ngs9lIDtNPJEg dynamical systems are typically stiff) and then solving the resulting matrix problem at each time step. This approach can be ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP disadvantageous for a parallel implementation, especially for MIMD parallel computers having a high communication latency, since the processors will have to synchronize repeatedly for each timestep. A ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP effective approach to solving the IVP with a parallel computer is to decompose the problem at the ODE level. That is, the large system is decomposed into smaller subsystems, each of which is assigned to a single processor. The IVP is solved iteratively by solving the smaller IVP's for each subsystem, using fixed values from previous iterations for the variables from other subsystems. This dynamic iteration process is known as waveform relaxation (WR) or sometimes as the Picard-Lindelof iteration. In this section, conjugate-direction methods are considered for accelerating the classical dynamic iteration methods. The approach is to first convert the IVP to a system of second-kind Volterra integral equations by using a ``dynamic preconditioner.'' Next, it is shown that the classical dynamic this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this iteration methods are obtained by applying the Richardson iteration to the integral equation system. Finally, conjugate-direction methods are developed for accelerating the classical dynamic iteration methods. This development is approached by considering conjugate-direction methods as Galerkin methods. Remark: Theorem only provides a description of the asymptotic behavior of the dynamic iteration. In Miekkala and Nevanlinna examine the dynamic iteration on the infinite interval. In that case, intuition about the convergence rate of the dynamic iteration can be obtained, although the extent to which that intuition applies for a given dynamic iteration on a finite interval depends on the stiffness of the problem. Another approach to solving is to apply a Galerkin method to solving a variational formulation of the problem. This approach leads directly to the conjugate-direction methods. ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP Consider the problem of numerically solving the linear time-varying initial value problem (IVP), is a given right-hand side, and computed over the simulation interval. There are several approaches to solving the IVP. The traditional numerical approach is to begin by discretizing a in time with an implicit integration rule (since large dynamical systems are typically stiff) and then solving the resulting matrix problem at each time step. This approach can be disadvantageous for a parallel implementation, especially for MIMD parallel computers having a high communication latency, since the processors will have to synchronize repeatedly for each timestep. A effective approach to solving the IVP with a parallel computer is to decompose the problem at the ODE level. That is, the large system is decomposed into smaller subsystems, each of which is assigned to a single processor. The IVP is solved iteratively by solving the smaller IVP's for each subsystem, using fixed values from previous iterations for the variables from other subsystems. This dynamic iteration process is known as waveform relaxation (WR) or sometimes as the Picard-Lindelof iteration. In this section, conjugate-direction methods are considered for accelerating the classical dynamic iteration methods. The approach is to first convert the IVP to a system of second-kind Volterra integral equations by using a ``dynamic preconditioner.'' Next, it is shown that the classical dynamic this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this iteration methods are obtained by applying the Richardson iteration to the integral equation system. Finally, conjugate-direction methods are developed for accelerating the classical dynamic iteration methods. This development is approached by considering conjugate-direction methods as Galerkin methods. Remark: Theorem only provides a description of the asymptotic behavior of the dynamic iteration. In Miekkala and Nevanlinna examine the dynamic iteration on the infinite interval. In that case, intuition about the convergence rate of the dynamic iteration can be obtained, although the extent to which that intuition applies for a given dynamic iteration on a finite interval depends on the stiffness of the problem. Another approach to solving is to apply a Galerkin method to solving a variational formulation of the problem. This approach leads directly to the conjugate-direction methods. ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP ivp ivP iVp Ivp iVP IVp IVP this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this this more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more more more more more more more more moremore more more more more more isis isi is is is is is is is is is is is is is is is is is is is is is is ISIS ISI IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS isis isi is is is is is is is is is is is is is is is is is is is is is is ISIS ISI IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS isis isi is is is is is is is is is is is is is is is is is is is is is is isis isi is is is is is is is is is is is is is is is is is is is is is is ISIS ISI IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS isis isi is is is is is is is is is is is is is is is is is is is is is is isis isi is is is is is is is is is is is is is is is is is is is is is is isis isi is is is is is is is is is is is is is is is is is is is is is is ISIS ISI IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS IS