On the behavior of high-order compact approximations in the one-dimensional sine–Gordon equation

Abstract

In this paper, a family of high-order compact finite difference methods in combination with Krylov subspace methods is used for solution of the nonlinear sine–Gordon equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite-difference equations is solved by Krylov subspace methods. The behavior of the compact finite-difference method is analyzed for error estimate and computational cost. Numerical results are presented to verify the behavior of high-order compact approximations for stability and convergence. The accuracy and efficiency of the proposed scheme are also considered.