University:

Halle

Department:

FB Mathematik und Informatik

Institut fuer Numerische Mathematik

Date: 01.10.2002

MSC:

65L05 Initial value problems

Language: written in GER

Abstract: We derive a new type of parallel time-integration methods for large stiff ODE

systems: parallel two-step W-methods, PTSW-methods. Our methods possess s

linearly implicit stages which can be computed in parallel using s processors.

Note that the parallelism is hidden within the solver (black-box) and

additional no effort from the user is needed.



We study the local errors of our methods by means of Taylor series expansion

and derive simplifying conditions which guarantee convergence. We can attain

the order s and stage order s with an s-stage PTSW-method. We show the

existence of L-stable PTSW-methods up to 12 stages. The construction of methods

for practical use with two, three and four stages is discussed in detail. We

describe the implementation of the PTSW-methods with step size control and

Krylov approximation. For parallelization we use OpenMP shared memory

directives. In numerical tests we compare with the implicit Runge-Kutta-Code

RADAU and the Krylov-methods VODPK based on BDF methods. We consider standard

test problems taken from the CWI IVP-testset as well as semidiscretized reaction



diffusion problems where we use Krylov approximation (fully orthogonalization

method, FOM). Our code performs well. PTSW-methods are competitive with the

references method already in sequential mode. Due to the parallelization we

outperform VODPK in most examples where we use Krylov approximation.