Continuous Block θ-Methods And Their Applications

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations,delay differential equations,neutral delay differential equations and integro-differential equations.In the past decades,some kinds of continuous extensions of Runge-Kutta methods and linear multistep methods have been defined.The motivation is to provide a dense output of the numerical solution of above kinds of equations.All these problems need approximate solutions defined not on the mesh points.In this thesis,we are concerned with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of blockθ-methods.Existence and uniqueness for the continuous extensions are discussed.Convergence and absolute stability of the continuous blockθ-methods for ordinary differential equations are studied.As applications, we adopt the continuous blockθ-methods to solve delay differential equations and neutral delay differential equations.It has been proven that the continuous blockθ-methods are GP-stable for DDEs and NGP-stable for NDDEs if and only ifθ∈[1/2,1].Several numerical experiments are given to illustrate the performance of the continuous blockθ-methods and confirm our theoretical results.