In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics.The two methods can preserve the intrinsic properties of original problems as much as possible.The splitting technique increases the computational efficiency.Meanwhile,the error estimation and some conservative properties are investigated.It is proved to preserve the charge conservation exactly.The global energy and momentum conservation laws can be preserved under several conditions.Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.
We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.
In this paper,we develop a multi-symplectic wavelet collocation method for three-dimensional(3-D)Maxwell’s equations.For the multi-symplectic formulation of the equations,wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration.Theoretical analysis shows that the proposed method is multi-symplectic,unconditionally stable and energy-preserving under periodic boundary conditions.The numerical dispersion relation is investigated.Combined with splitting scheme,an explicit splitting symplectic wavelet collocation method is also constructed.Numerical experiments illustrate that the proposed methods are efficient,have high spatial accuracy and can preserve energy conservation laws exactly.
