Delay Integro-differential Equation Of Multi-step Runge - Kutta Method And Parallel Implementation

Delay integro-differential equations(DIDEs) plays a very important role in many fields such as power engineering, ecology, automatic control, environmental science and so on. Some scholars pay wide attention to the theoretical researches about the delay integro-differential equations. In general, it is difficult to obtain an analytical solution of DIDEs. However, the researches on numerical methods can make the theoretical researches perfect. So it is necessary to study the numerical solutions of DIDEs.A large number of serial algorithms have spring up, but with the rapid development of technology, large-scale delay problems appear in many fields, unfortunately, it is hard to solve them via the traditional serial algorithms, so it is necessary to bring in the parallel algorithms Firstly, this paper introduces the background of theoretical study and some achievement of parallel predictor-corrector algorithms. Secondly, it is shown that A-stable multistep Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems. Finally, a class of parallel Runge-Kutta predictor-corrector algorithms to solve DIDEs is presented, we give the local truncation error analysis in detail. The principal truncation error estimation can be directly used to monitor the stepsize. Theoretical analysis and numerical experiments show the algorithm for the delay integro-differential equations has a good effect.