Advance Numerical Methods for Special Differential Equations with Structure

Abstract

The current work is concerned with structure-preserving numerical meth- ods for particular differential equations representing dynamically in classical mechanics from the Hamiltonian perspective. These differential equations involve quantities that should remain constant throughout the investiga- tion. These methods includes the symplectic Runge-Kutta methods and the G-symplectic general linear methods. They faithfully maintain the un- derline characteristic properties of a Hamiltonian system during numerical discritization of the problem, whereas conventional numerical methods do not account for the preservation of these invariants. The conservation of total energy and the symplecticity of flow are two physically significant invariant properties of a Hamiltonian system which we to preserve numeri- cally. This thesis contains four chapters. In chapter 1, the fundamental theory of ordinary differential equations is presented. The exposition combines definitions with the development and application of numerical methods for special differential equations, and it serves as the foundation for all subsequent investigations in this work. In chapter 2, numerical methods for the Hamiltonian system are thoroughly discussed. In chapter 3, the crucial concept of projection is introduced. The basic concept of the standard projection technique is generalised to create the heterogeneous class of numerical methods known as general lin- ear methods. In chapter 4, the symmetric general linear methods are used with the Runge-Kutta methods as starting methods.