Adams-Bashforth-Moulton Method
Source inspiration: (Mathew 2000-2019).
Description
The Adams-Bashforth-Moulton (ABM) method is a predictor-corrector multistep scheme for initial value problems \(y' = f(t,y)\). In the 4-step form, an explicit Adams-Bashforth predictor estimates \(y_{n+1}\) from recent history, and an implicit Adams-Moulton corrector immediately refines it.
Because ABM is not self-starting, initial points are generated with a one-step method (such as RK4). After startup, each new value uses prior function evaluations, giving high accuracy with low per-step cost.
Animations
The animation below reconstructs legacy Example 11 from the Mathews module: \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\), over \(0 \le t \le 5\), with the historical \(n=50\)-step setup (51 mesh points).
The dashed black curve is a high-accuracy reference trajectory computed with fixed-step RK4 at \(h=10^{-4}\), since no closed-form exact solution is used for this case.
Case 1 - ABM4 predictor-corrector, \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\), \(0 \le t \le 5\)
Behavior: The ABM trajectory tracks the reference curve closely over the full interval while keeping a smooth and stable descent toward the right endpoint.
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: Adams-Bashforth-Moulton Method (ok)
- Animation links found in module Animations paragraph: 4
- Unique animation portals: 1
Animation Portals
- Animation item links found: 1
Animation Items
- Main animated GIF count: 1
- http://localhost:8000/a2001/Animations/OrdinaryDE/Adams1/Adamsaa.gif