Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation - Almushaira, Mustafa (Applied mathematics and computation, 2023)
•Novel high-order accurate and energy-preserving compact finite difference schemes are developed for solving the sine-Gordon equation on a two-dimensional domain.•The schemes have eighth-order accuracy in space and second-order accuracy in time.•They are unconditionally long-time stable and preserve the discrete energy conservation law.•Based on the FFT-based implementation, the methods are designed to reduce the computing costs of matrix multiplication while also using less memory.•The structures of the schemes can be easily generalized to multi-dimensions based on tensorproducts.•Extensive numerical comparisons confirm that the proposed fast algorithms enjoy high efficiency in long-time simulations. In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.