Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parareal algorithm for coupled elliptic-parabolic problems

Iñigo Jimenez-Ciga, Francisco Gaspar, Kundan Kumar, Florin A. Radu

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.

Parareal algorithm for coupled elliptic-parabolic problems

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.
Paper Structure (14 sections, 8 theorems, 67 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 8 theorems, 67 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Parareal algorithm
  3. Formulation
  4. Convergence conditions
  5. Extension of the Biot's model and iterative solvers
  6. Quasi-static Biot model
  7. Iterative solvers for the coupled problem
  8. Convergence of Parareal for the coupled problem
  9. Monolithic approach and the FS method
  10. Multirate FS method
  11. Numerical experiments
  12. Manufactured problem
  13. Mandel's problem
  14. Conclusions

Key Result

Proposition 2.1

Let us assume that operator $M_G$ given by eq:notation_matrix2 is coercive, with coercivity constant $\alpha_{M_G}$, and that the inequality $\|M_G-M_F\| < \alpha_{M_G}$ holds. Then, the Parareal algorithm is convergent.

Figures (6)

  • Figure 1: Coarse and fine discretizations of the time interval $[0,T]$.
  • Figure 1: Flowchart for the FS method described in \ref{['alg1_new']}.
  • Figure 1: Relative error of the Parareal/FS-FS, the Parareal/FS-MFS, and the Parareal/MFS-MFS methods applied to the manufactured problem, for decreasing value of $K$ and an increasing number of iterations.
  • Figure 2: Flowchart for the multirate FS method described in \ref{['alg_2_new']}.
  • Figure 2: Mandel's problem.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Proposition 2.1
  • Proof 1
  • Proposition 2.2
  • Proof 2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1
  • ...and 14 more