Milne-Simpson’s Method
Source inspiration: (Mathew 2000-2019).
Description
Milne-Simpson’s method is a multistep predictor-corrector method for initial value problems \(y' = f(t,y)\). It uses a 4-step Milne predictor and a Simpson-style corrector:
\[ y_{n+1}^{(p)} = y_{n-3} + \frac{4h}{3}\left(2f_{n-2} - f_{n-1} + 2f_n\right), \]
\[ y_{n+1} = y_{n-1} + \frac{h}{3}\left(f_{n-1} + 4f_n + f(t_{n+1},y_{n+1}^{(p)})\right). \]
Because the method is not self-starting, several initial points are generated with one-step RK4 before predictor-corrector stepping begins.
Animations
The animation below reconstructs legacy Example 11 from the Mathews module using the IVP \(y' = t^2 + y^2\), \(y(0)=1\). A dashed high-accuracy RK4 trajectory (\(h=10^{-4}\)) is shown as reference.
Legacy archive note: the local mirror is missing several Example 11 image IDs; the equation and behavior were recovered from the module link and remaining assets, with missing image files fetched from the Fullerton archive.
Case 1 - Milne-Simpson predictor-corrector, \(y' = t^2 + y^2\), \(y(0)=1\), \(0 \le t \le 0.9\)
Behavior: The Milne-Simpson trajectory follows the RK4 reference closely in the pre-asymptotic range and shows the characteristic rapid growth trend of this Riccati equation as \(t\) approaches the blow-up region.
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: Milne-Simpson’s Method (ok)
- Animation links found in module Animations paragraph: 4
- Unique animation portals: 1
Animation Portals
- Animation item links found: 1
Animation Items
- Milne-Simpson Method (ok)
- Main animated GIF count: 1
- http://localhost:8000/a2001/Animations/OrdinaryDE/Milne1/Milneaa.gif