Implicit Fixed-Point Iteration

📐 Definition

Given an implicit update $u = G(u)$ (for example, $u = u^{n} + \Delta t\,F(u, t_{n+1})$ ), fixed-point iteration generates

u^{(m+1)} = G\!\big(u^{(m)}\big)

starting from an initial guess $u^{(0)}$ until convergence.

Domain and Codomain

Applies on Banach spaces (e.g., $\mathbb{R}^m$ or function spaces) where $G$ maps the space into itself and is contractive on the relevant neighborhood. The limit, when it exists, lies in the same space.

⚙️ Key Properties

Contraction Condition and Convergence

If $G$ is a contraction on a neighborhood,

\|G(u) - G(v)\| \le L\|u-v\|, \qquad 0 \le L < 1

then the iteration converges to the unique fixed point, with linear rate controlled by $L$ .

Cost and Implementation

Each iteration requires one evaluation of $G$ (often one drift evaluation and an algebraic update). No Jacobian is required, making Picard iteration attractive for matrix-free solves when contractivity holds.

Failure Modes

If $G$ is not contractive in the region of interest, the iteration may diverge, converge very slowly, or stagnate; damping or switching to Newton-type methods can help.

🎯 Special Cases and Limits

For backward Euler on $u_t = F(u,t)$ , $G(u) = u^{n} + \Delta t\,F(u, t_{n+1})$ ; convergence requires $\Delta t\,\|F'\|$ small enough for contractivity. For linear $G(u) = Bu + c$ , convergence occurs iff the spectral radius $\rho(B) < 1$ , with error decaying as $\|B\|^m$ .

Provides the nonlinear solve for implicit integrators such as backward Euler and Crank–Nicolson; Newton or quasi-Newton iterations accelerate convergence when contractivity is weak.

Usage in Oakfield

Oakfield uses fixed-point (Picard) iteration internally to implement its implicit integrators:

Backward Euler (backward_euler) and Crank–Nicolson (crank_nicolson) iterate a fixed number of times (or until the residual falls below tolerance), starting from an explicit predictor.
No Jacobians/Newton: the current implementations are matrix-free and rely on repeated drift evaluations rather than derivative-based solves.
Adaptive mode can reduce dt between attempts when convergence is not achieved within the iteration budget.

Historical Foundations

📜 Picard Iteration

Picard iteration emerged alongside foundational existence and uniqueness theory for ODEs, where successive approximations provide both a constructive method and a proof technique.

🔬 Fixed-Point Theorems

Banach’s fixed-point theorem (contraction mapping principle) formalized sufficient conditions for convergence and uniqueness in complete metric spaces, directly supporting modern convergence analysis.

📚 References

Ortega & Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables
Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations II