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.
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