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This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter ε on error dynamics is investigated.

For differential equations with multiple order spatial derivatives, there are some shortcomings by the classical high order compact (HOC) discretization. At least one of them is reducing the computational efficiency due to the multiple inverse manipulation of matrices. This motivates us to design a new kind of compact method what is called combined high order compact methods. The basic idea lying in this kind of method is to solve all the spatial derivatives simultaneously. Then, it is used to solve coupled nonlinear Schrödinger (CNLS) equations which contain both the first and second order derivatives. This scheme is not only more compact and accurate than standard HOC scheme and standard finite difference method with the same order, but also it can construct structure-preserving schemes. It preserves the symplectic structure and mass, and sometimes energy and momentum. Numerical experiments indicate that the new scheme can simulate the CNLS equations very accurately and efficiently. The mass and momentum are exactly preserved. The energy is preserved in some especially cases.