GuideODEsadvanced

Predictor-corrector methods

Combining an explicit method (predictor) with an implicit one of the same order (corrector) to get the accuracy and stability of the implicit without solving equations: ABM2 and ABM4.

Predict and correct

Instead of solving the implicit equation of Adams-Moulton, one predicts yk+1y_{k+1} with an explicit method and uses that prediction inside the implicit corrector, which is thus evaluated only once. With AB2 as predictor and AM2 as corrector one obtains ABM2:

Predictor (AB2):yk+1(p)=yk+h2(3f(tk,yk)−f(tk−1,yk−1))Corrector (AM2):yk+1=yk+h2(f(tk+1,yk+1(p))+f(tk,yk))\begin{aligned} \text{Predictor (AB2):}\quad & y_{k+1}^{(p)}=y_k+\tfrac{h}{2}\bigl(3f(t_k,y_k)-f(t_{k-1},y_{k-1})\bigr)\\ \text{Corrector (AM2):}\quad & y_{k+1}=y_k+\tfrac{h}{2}\bigl(f(t_{k+1},y_{k+1}^{(p)})+f(t_k,y_k)\bigr) \end{aligned}