Chapter 46 - Adaptive Time Domain Decomposition for systems of ODEs on Grid architecture

Abstract

This chapter describes adaptive time domain decomposition for systems of ODEs on grid architecture. Modeling complex systems can lead to solve ODE, and/or DAE systems to understand the dynamic behavior of the solutions. The main difficulty in terms of parallel implementation of such systems of equations is the poor granularity of the computations, especially, for the grid computing architecture, which is characterized by a large number of processors. Nevertheless, the development of low cost parallel computer changes the concept of an efficient parallel implementation. The number of available processors allows considering these computational resources to improve the confidence in the solution by combining several schemes of different orders and different discretizations, to validate and to verify the result. The chapter introduces some adaptivity in the parareal parallel ODEs solvers to apply its concepts to stiff ODEs. The main objective of such solver is to reduce the elapse time and not actually to have the best parallel efficiency. Some improvement in the method is shown with defining the fineness of the grids on the relative tolerance of the time slice integrator instead of some multiple of the time step of a reference time fine grid. Moreover, it is shown that the time slice size should be connected to the dynamic of the solution.