High-order solution of one-dimensional sine–Gordon equation using compact finite difference and DIRKN methods

In this work we propose a high-order and accurate method for solving the one-dimensional nonlinear sine–Gordon equation. The proposed method is based on applying a compact finite difference scheme and the diagonally implicit Runge–Kutta–Nyström (DIRKN) method for spatial and temporal components, respectively. We apply a compact finite difference approximation of fourth order for discretizing the spatial derivative and a fourth-order AA-stable DIRKN method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables and is unconditionally stable. The results of numerical experiments show that the combination of a compact finite difference approximation of fourth order and a fourth-order AA-stable DIRKN method gives an efficient algorithm for solving the one-dimensional sine–Gordon equation.