Exponentially-Fitted Methods For Constrained Hamiltonian Systems

With the advancement of sciences and engineering technologies, ordinary differential equations alone are not enough for modeling some complex systems correctly. This forces people to study coupled systems of ordinary differential equations and algebraic equations, called systems of differential-algebraic equations (DAEs). On the other hand, many applications require urgently that numerical methods preserve the intrinsic physical or geometrical properties of the exact solution of problems in a long term, which leads to investigation of structure-preserving algorithms.All real and non-dissipative physical processes can be represented by Hamiltonian systems. Theoretical and practical researches show that symplectic methods are much superior to non-symplectic methods in preserving qualitative properties of the systems, especially in long term. Constrained Hamiltonian systems form a significant category of differential-algebraic systems, where the orbits of the solutions to the differential equations always lie on the manifolds defined by the constraining algebraic conditions. The numerical solution to this class of constrained problems has been investigated systematically by many authors. For instance, Ryckaert, Ciccotti and Berendsen developed the SHAKE method, L. Jay studied symplectic partitioned Runge-Kutta methods (PRK). On the other hand, a new type of technique, called exponentially-fitted methods, emerge. These methods can integrate exactly some basic oscillatory functions and, when applied to oscillatory differential equations, have satisfactory phase properties and are more efficient than traditional integrators.This thesis is mainly concerned with exponentially-fitted methods for constrained Hamiltonian systems with oscillatory solutions.This thesis consists of four chapters.Chapter1provides preliminaries including a survey of elementary knowledge of differential equations like the existence and uniqueness of the solution of initial value problems of ordinary differential equations (ODEs), consistency, convergence and stability of numerical methods. We introduce symplectic Runge-Kutta methods and symplectic partitioned Runge-Kutta methods solving Hamiltonian systems, and exponentially fitted methods which are one of the most frequently used techniques in structure-preserving methods for oscillatory problems.Chapter2discusses Runge-Kutta methods solving differential-algebra equations (DAEs). After a brief introduction to the concept of differential-algebraic equations, we present RK methods for index-1, index-2and index-3DAEs, respectively, and the simplified Newton method in the implementation of implicit methods. For index-1DAEs and their numerical methods, we build systematically the bi-colored rooted-tree theory and the related B-series theory and derive order conditions for RK methods solving index-1DAEs.Chapter3focuses on symplectic methods for constrained Hamiltonian systems. Among the numerical methods preserving some important properties of the exact flows of Hamiltonian systems, we pay special attention to the analysis of the order of dispersion of the Lobatto ⅢA-ⅢB method. This method is zero-dissipative due to sympleticity.In Chapter4, we investigate the exponentially-fitted methods for constrained Hamiltonian systems. By the symplecticness conditions and the exponential fitting conditions, we construct a three stage symplectic exponentially-fitted PRK method of maximized phase-lag order. Numerical results show that the new method preserves the Hamiltonian energy more precisely and is more efficient than the traditional PRK methods.Finally we summarize the main contributions of this thesis and foresee some prospect topics of further work concerning exponentially-fitted methods for constrained oscillatory Hamiltonian systems.