Keywords: time domain decomposition, parallel computing, Schwarz domain decomposition, spectral deferred correction, Schur domain decomposition.

This review chapter focuses on several developments for the parallelizing in time through domain decomposition methods for systems of first order ordinary differential equations. The time interval is split into time slices and the attempt to solve the system in parallel on time slices is investigated. The first approach proposed consists of some variations of the multiple shooting method without modifying the initial value problem. The lack of knowledge of the value of the solution at the end of the interval leads to having a propagation of information starting from the initial time until the time interval end. We proposed a pipeline of the iterations of the spectral deferred correction on the time slices. In the second approach we changed the initial value problem into a boundary value problem. This modification allows the propagation of the information in the time forward and time backward directions. Then we show how to select nonlinear transmission conditions between time slices to apply classical Schwarz domain decomposition methods with no overlap and how to derive a dual Schur domain decomposition method required in the Newton's nonlinear solver step.

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