Predictor–corrector method

In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:

When considering the numerical solution of ordinary differential equations (ODEs), a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step.

A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method).

Consider the differential equation

and denote the step size by ${\displaystyle h}$.

First, the predictor step: starting from the current value ${\displaystyle y_{i}}$, calculate an initial guess value ${\displaystyle {\tilde {y}}_{i+1}}$ via the Euler method,

...
Wikipedia