General Linear Methods With Projection

Abstract

The present work deals with the structure-preserving numerical methods for the solution of ordinary differential equations representing dynamic systems from the Hamiltonian perspective of classical mechanics. These differential systems involve quantities that should remain constant over the course of enquiry. Whereas the usual numerical methods do not account for the preservation of these invariants, structure-preserving methods such as the symplectic Runge-Kutta methods and the G-symplectic general linear methods faithfully maintain these characteristic properties of a Hamiltonian system during the numerical discretization of the problem. Physically significant invariant properties of a Hamiltonian system include the conservation of total energy and the symplecticity of flow. In Chapter 1, basic theory of ordinary differential equations is introduced. The exposition combines definitions with the construction and implementation of numerical methods for ordinary differential equations and is central to all the subsequent investigations carried out in this work. In Chapter 2, long - term integrators for a Hamiltonian system are treated in detail. The general linear methods subsume the Runge-Kutta methods as well as the linear multi-step methods introduced previously. In Chapter 3, the important notion of projection is also introduced here. The basic idea of the standard projection technique is generalized to develop the heterogeneous class of numerical methods collectively known as the general linear methods. Projection technique is applied to the general linear methods and the similarity of the symplectic Runge-Kutta methods to the G-symplectic general linear methods is discussed.