Study On Two Classes Of Structure-Preserving Methods For Hamiltonian Partial Differential Equations

Hamiltonian partial differential equations(PDEs) arise as important models in molecular dynamics, electromagnetic, celestial mechanics and quantum mechanics. The development of robust stable numerical algorithms for Hamiltonian PDEs is one of the great challenges in the numerical analysis of PDEs. Currently, the geometric integration of Hamiltonian ODEs is well-developed. However, there is a principal difficulty that arises when generalizing from Hamiltonian ODEs to Hamiltonian PDEs. It’s of great importance to study the structure-preserving methods for Hamiltonian PDEs.The thesis is devoted to investigating the invariants preserving methods and multisymplectic methods for some Hamiltonian PDEs. The main achievements are as follows.1. Regarding invariants of the coupled Schr?dinger-KdV equations, two energy preserving schemes are constructed by using Fourier pseudospectral method in space discretization and average vector field method in time discretization. The split-step technique is used to accelerate the simulation. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it’s not as good as the non-splitting scheme in preserving the invariants2. The multi-symplectic Hamiltonian formulation of the coupled Schr?dinger-KdV equations is presented for the first time. Then we develop a novel multi-symplectic Fourier pseudospectral scheme for the equations. In numerical experiments, the multi-symplectic Fourier pseudospectral scheme is compared with the Crank–Nicholson scheme. The results show that multi-symplectic Fourier pseudospectral method has a good ability of conserving the invariants and has high accuracy.3. The projection method for ODEs is generalized to PDEs and applied to the KdV equation. We compare the projection method with the ZK scheme, multi-symplectic Fourier pseudospectral scheme, simplified multi-symplectic scheme, average vector field scheme and an implicit scheme, the results show that the projection method can well preserve the invariants.