Numerical solution of nonlinear Sine-Gordon equation using modified cubic B-spline-based differential quadrature method

In this article, we discuss the numerical solution of the nonlinear Sine-Gordon equation in one and two dimensions and its coupled form. A differential quadrature technique based on a modified set of cubic B-splines has been used. The chosen modification possesses the optimal accuracy order four in the spatial domain. The spatial derivatives are approximated by the differential quadrature technique, where the weight coefficients are calculated using this set of modified cubic B-splines. This approximation will lead to the discretization of the problem in the spatial domain that gives a system of first-order ordinary differential equations. This system is then solved using the SSP-RK54 scheme to progress the solution to the next time level. The convergence of this numerical scheme solely depends on the differential quadrature and is found to give a stable solution. The order of convergence is calculated and is observed to be four. The entire computation is performed up to a large time level with an efficient speed. It is found that the computed solution is in good agreement with the exact one and the error comparison with similar works in the literature indicates the scheme outperforms.