This paper endeavors to compute the soliton solution for the sine-Gordon (SG) equation using a hybrid numerical method. The present method has been developed using a modified trigonometric B-spline function with the differential quadrature method (DQM). The method is capable of reducing the partial differential equation into a system of ordinary differential equations which is further stimulated with SSP-RK43, which is a form of the Runge–Kutta method. The present method has been established for its applicability by the numerical simulation of one-soliton and interaction of two solitons for one- and two-dimensional SG equation. The error norms have been calculated and the obtained results are found to approach the exact solutions. The stability of the method is demonstrated with the help of eigenvalues. The results are found encouraging and thus the method can be implemented to solve the similar nonlinear partial differential equations.

本文试图使用混合数值方法计算正弦-戈登 (SG) 方程的孤子解。本方法是使用改进的三角 B 样条函数和微分求积法 (DQM) 开发的。该方法能够将偏微分方程简化为常微分方程组，并进一步用 SSP-RK43 进行刺激，这是龙格-库塔方法的一种形式。通过对一维和二维 SG 方程的单孤子和两个孤子相互作用的数值模拟，确立了本方法的适用性。已经计算了误差范数，并且发现获得的结果接近精确解。该方法的稳定性是在特征值的帮助下证明的。