Some Topics In Structure-Preserving Methods For Differential Equations

In the classical sense, the attention in the numerical solutions of ordinary and partial differential equations is mainly absorbed in the constructing numerical methods, the analysis of accuracy, convergence and numerical stability of the numerical methods and so on. The proposed methods are usually regarded as the all-purpose ones, that is, they can be applicable to any differential equation. However, many difficulties arisen in the implementation of these all-purpose methods address us the thought that a scheme can solve all problems is incorrect. In fact, many differential equations have their particular qualitative properties, or so-call geometric structure. Gradually, one realizes that it is very important that a numerical method can capture some qualitative properties of the differential equations solved or not. The capturing ability has been regarded as a criterion of the success of numerical simulations or not. The idea leads to the study of the geometric numerical integration, or structure-preserving algorithms in recent years. The main desire of geometric numerical integration is that the methods should preserve as much of the qualitative solution behavior as possible when solving differential equations numerically.The thesis contains three topics in the structure-preserving methods for ordinary and partial differential equations. They are respectively: Two-step explicit P-stable methods for solving periodic IVPs; Multisymplectic integrators for Hamiltonian partial differential equations; Energy and momentum conserving integrators for Newtonian motion equations.Firstly, we consider the initial value problems of second order ordinary differential equations (ODEs) which have periodic or oscillatory solutions (for short, call them periodic IVPs). This class of problems usually occur in many scientific and engineering fields. When designing a numerical method for these problems, we have to consider three important features: algebraic order, periodicity stability and phase-lag, especially the phase-lag property. In the second chapter, we propose some two-step explicit P-stable nonlinear methods of high phase-lag order for one-dimensional periodic IVPs. These methods are only component-applicable. With the aid of a special vector operator, the methods can be extended to the vector case directly. The stability behavior of these methods and their vector form are thoroughly discussed. Some numerical results are presented to illustrate the efficiency and disadvantage of these methods.Secondly, we study the multisymplectic integrators for multisymplectic Hamiltonian partial differential equations (PDEs). It is well-known that the symplectic structure is the most essential qualitative property for Hamiltonian ODEs. When solving the Hamiltonian ODEs numerically, naturally, we expect that the symplectic structure can be captured, and it leads to the symplectic integration which has been thoroughly analyzed in recent years. The practice suggests the results given by symplectic integrators are better than that by nonsymplectic integrators for solving the Hamiltonian ODEs, especially over long times. The Hamiltonian PDEs are proposed as a generalization of Hamiltonian ODEs in time and space, namely, Hamiltonian PDEs have the multisymplectic structure with respect to time and space. One of great challenges in the numerical solutions of PDEs is the development of robust stable numerical algorithms for Hamiltonian PDEs. Similarly,...