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Anderson acceleration of a Picard solver for the Oldroyd-B model of viscoelastic fluids

Duygu Vargun, Igor O. Monteiro, Leo G. Rebholz

TL;DR

This work addresses the numerical challenge of solving the steady Oldroyd-B viscoelastic flow using a Picard fixed-point iteration that becomes non-contractive at larger Weissenberg numbers. By embedding Anderson acceleration within the Picard scheme and establishing the necessary smoothness and Lipschitz properties of the fixed-point operator, the authors provide both theoretical justification and empirical evidence for accelerated convergence. Theoretical results show AA preserves contraction characteristics under suitable conditions, while numerical experiments demonstrate that AA-Picard extends convergence to higher $\lambda$ and substantially reduces iteration counts, with robust performance across mesh refinements. The findings have practical impact for efficient, robust nonlinear solvers in non-Newtonian fluid simulations and can be adapted to time-dependent problems and other viscoelastic models.

Abstract

We study an iterative nonlinear solver for the Oldroyd-B system describing incompressible viscoelastic fluid flow. We establish a range of attributes of the fixed-point-based solver, together with the conditions under which it becomes contractive and examining the smoothness properties of its corresponding fixed-point function. Under these properties, we demonstrate that the solver meets the necessary conditions for recent Anderson acceleration (AA) framework, thereby showing that AA enhances the solver's linear convergence rate. Results from two benchmark tests illustrate how AA improves the solver's ability to converge as the Weissenberg number is increased.

Anderson acceleration of a Picard solver for the Oldroyd-B model of viscoelastic fluids

TL;DR

This work addresses the numerical challenge of solving the steady Oldroyd-B viscoelastic flow using a Picard fixed-point iteration that becomes non-contractive at larger Weissenberg numbers. By embedding Anderson acceleration within the Picard scheme and establishing the necessary smoothness and Lipschitz properties of the fixed-point operator, the authors provide both theoretical justification and empirical evidence for accelerated convergence. Theoretical results show AA preserves contraction characteristics under suitable conditions, while numerical experiments demonstrate that AA-Picard extends convergence to higher and substantially reduces iteration counts, with robust performance across mesh refinements. The findings have practical impact for efficient, robust nonlinear solvers in non-Newtonian fluid simulations and can be adapted to time-dependent problems and other viscoelastic models.

Abstract

We study an iterative nonlinear solver for the Oldroyd-B system describing incompressible viscoelastic fluid flow. We establish a range of attributes of the fixed-point-based solver, together with the conditions under which it becomes contractive and examining the smoothness properties of its corresponding fixed-point function. Under these properties, we demonstrate that the solver meets the necessary conditions for recent Anderson acceleration (AA) framework, thereby showing that AA enhances the solver's linear convergence rate. Results from two benchmark tests illustrate how AA improves the solver's ability to converge as the Weissenberg number is increased.

Paper Structure

This paper contains 12 sections, 10 theorems, 69 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Finite element discretization of Oldroyd-B model
  4. Analysis of fixed-point iteration of Oldroyd-B model
  5. Acceleration of the Picard iteration for Oldroyd-B model
  6. Anderson acceleration background
  7. Fixed-point operator $G$ and associated properties
  8. Numerical experiments
  9. Flow past a cylinder experiment
  10. Flow of Oldroyd-B fluid in a contracted L-shaped geometry
  11. Conclusions
  12. Appendix

Key Result

Lemma 3.2

(Stability) Assume the true velocity solution $u$ to OldroydB and previous iterate $u_k$ to divfreeOldroydBPicardScheme is sufficiently close to $u$ so that Then the iteration divfreeOldroydBPicardScheme at step $k+1$ is bounded:

Figures (4)

  • Figure 1: Flow of an Oldroyd-B past a circular cylinder domain.
  • Figure 2: Flow of an Oldroyd-B past a circular cylinder mesh.
  • Figure 3: Flow of an Oldroyd-B past a circular cylinder mesh.
  • Figure 4: The 32x64 L-shaped geometry mesh.

Theorems & Definitions (22)

  • Lemma 3.2
  • proof
  • Lemma 3.7
  • proof
  • Theorem 3.9
  • Remark 3.11
  • proof
  • Theorem 4.6
  • Lemma 4.9
  • proof
  • ...and 12 more