We developed parallel time domain decomposition methods to solve systems of linear ordinary differential equations (ODEs) based on the Aitken-Schwarz [5, 3] or primal Schur complement domain decomposition methods [2]. The methods require the transformation of the initial value problem in time defined on \(]0, T]\) into a time boundary values problem. Let \(f(t,y(t))\) be a function belonging to \(\mathcal{C}^1(\mathbb{R}^+, \mathbb{R}^d)\) and consider the Cauchy problem for the first order ODE: \[\begin{cases} \dot{y}=f(t,y(t)),\, t \in ]0,T], \; y(0)=\alpha \in \mathbb{R}^d. \end{cases} \label{tromeur_contrib_IVP}\]

The time interval \([0,T]\) is split into \(p\) time slices \(S^{(i)}=[T_{i-1}^+,T_{i}^-]\), with \(T_0^+=0\) and \(T_p^-=T^-\). The difficulty is to match the solutions \(y_i(t)\) defined on \(S^{(i)}\) at the boundaries \(T_{i-1}^+\) and \(T_{i}^-\). Most of time domain decomposition methods are shooting methods [1] where the jumps \(y_i(T_{i}^-)-y_{i+1}(T_{i}^+)\) are corrected by a sequential process which is propagated in the forward direction (i.e. the correction on the time slice \(S^{(i-1)}\) is needed to compute the correction on time slice \(S^{(i)}\)). Our approach consists in breaking the sequentiality of the update of each time slice initial value. To this end, we transform the initial value problem (IVP) into a boundary values problem (BVP) leading to a second order ODE:

Then classical domain decomposition methods apply such as the multiplicative Schwarz method with no overlapping time slices and Dirichlet-Neumann transmission conditions (T.C.) for linear system of ODE (or PDE [4]). As proved in [3] the convergence/divergence of the error at the boundaries of this Schwarz time DDM can be accelerated by the Aitken technique to the right solution when \(f(t,y(t))\) is linear. Nevertheless, the difficulty in solving equation [tromeur_contrib_BVP] is that \(\beta\) is not given by the original IVP. In this talk, we investigate the proposed approach for nonlinear ODE.

References

1. Bellen, A., Zennaro, M.: Parallel algorithms for initial value problems for difference and differential equations. J. Comput. Appl. Math. 25(3), 341–350 (1989). doi 10.1016/0377-0427(89)90037-X

2. Linel, P., Tromeur-Dervout, D.: Aitken-schwarz and schur complement methods for time domain decomposition. In: B. Chapman, F. Desprez, G. Joubert, A. Lichnewsky, F. Peters, T. Priol (eds.) Parallel Computing: From Multicores and GPU’s to Petascale, Advances in Parallel Computing, vol. 19, pp. 75–82. IOS Press (2010)

3. Linel, P., Tromeur-Dervout, D.: Une méthode de décomposition en temps avec des schémas d’intégration réversible pour la résolution de systèmes d’équations différentielles ordinaires. C. R. Math. Acad. Sci. Paris 349(15-16), 911–914 (2011). doi 10.1016/j.crma.2011.07.002

4. Linel, P., Tromeur-Dervout, D.: Analysis of the time-schwarz ddm on the heat pde. Computers & Fluids 80, 94–101 (2013). doi 10.1016/j.compfluid.2012.04.023.

   http://www.sciencedirect.com/science/article/pii/S0045793012001624

5. Tromeur-Dervout, D.: Meshfree Adaptive Aitken-Schwarz Domain Decomposition with application to Darcy Flow. In: Topping, BHV and Ivanyi, P (ed.) Parallel, Distributed and Grid Computing for Engineering, Computational Science Engineering and Technology Series, vol. 21, pp. 217–250 (2009). doi 10.4203/csets.21.11