Parareal algorithm for coupled elliptic-parabolic problems

TL;DR

This paper extends the Parareal parallel-in-time algorithm to complex coupled elliptic-parabolic systems from Biot models, providing convergence analysis, specific conditions, and demonstrating computational savings in multiphysics simulations.

Contribution

The paper introduces a novel extension of the Parareal algorithm tailored for degenerate differential-algebraic systems in poroelasticity, with convergence conditions and multiple coarse propagators.

Findings

Convergence conditions for each coarse propagator.

Explicit time step restrictions for contractivity.

Demonstrated computational savings in multiphysics simulations.

Abstract

We present a convergence analysis of the parallel-in-time integration method known as the Parareal algorithm for degenerate differential-algebraic systems arising from quasi-static Biot models, which govern coupled flow and deformation in porous media. The underlying system exhibits a saddle-point structure and degeneracy due to the quasi-static assumption. We extend the Parareal algorithm to this setting and propose three coarse propagators: monolithic, fixed-stress, and multirate fixed-stress schemes. For each, we derive sufficient conditions for convergence and establish explicit time step restrictions that guarantee contractivity of the iteration matrix. Numerical experiments show computational savings accrued by using a parareal solver in multiphysics simulations involving poroelasticity and other coupled systems.