Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs (Communications in Computational Physics, 2020)
This paper explores the discrete singular convolution method for HamiltonianPDEs. The differential matrices corresponding to two delta type kernels of thediscrete singular convolution are presented analytically, which have the properties ofhigh-order accuracy, band-limited structure and thus can be excellent candidates for thespatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equationand the coupled Schrödinger equations for example, we construct two symplecticintegrators combining this kind of differential matrices and appropriate symplectictime integrations, which both have been proved to satisfy the square conservationlaws. Comprehensive numerical experiments including comparisons with the centralfinite difference method, the Fourier pseudospectral method, the wavelet collocationmethod are given to show the advantages of the new type of symplectic integrators.