High-order Compact Structure-preserving Algorithms For Multi-symplectic Hamlltonian System

Methods that conserve at least some of structural properties of systems are called geometric integrators or structure-preserving algorithms. The idea of structure-preserving algorithm was firstly presented by Feng systematically. Symplectic al-gorithms and multi-symplectic algorithms have the special advantages on simulat-ing PDEs with Hamiltonian structure. Much attention has been paid to high-order compact finite difference schemes due to their high efficiency and accuracy. Com-pared with traditional finite difference schemes, compact finite difference schemes need fewer nodes and calculations to achieve the same accuracy. In this thesis, based on the multi-symplectic form of PDEs, we discuss the high-order compact structure-preserving algorithms. Firstly, we use the high-order compact finite difference method to discrete multi-symplectic PDEs in space resulting in a semi-discrete system which still preserve the energy conservation law or the multi-symplectic conservation law. Secondly, we apply the average vector field (AVF) method, the mid-point method and the Euler-box method in time to integrate the resulted ODEs, respectively, and then obtain several energy-preserving schemes or multi-symplectic schemes for multi-symplectic PDEs. These schemes not only preserve the Hamiltonian energy or the multi-symplectic conservation law of the original system, but also possess the superi-ority of the compact finite difference method. Last but not least the proposed schemes are illustrated by simulating the Korteweg-de Vries(KdV) equation and the nonlinear schrodinger equation. Especially, using the idea of composition method, we construc-t a new class of multi-symplectic algorithms for the nonlinear schrodinger equation which are of 2-order. Numerical experiments are also presented to verify the theoret-ical results.