Euler Method

📐 Definition

For an ODE $u_t = F(u,t)$ with timestep $\Delta t$, the forward Euler update advances the state by evaluating the drift at the current time level:

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

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Domain and Codomain

The method applies to states $u^n$ in $\mathbb{R}^m$ (or a function space) with locally Lipschitz $F$. The update returns $u^{n+1}$ in the same space.

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⚙️ Key Properties

Order and Local Error

Forward 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 $u_t = \lambda u$, the update is

$$u^{n+1} = (1 + z)u^n, \qquad z = \lambda \Delta t$$

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Stability requires $|1+z| < 1$, which yields a CFL-type timestep restriction for many PDE semi-discretizations.

Cost

One drift evaluation $F(u^n,t_n)$ per step; no linear or nonlinear solves.

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🎯 Special Cases and Limits

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

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

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As $\Delta t \to 0$, the method converges to the exact flow under standard smoothness assumptions.

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🔗 Related Functions

Heun’s method improves accuracy by averaging drift estimates; backward Euler trades explicitness for strong stability on stiff problems.

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Usage in Oakfield

Oakfield provides Euler stepping as a built-in context integrator:

• Integrator names: euler and euler_deterministic (Lua: sim.sim_create_context_integrator(ctx, "euler", {...})).
• Optional adaptivity: the Euler integrator can run in adaptive mode using a full-step vs two half-step estimate (step rejection up to a small fixed attempt count).
• Stochastic increments: when enabled, Oakfield applies stochastic updates after the deterministic step using the integrator’s noise hook.
• Scope: integrators advance the context by evolving the primary field state (field index 0) while drift evaluation executes the operator graph.

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Historical Foundations

📜 Origins

Euler’s method is one of the earliest systematic numerical schemes for integrating differential equations, arising from 18th-century work on differential calculus and approximation.

🌍 Modern Perspective

Today it serves as a canonical explicit first-order method: simple, interpretable, and a building block for stability and error analysis of more sophisticated integrators.

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📚 References

• Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I
• Butcher, Numerical Methods for Ordinary Differential Equations