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Accelerated implicitization: Robust fixed-point iterations arising from an explicit scheme

Nicolas A. Barnafi, Felipe Galarce, Pablo Brubeck

TL;DR

The paper addresses solving nonlinear systems arising from implicit time discretizations by reformulating them as explicit-like fixed-point iterations that lag implicit terms, i.e. $x^{n,\\ell}-x^{n-1}=\\Delta t\\,f(t^n,x^{n,\\ell-1})$. It enhances this implicitization with Anderson acceleration, turning the method into an effective, quasi-Newton/multi-secant approach with manageable overhead. A convergence analysis shows CFL-type stability bounds (e.g., $\\Delta t \\le 1/L$ for Lipschitz $f$, and $\\Delta t \\le ilde{C} h^2$ in space-time discretizations), and numerical experiments on FitzHugh–Nagumo, heat equation, and Navier–Stokes demonstrate robustness and efficiency, especially when explicit schemes would be unstable or preconditioning is weak. The framework is simple to implement, scales well on large problems, and is well suited to high-performance hardware, including GPUs, offering a practical alternative for challenging nonlinear PDE simulations.

Abstract

This work proposes a general strategy for solving possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. The resulting sequence may exhibit instabilities similar to those of the base explicit scheme, which can be mitigated through Anderson acceleration. The approach uses explicit fixed-point subiterations for nonlinear problems, combined with Anderson acceleration to improve convergence and computational efficiency. Its usability and scalability are verified on three nonlinear differential equations. An error analysis is presented to establish the expected properties of the proposed strategy for both time and space-time formulations. Several examples illustrate the simplicity of the implementation and reveal the influence of parameter choices. The method proves simple to implement and performs well across a range of problems, particularly when matrix assembly is expensive or a good preconditioner for the implicit system is unavailable, such as in highly convective fluid flows. This work formalizes the delay of implicit terms in time discretization, provides a concise error analysis, and enhances the approach using Anderson acceleration. The results are encouraging and well supported by existing theory, laying the groundwork for further research.

Accelerated implicitization: Robust fixed-point iterations arising from an explicit scheme

TL;DR

The paper addresses solving nonlinear systems arising from implicit time discretizations by reformulating them as explicit-like fixed-point iterations that lag implicit terms, i.e. . It enhances this implicitization with Anderson acceleration, turning the method into an effective, quasi-Newton/multi-secant approach with manageable overhead. A convergence analysis shows CFL-type stability bounds (e.g., for Lipschitz , and in space-time discretizations), and numerical experiments on FitzHugh–Nagumo, heat equation, and Navier–Stokes demonstrate robustness and efficiency, especially when explicit schemes would be unstable or preconditioning is weak. The framework is simple to implement, scales well on large problems, and is well suited to high-performance hardware, including GPUs, offering a practical alternative for challenging nonlinear PDE simulations.

Abstract

This work proposes a general strategy for solving possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. The resulting sequence may exhibit instabilities similar to those of the base explicit scheme, which can be mitigated through Anderson acceleration. The approach uses explicit fixed-point subiterations for nonlinear problems, combined with Anderson acceleration to improve convergence and computational efficiency. Its usability and scalability are verified on three nonlinear differential equations. An error analysis is presented to establish the expected properties of the proposed strategy for both time and space-time formulations. Several examples illustrate the simplicity of the implementation and reveal the influence of parameter choices. The method proves simple to implement and performs well across a range of problems, particularly when matrix assembly is expensive or a good preconditioner for the implicit system is unavailable, such as in highly convective fluid flows. This work formalizes the delay of implicit terms in time discretization, provides a concise error analysis, and enhances the approach using Anderson acceleration. The results are encouraging and well supported by existing theory, laying the groundwork for further research.
Paper Structure (11 sections, 21 equations, 5 figures, 2 tables)

This paper contains 11 sections, 21 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Implicitization formulation
  3. Convergence analysis
  4. On the spatial dependency of implicitization
  5. Fixed-point iterations and Anderson acceleration
  6. Numerical Examples
  7. Electrophysiology model for cell-membrane potential
  8. Application to heat equation
  9. Application to Navier-Stokes
  10. Conclusions and Outlook
  11. Acknowledgments

Figures (5)

  • Figure 1: Normalized numerical solutions to \ref{['eq:fgn']} using fully implicit scheme. The parameter configuration is set to $a=0.7$, $b=0.8$, $\tau=5$, $R=1$ and $I_\text{ext}=0.5$.
  • Figure 2: Numerical solutions with $\Delta t = 0.1$.
  • Figure 3: Convergence study (left) and sensitivity study to AA depth.
  • Figure 4: Numerical solutions of \ref{['eq:nse']} using implicified scheme with coarse $\Delta t = 0.15$ s and AA with $m=3$. We emphasize the success of the tandem implicitization-AA, as we observe how the solver diverges for $m=0$, numerically verifying the broader convergence radius when using AA lupo2019convergence.
  • Figure 5: Picard iterations as time goes by. Optimal convergence speed is achieved in this numerical example for AA with $m=1$, and we highlight that the non-accelerated case $m=0$ fails to converge.