Backward Euler

📐 Definition

For $u_t = F(u,t)$ with timestep $\Delta t$ , backward Euler computes $u^{n+1}$ from

u^{n+1} = u^{n} + \Delta t\,F(u^{n+1}, t_{n+1})

typically via a nonlinear solve each step.

Domain and Codomain

Applies to states $u^n \in \mathbb{R}^m$ (or function spaces) with Lipschitz $F$ to ensure well-posed implicit solves. Output $u^{n+1}$ remains in the same space after solving the implicit equation.

⚙️ Key Properties

Order and Local Error

Backward Euler is first-order accurate in time, with local truncation error $\mathcal{O}(\Delta t^2)$ .

Linear Stability (Test Equation)

For the linear test equation $u_t = \lambda u$ ,

u^{n+1} = \frac{1}{1 - z}\,u^n, \qquad z = \lambda \Delta t

The method is A-stable and strongly damping for $\Re(z) \ll 0$ , making it effective on stiff dissipative dynamics.

Implicit Solve Requirement

Each step requires solving a nonlinear (or linearized) system for $u^{n+1}$ ; fixed-point iteration and Newton-type solvers are common choices.

🎯 Special Cases and Limits

For linear drift with forcing $F(u,t) = Au + g(t)$ , the update satisfies

(I - \Delta t\,A)u^{n+1} = u^{n} + \Delta t\,g(t_{n+1})

As $\Delta t \to 0$ , the scheme converges to the exact flow; for large $\Delta t$ on stiff stable $A$ , the method damps high-frequency modes without oscillation.

Forward Euler avoids implicit solves but is only conditionally stable; Crank–Nicolson raises the order while retaining implicitness; fixed-point iteration is a common solver for the implicit step.

Usage in Oakfield

Oakfield exposes backward Euler as an implicit built-in integrator implemented via fixed-point iteration:

Integrator name: backward_euler (Lua: sim.sim_create_context_integrator(ctx, "backward_euler", {...})).
Solve strategy: uses a simple Picard/fixed-point loop with an explicit Euler prediction as the initial guess and a residual check against the integrator tolerance (no Jacobians/Newton).
Adaptive fallback: when adaptive = true, Oakfield reduces dt across a small number of attempts if the fixed-point solve does not converge.
Complex + stochastic support: works for complex primary fields and applies stochastic increments after the implicit update when enabled.

Historical Foundations

📜 Origins

Implicit variants of Euler’s time stepping appear naturally when the drift is evaluated at the next time level to improve stability on dissipative dynamics.

🔬 Stiffness and Stability

20th-century stability theory (notably Dahlquist’s work) formalized why backward Euler is robust on stiff problems and helped establish A-stability as a guiding design principle for implicit methods.

📚 References

Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations II
Ascher & Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations