Euler’s Method for ODE’s

Source inspiration: (Mathew 2000-2019).

Description

Euler’s method is the simplest one-step numerical solver for an initial value problem \(y' = f(t,y)\), \(y(t_0)=y_0\). It advances the solution by following the local tangent at each grid point:

\[ y_{n+1} = y_n + h\,f(t_n,y_n), \qquad t_{n+1}=t_n+h. \]

For smooth problems, Euler has local truncation error \(O(h^2)\) and global error \(O(h)\), so reducing the step size improves accuracy linearly. The modified Euler (Heun) predictor-corrector method averages endpoint slopes and improves global accuracy to \(O(h^2)\).

Animations

Each animation below shows the numerical solution trace for the IVP \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\) on \(0 \le t \le 5\), using the legacy module’s \(n=41\) point setup (\(h=0.125\)). Each frame adds one new computed point and segment. The dashed black curve is a high-accuracy reference trajectory computed with fixed-step RK4 at \(h=10^{-4}\) and shown for direct visual error comparison.

Legacy archive note: the original Euleraa.gif and MEuleraa.gif files were missing from the recovered animation folders, so these are reconstructed from the module’s worked-example equations and parameters (Examples 11-12).

Case 1 - Forward Euler, \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\), \(0 \le t \le 5\), \(n=41\)

Behavior: Forward Euler accumulates first-order global error and drifts low near \(t=5\) relative to the dashed RK4 reference trajectory.

Julia source

Forward Euler animation for y’ = 1 - t cbrt(y), y(0)=1 over 0 <= t <= 5 using 41 points

Case 2 - Modified Euler (Heun), same IVP and grid

Behavior: The predictor-corrector slope average reduces drift and stays closer to the dashed RK4 reference trajectory at the right endpoint.

Julia source

Modified Euler animation for y’ = 1 - t cbrt(y), y(0)=1 over 0 <= t <= 5 using 41 points

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

Worked Example (Planned)

A compact numerical example with intermediate steps will be included.

Implementation Notes (Planned)

Implementation details, numerical stability notes, and practical pitfalls will be added.

Legacy Animation Inventory (Stub)

Legacy module page: Euler’s Method for ODE’s (ok)
Animation links found in module Animations paragraph: 8
Unique animation portals: 2

Animation Portals

Euler’s Method (ok)
Modified Euler’s Method (ok)

Animation item links found: 2

Animation Items

Euler’s Method (ok)

Main animated GIF count: 1
http://localhost:8000/a2001/Animations/OrdinaryDE/Euler1/Euleraa.gif

Modified Euler’s Method (ok)

Main animated GIF count: 1
http://localhost:8000/a2001/Animations/OrdinaryDE/MEuler1/MEuleraa.gif

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.