An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs (Communications in Computational Physics, 2020)
In this paper, we develop a novel energy-preserving wavelet collocationmethod for solving general multi-symplectic formulations of Hamiltonian PDEs. Basedon the autocorrelation functions of Daubechies compactly supported scaling functions,the wavelet collocation method is conducted for spatial discretization. The obtainedsemi-discrete system is shown to be a finite-dimensional Hamiltonian system, whichhas an energy conservation law. Then, the average vector field method is used fortime integration, which leads to an energy-preserving method for multi-symplecticHamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödingerequation and the Camassa-Holm equation. Since differentiation matrix obtained bythe wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transformto solve equations in numerical calculation. Numerical experiments show the highaccuracy, effectiveness and conservation properties of the proposed method.