A collocation method based on extended cubic B-splines for numerical solutions of the Klein-Gordon equation

TL;DR

This paper introduces an extended cubic B-spline collocation method combined with Crank-Nicolson time integration to numerically solve the nonlinear Klein-Gordon equation, demonstrating its accuracy and conservation properties.

Contribution

It develops a generalized cubic B-spline collocation approach with parameter extension for solving Klein-Gordon equations, including linearization and conservation law analysis.

Findings

The method achieves high accuracy in numerical solutions.

Conservation laws for energy and momentum are well preserved.

Numerical results agree closely with analytical solutions.

Abstract

A generalization of classical cubic B-spline functions with a parameter is used as basis in the collocation method. Some initial boundary value problems constructed on the nonlinear Klein-gordon equation are solved by the proposed method for extension various parameters. The coupled system derived as a result of the reduction of the time order of the equation is integrated in time by the Crank-Nicolson method. After linearizing the nonlinear term, the collocation procedure is implemented. Adapting the initial conditions provides a linear iteration system for the fully integration of the equation. The validity of the method is investigated by measuring the maximum errors between analytical and the numerical solutions. The absolute relative changes of the conservation laws describing the energy and the momentum are computed for both problems.