinear Klein-Gordon equation which is obtained from using the implicit finite difference method. The convergence analysis of the solution is discussed. Numerical computations are conducted and the solutions are stable and convergent when the sine function is used as an initial condition.

Highlights

Let us consider the following nonlinear Klein-Gordon equation (NKG) [2,4] 2 t 2 − + c 2 p −2 =0 (1)(t,0) = 0 = (t, L), and initial conditions 0 tConvergence Analysis of ...t=0 = u0, t t=0 = v0, 0 x L let p=4 eq(1) becomes (2)
We will derive the matrix equation for the onedimensional Klein-Gordon equation which is obtained from using finite difference method by using the implicit time discretization method, we will prove the existence of the solution of the matrix equation, and give two examples for the Klein-Gordon equation in order to show the convergence analysis for the numerical results
2-Derivation of the matrix equation using the finite difference method We introduce a uniform grid by defining the following discrete set of points in the x, t plane: xi = ih, i=0,1,...,n-1,n tj = jk, j=0,1,...,m-1,m

Summary

1-Introduction

The Klein-Gordon equation is one of the nonlinear extensions of the wave equation. Dimitri Mugnai [4] established the existence of infinitely many nontrivial radially symmetric solitary waves for the nonlinear Klein-Gordon equation, coupled with a Born-Infeld type equation under general assumptions. Difference method by using the implicit time discretization method, we will prove the existence of the solution of the matrix equation, and give two examples for the Klein-Gordon equation in order to show the convergence analysis for the numerical results. 2-Derivation of the matrix equation using the finite difference method We introduce a uniform grid by defining the following discrete set of points in the x, t plane: xi = ih , i=0,1,...,n-1,n tj = jk , j=0,1,...,m-1,m. The discretized solution of equation (2), by using the implicit finite difference method is. Since A ( j (x i )) is a constant the matrix ( j ) is symmetric

In each time step jЄ

Thus the is positive definite and bounded away from zero since

The values of t

The values of x