Symplectic And Multi-symplectic Methods Based On Wavelets

Large physical phenomenons in nature can be represented by infinite Hamiltoniansystemsandmulti-symplecticPDEs,whichhavesymplecticstructuresandmulti-symplecticstructures respectively. Symplectic and multi-symplectic methods can preserve these twokinds of structures respectively, and have superiority in long time simulation and preserv-ing invariants of original systems. On the other hand, wavelet-based methods, which takethe advantages of both finite difference method and spectral method, have superiority insolving singular problems. In this paper, based on wavelet-based methods, we constructsymplectic and multi-symplectic methods, and generalize them to high dimensions. Themain contributions of this paper go as follows:1. Symplecticwaveletcollocationmethod(SWCM)isconstructedforinfiniteHamil-toniansystem. BasedontheautocorrelationfunctionsofDaubechiescompactlysupportedscaling functions, collocation method is conducted for the spatial discretization, whichleads to a finite-dimensional Hamiltonian system. Then, symplectic Runge–Kutta methodis used for the integration of the Hamiltonian system. Moreover, the properties of the re-sulted space differentiation matrix are analyzed in detail. The convergence and invariantsconservation properties of the proposed methods are also investigated. Various numericalexperiments for the nonlinear wave (NLW) equation and nonlinear Schr(o|¨)dinger (NLS)equation show that the method has high accuracy in space, is suitable for long time simu-lation, can capture singularity efficiently and preserve the invariant quantities very well.2. Multi-symplecticwaveletcollocationmethod(MSWCM)isconstructedformulti-symplectic PDEs. Wavelet collocation method is conducted for the spatial discretizationof multi-symplectic PDEs and symplectic scheme is used for time discretization. The ob-tained semi-discrete system is proved to have N semi-discrete multi-symplectic conser-vation laws and N semi-discrete energy conservation laws. The method has full-discretemulti-symplectic conservation laws and conserve global symplecticity. Numerical exper-iments for the NLS equation and Camassa-Holm equation show that the method has thegood properties of symplectic wavelet collocation method and can preserve local conser-vation laws well.3. Symplectic and multi-symplectic wavelet spectral methods are constructed for nonlinearSchr(o|¨)dingerequation. Waveletspectraldiscretization,whichcanusefastFouriertransform to decrease computations, is used for spatial discretization. Then, symplecticRunge–Kutta method is used for time integration to construct multi-symplectic waveletspectral methods (MSWSM). In addition, splitting scheme is combined to construct ex-plicit splitting symplectic and multi-symplectic wavelet spectral methods (ES-SWSM andES-MSWSM). Numerical results show that these methods have similar numerical resultswith SWCM and MSWCM, but take less computations.4. Symplectic and multi-symplectic methods are generalized to two-dimensional (2-D) case. The 2-D NLS equation and the 2-D time-dependent linear Schr(o|¨)dinger equationin quantum physics are simulated. The stability, invariant quantities preserving and thetradeoff between numerical error and CPU time are analyzed. Numerical results showthat the proposed methods for 2-D case maintain the good properties of 1-D case. Thosemethods are compared with other methods which use central finite difference scheme andFourier spectral scheme for spatial discretization.