Continuous parallel block methods and their applications

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations, delay differential equations, delay differential–algebraic equations, neutral delay differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied. As applications, we adopt the continuous parallel block methods to solve delay differential equations, neutral delay differential equations and delay differential–algebraic equations, and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable. Several numerical experiments are given to illustrate the performance of the continuous parallel block methods.