Abstract

We propose a variant of the parareal method for parallel-in-time computation of time dependent problems. We call it the θ-parareal schemes because it formally resembles the θ-schemes for discretizing time dependent partial differential equations.  

We analyze the linear stability of the resulting parareal iterations and consider furthere generalizations where θ can be more complicated operators that can improve the stability and accuracy of the parareal schemes using the computations performed in the past iterations. 

This work provides a systematic way of using the parareal iterations as a device to perform heterogeneous and multiscale couping, where solutions of different but similar equations are coupled together to achieve more efficient and accurate simulations. 

We present some numerical examples, including some widely studied nonlinear Hamiltonian systems,to demonstrate the potential of the proposed schemes.