Researches On Structure-preserving Algorithms For The Infinite Dimensional Hamiltonian Systems

Many partial differential equations arising in physics can be seen as infinite di-mensional Hamiltonian systems. One of the great challenges in the numerical anal-ysis of partial differential equations (PDEs) is the development of stable numerical algorithms for Hamiltonian PDEs. It is well known that symplectic algorithms can simulate the Hamiltonian ODEs very well. As a direct generalization of the symplec-tic algorithm, multisymplectic algorithms have good performance on conserving all local conservation laws of the Hamiltonian system which guarantees the efficiency of long-time simulations. In this dissertation, we further study the multisymplectic algo-rithm. Based on different equations, we construct a series of multisymplectic schemes respectively and analyze their properties. Numerical experiments verify the efficiency of the schemes as well as the invariants preservation.The classical Crank-Nicolson scheme has good behaviours on simulating the nonlinear Schrodinger equation. However, there rarely exist studies on the connec-tion between this scheme and the symplectic geometric algorithm. By the method of lines, we prove this scheme is symplectic. Furthermore, from the sides of concatenat-ing method and variational integrator, we prove this scheme is also multisymplectic. Therefore, we successfully explain the nice performance of such scheme. We also prove that the scheme satisfies the discrete mass conservation law and present the theoretical analysis on the convergence. Numerical comparisons with the multisym-plectie:Preissmann scheme are made to show the superiority of the Crank-Nicolson scheme.For the three dimensional Maxwell’s equations, we construct a semi-explicit mul-tisymplectic Euler-box scheme and present the detail algorithm. We also prove this scheme can preserve not only the discrete local and global energy, but also the discrete divergence. Furthermore, we analyze some properties of this scheme from the point of numerical dispersion relation.As a b-family equation, Degasperis-Proseci equation has very rich geometry properties. Following the idea of structure-preserving algorithm, we derive two nu-merical integrators. Starting from the bi-Hamiltonian structure, we construct a sym- plectic pseudospectral scheme which can simulate the peakon solutions well. To catch the shock peakon, we derive a new integrator by the splitting technique. The new in-tegrator can handle with all kinds of solutions the equation. Numerical examples give the feasibility of the two schemes.