Numerical Simulation of Two Dimentional sine-Gordon Solitons Using the Modified Cubic B-Spline Differential Quadrature Method

TL;DR

This paper presents a novel numerical approach using a modified cubic B-spline differential quadrature method to simulate two-dimensional sine-Gordon solitons, effectively handling boundary conditions and converting PDEs into ODEs for efficient solution.

Contribution

The paper introduces a modified cubic B-spline based differential quadrature method for 2D sine-Gordon equations, enhancing accuracy and stability over existing methods.

Findings

Results agree well with exact solutions.

Method effectively handles damped and undamped cases.

Provides a reliable numerical tool for sine-Gordon equations.

Abstract

In this article, a numerical simulation of two dimensional nonlinear sine-Gordon equation with Neumann boundary condition is obtained by using a composite scheme referred to as a modified cubic B spline differential quadrature method. The modified cubic B-spline serves as a basis function in the differential quadrature method to compute the weighting coefficients. Thus, the sine-Gordon equation is converted into a system of second order ordinary differential equations (ODEs). We solve the resulting system of ODEs by an optimal five stage and fourth-order strong stability preserving Runge Kutta scheme. Both damped and undamped cases are considered for the numerical simulation with Josephson current density function with value minus one. The computed results are found to be in good agreement with the exact solutions and other numerical results available in literature.