AdamsBashforthMoulton

Adams-Bashforth-Moulton methods are a family of predictor-corrector multistep methods used to numerically solve ordinary differential equations of the form y' = f(x, y) with a given initial value y(x0) = y0. They combine an explicit Adams-Bashforth predictor with an implicit Adams-Moulton corrector to estimate the solution at the next grid point x_{n+1}.

In a typical AB-M pair, the Adams-Bashforth predictor uses only past values of f to predict y_{n+1} before the actual evaluation at x_{n+1}. For example, the four-step Adams-Bashforth predictor (AB4) is y_{n+1}^p = y_n + h/24 [55 f_n - 59 f_{n-1} + 37 f_{n-2} - 9 f_{n-3}], where h is the step size and f_k = f(x_k, y_k). The corresponding Adams-Moulton corrector (AM5) then refines this estimate using the predicted f_{n+1}: y_{n+1} = y_n + h/720 [251 f_{n+1} + 646 f_n - 264 f_{n-1} + 106 f_{n-2} - 19 f_{n-3}]. Here f_{n+1} is evaluated at the predicted y_{n+1}^p, and the result can be used directly or subjected to further correction iterations.

These methods require starting values obtained by another method, such as a Runge-Kutta scheme, to generate the first few steps. They are popular for non-stiff problems due to their high order and moderate cost, though their stability depends on the step size and problem structure; high-order AB-M methods are typically not A-stable for stiff equations.

The AB-M family includes several predictor-corrector pairs, commonly AB4 with AM5, and higher-order variants like ABm/AMm+1 for m ≥ 4. The methods are named after their developers: Adams, Bashforth, and Moulton.