Continuous Block $\theta$-Methods for Ordinary and Delay Differential Equations

Abstract

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations (ODEs), delay differential equations, neutral differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ODEs generated by a class of block $\theta$-methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous block $\theta$-methods for ODEs are studied. As an application, we adopt the continuous block $\theta$-methods to solve delay differential equations and prove that the continuous block $\theta$-methods are $GP$-stable if and only if they are $A_{\omega}$-stable for ODEs. Several numerical experiments are given to illustrate the performance of the continuous block $\theta$-methods.