Some energy-preserving schemes for fractional Hamiltonian system with fractional Laplacian

In the paper, the energy-preserving scheme is presented for a class of fractional Hamiltonian system with fractional Laplacian. First, we show an equivalent form of the fractional Hamiltonian system by introducing some new auxiliary variables. The new system is discretized by the scalar auxiliary variable scheme in time, and a linear semi-discrete system is obtained, which can conserve the energy conservation law. Second, we show numerical schemes for one dimensional and two dimensional fractional Laplacian based on hyper-singular integral definition by quadratic interpolation function and linear interpolation function, and it finds that the differential matrices of the above schemes are symmetric Toeplitz matrices. Then, we use above scalar auxiliary variable scheme in time, and the above numerical scheme of fractional Laplacian in space to solve some fractional systems, and prove that the schemes can preserve energy conservation laws. Finally, the numerical experiments of some fractional Hamiltonian systems are given to verify the correctness of theoretical results.