Stability Analysis And Numerical Simulation Of Rosenbrock Methods For Stiff Delay Systems

Delay differential equations (DDEs) arise widely in many fields such as control engineering, aerospace, aviation, physics etc. In recent years, the research of numerical methods for those questions is very deep and popular. Much discussion about Runge-Kutta methods, linear multistep methods, Rosenbrock methods and general linear methods for DDEs is found in majority relevant literatures while little discussion deals with parallel methods for DDEs and few more complicated practical systems have been computed. On account of the rapid development of parallel computer and the rigorous demand of celerity of numerical methods in many fields, parallel methods have been frequently researched and have become one of the most important means in solving stiff differential equations. Therefore, it is urgent and significant in theory and practice to study the parallel methods for DDEs under those backgrounds.In Chapter 2, we analyze the GPm-stability of Rosenbrock methods for general differential equations with several delays, and show that the Rosenbrock methods are GPm-stable if and only if they are A-stable. Moreover, a PCF laser and a kind of linear stiff feedback control system with two delay states are numerically simulated.In Chapter 3, we deal with the stability analysis of parallel two-step ROW-methods (PTSROW methods) for the numerical solution to stiff delay differential equations. We prove that PTSROW method is GP-stable or GPL-stable if and only if it is A-stable or L-stable respectively.As the development of scientific research and large engineering design, numerical simulation has been researched and applied in many fields. However, a majority of research about efficient simulation methods are focused on ordinary differential equations (ODEs) while little discussion deals with efficient simulation methods for DDEs. But it is significant in theory and practice to study the efficient simulation methods for DDEs. Considering the easy realization of parallel methods in efficient simulation and favorable stability of PTSROW methods for DDEs shown in Chapter 3, we make use of tree theory and B-series to get the order and stage order conditions of PTSROW methods and it is proved that PTSROW methods satisfy order and stage-order conditions if they satisfy the simplifying conditions given in literature. Moreover, we construct some efficient parallel two-step ROW-methods (RTPTSROW methods) in Chapter 4 and show the efficiency of these RTPTSROW methods for stiff ODEs in addition to stiff DDEs by the numerical simulation experiments in parallel environment.