An improvised B-spline collocation method using a fourth-order implicit scheme for solving the nonlinear Klein-Gordon equation

In this paper, an improvised B-spline collocation method is proposed to numerically solve the nonlinear Klein-Gordon (KG) equation. The original equation is initially transformed into a system of partial differential equations (PDEs) that is first-order with respect to time by introducing an auxiliary variable v = u t . The improvised cubic B-spline collocation method is then employed to perform spatial discretization on the resultant system, while a fourth-order implicit scheme is adopted for temporal discretization. The stability of the proposed method is examined via von-Neumann analysis, and its convergence is also discussed. Five numerical tests are carried out to further validate the accuracy and stability of the proposed method.