Model Reduction Of Structure-Preserving For Two Kinds Of Spatial Fractional Equations

Since the spatial fractional Laplace operator(-?)^?/2(1<??2)is a nonlo-cal operator,its difference discrete often results in full matrix.For a fractional partial differential equation in space containing this operator,the ordinary dif-ferential equations obtained after semi-discretization of space usually have very high dimensions,resulting in strong stiffness and other computational difficul-ties.In addition,in order to maintain some important characteristics or good numerical stability of the original system,the implicit method is generally used for time discretization.If this kind of equation needs to be simulated for a long time,it needs to solve a large number of high-dimensional algebraic equations,which requires a huge amount of calculation.Therefore,for this kind of equation,improving the computational efficiency becomes an important problem.In this paper,we consider model reduction methods for two kinds of typi-cal fractional equations,namely spatial fractional semilinear wave equations and Schr¨odinger equations.We first restate the fractional equation as a continuous Hamiltonian system with symplectic structure by using variational principle.Then these two kinds of equations are discretized in space respectively.We discretize the (-?)^?/2 operator in spatial fractional wave equation by finite dif-ference,while in the spatial fractional Schr¨odinger equation with periodic bound-ary conditions by Fourier spectral method.Thus,we obtain two corresponding Hamilton ordinary differential equations.In order to reduce the dimension of spatial direction while maintaining the symplectic geometry of the system,we applied cotangent lift method and com-plex singular value decomposition method to get two mapping matrix,two kinds of equations are symplectic reduced to low dimensional symplectic space by prop-er symplectic decomposition technique,then spatial semi-discrete Hamiltonian forms after symplectic reduction for two kinds of equations are obtained.Finally,the symplectic implicit Runge-Kutta method is used to solve the spatial semi-discrete symplectic reduced Hamiltonian system.Through comparative analysis of numerical experiments,it is found that when the spatial dimension of the original space fractional Hamiltonian system is large,both the cotangent lift method and the complex singular value decomposition method can effectively simulate the original system for a long time while maintaining the symplectic structure and stability of the system.Numerical experiments demonstrate the effectiveness of the proposed algorithm.