Two Structure-preserving Methods For Some Nonlinear Evolution Equations

In recent ten years, structure-preserving algorithms for nonlinear evolution equations have obtained rapid development and achieved a large number of re-search results. Symplectic and multi-symplectic algorithms are the important parts of structure-preserving algorithms, which are mainly applied to solve Hamil-tonian partial differential equations (PDEs). In this paper, we use multi-symplectic collocation method and energy-preserving Galerkin finite element method to solve some nonlinear partial differential equations.On the one hand, we establish the multi-symplectic collocation method based on the multi-symplectic formulation of Hamiltonian PDEs. B-spline collocation method is used in space and symplectic method in time. Then, we analyze the conservative properties of semi-discrete and full-discrete schemes. Furthermore, the numerical method is applied to solve KdV equation and combined KdV and mKdV equation. Finally, numerical experiments are presented to show the con-servative properties of multi-symplectic B-spline collocation method.On the other hand, energy-preserving Galerkin finite element method(FEM) is built according to the variational derivative form of partial differential equa-tion. Firstly, we use discrete variational derivative method in time and obtain semi-discrete scheme, and use Galerkin method in space and obtain full-discrete scheme. Secondly, analysis of energy conservation of the full-discrete scheme is given. Thirdly, we use the energy-preserving Galerkin FEM with low-order mixed finite elements for solving KdV equations and nonlinear Schrodinger equations. At last, numerical experiments confirm the conservative properties of the energy-preserving Galerkin FEM.