The Application Of Parallel Diagonal Implicit Runge-Kutta Methods For Differential Equations

Numerical methods for the solution of initial value problems in differential equa-tions appear in many fields of important scientific researches and engineering projects.Considered that the problems are very large in practice, and the response time is muchsignificant, so the needs for the parallel methods of initial value problems for differ-ential equations are increasing greatly.The major work of this thesis is to extend parallel methods from ODEs to DDEs.Using the idea of parallel diagonal implicit Runge-Kutta (PDIRK) methods for ODEs,we build different PDIRK methods for the initial value problems of integro-differentialequations(IDEs), DDEs and delay integro-differential equations(DIDEs), respectively.At first, we review the PDIRK methods for ODEs.Secondly, we study the initial value problem for IDEs. We replace the integralterm by a quadrature, so that IDEs turn into OEDs with high-order. Then we applyPDIRk methods to it.Finally, we extend PDIRK methods for ODEs into differential equations withconstant delay, and establish PDIRK methods for DDEs and DIDEs. Since the initialconditions of DDEs and DIDEs are continuous functions in the interval [t₀-Ï„,t₀],they are seen as ODEs. So we could apply PDIRK methods to them, and get thenumerical solutions in the interval [t₀,t₀ +Ï„]. Similarly, we have numerical solutionin the whole interval successively.We derive the convergence and stability strictly for different differential equa-tions. Computing specific test equations show that these methods are effective.