Predictor-Corrector Methods
Source inspiration: (Mathew 2000-2019).
Description
Predictor-corrector methods combine an explicit predictor step with an implicit or semi-implicit corrector step. The predictor advances using previously known values, then the corrector immediately refines that provisional value with a more accurate slope model.
In this page, we use an AB2-AM2 PECE pair:
\[ y_{n+1}^{(p)} = y_n + \frac{h}{2}\left(3f_n - f_{n-1}\right), \]
\[ y_{n+1} = y_n + \frac{h}{2}\left(f(t_{n+1},y_{n+1}^{(p)}) + f_n\right). \]
This preserves multistep efficiency while reducing the drift that appears in purely explicit methods.
Animations
The animation below uses the same IVP as the Euler and Adams-Bashforth-Moulton pages: \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\), over \(0 \le t \le 5\), with 50 steps. The dashed black curve is a high-accuracy RK4 reference trajectory at \(h=10^{-4}\).
Legacy archive note: a dedicated Predictor-Corrector module asset set was not found in the recovered baseline, so this case is reconstructed from the method equations and the same worked IVP family used in adjacent ODE pages.
Case 1 - AB2-AM2 predictor-corrector, \(y' = 1 - t\sqrt[3]{y}\), \(y(0)=1\), \(0 \le t \le 5\)
Behavior: The PECE update tracks the reference curve more closely than forward Euler while remaining computationally light through one predictor and one corrector evaluation per step.
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.