Two-Step and Four-Step Adams Predictor-Corrector Method

Consider the initial value problem , with . This Demonstration uses the two-step and four-step Adams predictor-corrector method to find the estimated solution of this first-order ordinary differential equation. In addition, the relative error is calculated for selected values of , where (i.e., we compare Adams method's solution with the result obtained using NDSolve, ). Finally, the Euclidean norm of the absolute error vector is given (i.e., ).

The predictor-corrector method is a two-step technique. First, the prediction step calculates a rough approximation of the desired quantity, typically using an explicit method. Second, the corrector step refines the initial approximation in another way, typically with an implicit method.

The two-step Adams predictor-corrector method:

(predictor step: two-step Adams–Bashforth)

(corrector step: two-step Adams–Moulton)

The four-step Adams predictor-corrector method uses the four-step Adams–Bashforth and Adams-Moulton methods together:

(predictor step) (corrector step)

The two-step and four-step Adams methods require two and four initial values to start the calculation, respectively. These later can be obtained by using other methods, for example Euler or Runge–Kutta.