A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.
The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations(FDEs)with Caputo derivatives,is an efficient and stable scheme.We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order indexα>0.
The method of splitting a plane-wave finite-difference time-domain(SP-FDTD) algorithm is presented for the initiation of plane-wave source in the total-field / scattered-field(TF/SF) formulation of high-order symplectic finite-difference time-domain(SFDTD) scheme for the first time.By splitting the fields on one-dimensional grid and using the nature of numerical plane-wave in finite-difference time-domain(FDTD),the identical dispersion relation can be obtained and proved between the one-dimensional and three-dimensional grids.An efficient plane-wave source is simulated on one-dimensional grid and a perfect match can be achieved for a plane-wave propagating at any angle forming an integer grid cell ratio.Numerical simulations show that the method is valid for SFDTD and the residual field in SF region is shrinked down to—300 dB.
