The solution of implicit differential equations on parallel computers

Abstract

We construct and analyze parallel iterative solvers for the solution of the linear systems arising in the application of Newton's method to ;;-stage implicit Runge-Kutta (RK) type discretizations of implicit differential equations (ID Es). These 1 incar solvers arc partly iterative and partly direct. Each linear system iteration again requires the solution of linear subsystems. hut now only of IDE dimension, which is ;; times less than the dimension of the linear system in Newton's melhod. Thus. the effective costs on a parallel computer system arc only one LU-decomposition of !DE dimension for each Jacobian update, yielding a considerable reduction of the effective LU-costs. The method parameters can he chosen such that only a few iterations by the linear solver are needed. The algorithmic properties arc illustrated hy solving the transistor problem (index 1) and the car axis problem (index 3) taken from the CW! test set.1997 Elsevier Science B. V.Kn·words: Numerical analysis; Implicit differential equations; DAEs: Runge-Kutta methods; Parallelism IntroductionWe consider initial value problems (lVPs) for systems of implicit differential equations (IDEs or DAEs) </{iJ(t),y(t))It will be assumed that the initial conditions arc consistent and that the IVP has a unique solution. Furthermore, we define the Jacobian matrices I<:= (/Ju(u,v) and ;v(u.v). In the case of txplicit ordinary differential equations (.T denotes the Jacobian of the right-hand side function f of the ODE.In this paper, we construct and analyze parallel iterative solvers for the solution of the .-;d-dirnensional linear systems arising in the application of Newton's method to s-stage implicit Runge-Kutta (RK) · The research reported in this paper was partially supported hy the Tec:hn{)logy Foundation (STWJ in the Netherlands. ' Corresponding author.0168-9274/97/$17.00 <0 1997 Elsevier Science B. V. All rights reserved. PI/ SO 168-9274(97)00071-8

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“…In [13], amplification matrices of the type M 1 have been analysed and led to the following definition and convergence theorem:…”

“…[12,13]). In order to see the intrinsic parallelism of the inner iteration process, we proceed as in the preceding section.…”

“…An estimate for ρ(M 2 ) can be obtained along the lines of a similar approach as in [13]. Theorem 3.2 presents conditions for convergence using the logarithmic matrix norm µ[·] associated with the Euclidean norm · , i.e., for any square matrix S, we have µ[S] = (1/2)λ max (S + S H ), where S H is the complex transposed of S and λ max (·) denotes the algebraically largest eigenvalue.…”

“…, 0, e), e being the s-dimensional vector with unit entries (see [5]). In [13], parallel iteration methods for solving the stage vector Y n from the nonlinear system (1.2) have been proposed. In this paper, we want to combine these parallel iteration techniques with the waveform relaxation (WR) approach.…”