Noha Makhoul-Karam
Co-authors : Nabil R. Nassif, Jessy Haykal
Title :
Rescaling and Parallel Time Integration for Systems of Ordinary Differential Equations
Abstract :
A new method for parallel time integration has been presented in [1]: it automatically
generates time-slices and uses a rescaling methodology (introduced by Nassif et al in [2]).
The result of such approach is a Ratio-Based Time Integration (RaPTI) algorithm. This
algorithm has been successfully tested, in [1] on non-linear diffusion-reaction PDE's of
which the solution was monotonously exploding in infinite time.
We consider here a membrane second order initial value problem, given by:
y” + b |y’|q-1 y’+|y|p-1 y = 0,
where p ≤ q ≤ 2p/(p+1) and of which the solution y(t) exhibits an oscillatory behavior and
blows-up as t tends to infinity.
When q = 2p/(p+1), the rescaling methodology used in [3], led to invariance with respect
to the time slices, in the sense that the computation of y(t) has been reduced to its finding
over one slice, yielding a perfect parallelism when applying RaPTI Algorithm.
We consider now the case p ≤ q < 2p/(p+1) that results in similar rescaled systems (not
invariant) and we show that RaPTI Algorithm can also show a good efficiency when applied
to such problems having oscillatory and explosive solution. Instead of tracking the ratios
stabilization, the ratio-based prediction procedure uses, in this case, a backward analysis
on previous exact ratios.
References :
[1] N.Nassif, N. Makhoul-Karam, Y. Soukiassian. A New Approach for Solving Evolution
Problems in Time-Parallel Way. International Conference on Computational Science ICCS
06, Proceedings, Part I. Springer-Verlag Berlin / Heidelberg. Volume 3991 / 2006 - pp. 148
- 155.
[2] N.Nassif, D.Fayad, M.Cortas. Sliced-Time Computations with Re-scaling for Blowing-Up
Solutions to Initial Value Differential Equations. V.S. Sunderam. et al. (Eds): ICCS 2005,
LNCS 3514, pp. 58-65. Springer-Verlag 2005.
[3] N.Nassif, N. Makhoul-Karam, J. Haykal. Parallel Time Integration for a Membrane
Problem. Journal of Computational and Applied Mathematics, 2008. Submitted.