Abstract:
The numerical solution of differential equations is important in many branches of science, and accurate and stable algorithms are needed. The derivation and analysis of these methods is quite sophisticated and uses intricate mathematical tools such as B-series and Lie Algebras and Lie Groups. Our aim is to develop new numerical methods, and new methods of analysis, for Hamiltonian problems and other problems for which good qualitative behaviour is essential. The starting point in this thesis is a new class of effective order five methods based on the algebraic structure associated with Runge–Kutta methods. This leads us to another class of symplectic effective order methods which are designed for the solution of differential equations with quadratic invariants. G-symplectic general linear methods, for which parasitic growth factors are zero, are a possible alternative to symplectic Runge–Kutta methods. They have a similar ability to preserve quadratic invariants over extended time intervals. They have lower implementation costs and are designed to approximately preserve the quadratic invariants of the Hamiltonian systems. In this thesis, G-symplectic general linear methods are investigated theoretically and computationally. As for Runge–Kutta methods, there is an interaction between order conditions and the symplectic conditions, resulting in significant simplifications. To evaluate the order conditions for the G-symplectic general linear methods and the possible extension to any order is presented in this thesis. Finally the construction of sixth order G-symplectic general linear method with the additional properties of time reversal symmetry and freedom from parasitism is presented.
