Continuous Parallel Block Methods And Their Applications

Continuous numerical methods have many applications in the numerical solution of discon-tinuous ordinary differential equations, delay differential equations, delay differential-algebraicequations, neutral delay differential equations, integro-differential equations, etc. In the pastdecades, some kinds of continuous extensions of Runge-Kutta methods and linear multistep meth-ods have been defined. The motivation is to provide a dense output of the numerical solution ofabove kinds of equations. All these problems need approximate solutions defined not on the meshpoints.This paper deals with a continuous extension for the discrete approximate solution of ordinarydifferential equations generated by a class of parallel block methods. Existence and uniquenessfor the continuous extension are discussed. Convergence and absolute stability of the continuousparallel block methods for ordinary differential equations are studied. As applications, we adoptthe continuous parallel block methods to solve delay differential equations, neutral delay differ-ential equations and delay differential-algebraic equations, and obtain sufficient and necessaryconditions for the continuous parallel block methods to be asymptotically stable.Several numerical experiments are given to illustrate the performance of the continuous par-allel block methods.