A numerical scheme has linear\convex combinations to weight the slopes at different time steps. How are these weights determined? It is through some mathematical techniques or through trial & error?
This example has the description and algorithm for the Adams-Bashford-Moulton predictor-corrector method. In the Algorithm box, how the weights $55, 59, 37, ...$ are determined?
Consider a simple example of second order Adams-Bashford scheme for $y' = f\left( {x,y} \right)$,
\begin{equation} \label{wt1} {y_{n + 1}} = {y_n} + h\left( {\frac{3}{2}{f_n} - \frac{1}{2}{f_{n - 1}}} \right) \end{equation}
where $n$ is the grid point, $h$ is the step size and ${f_n} \equiv f\left( {{x_n},{y_n}} \right)$. The co-efficients $\frac{3}{2}$ and $\frac{1}{2}$ are called the weights of a numerical scheme.
To derive an explicit scheme of any order, lets assume that $y_{n + 1}$ can be obtained from adding the integral of a polynomial approximation to $y_n$, i.e. ${y_{n + 1}} = {y_n} + \int_{{x_n}}^{{x_{n + 1}}} {P\left( x \right)dx} $ where $P\left( x \right)$ is some polynomial.
\begin{equation} \label{wt2} {y_{n + 1}} = {y_n} + \int_{{x_n}}^{{x_{n + 1}}} {\sum\limits_{i = n}^{n - k} {{L_i}\left( x \right){f_i}} \text{ } dx} \end{equation}
where \begin{equation} \label{wt3} {L_i}\left( x \right) = \mathop {\mathop \prod \limits_{j = n}^{n - k} }\limits_{j \ne i} \frac{{x - {x_j}}}{{{x_i} - {x_j}}} \end{equation}
here $k + 1$ gives the order of accuracy and ${L_i}\left( x \right)$ are called cardinal functions. The term under the integral is the Lagrangian polynomial, $P\left( x \right)$ of order $k$.
For a multi-step explicit method of order 2 with the information at $x_n$ and $x_{n - 1}$, $y_{n+1}$ becomes,
\begin{equation} \label{wt4} {y_{n + 1}} = {y_n} + \int_{{x_n}}^{{x_{n + 1}}} {\left[ {{L_n}\left( x \right){f_n} + {L_{n - 1}}\left( x \right){f_{n - 1}}} \right]} \text{ } dx \end{equation}
By expanding the cardinal functions, we get,
\begin{equation} \label{wt5} {y_{n + 1}} = {y_n} + {f_n}\int_{{x_n}}^{{x_{n + 1}}} {\frac{{x - {x_n}}}{h} \text{ } dx} - {f_{n - 1}}\int_{{x_n}}^{{x_{n + 1}}} {\frac{{x - {x_{n - 1}}}}{h} \text{ } dx} \end{equation}
where
\begin{equation} \label{wt6} {L_n}\left( x \right) = \frac{{x - {x_n}}}{h} \end{equation}
\begin{equation} \label{wt7} {L_{n - 1}}\left( x \right) = \frac{{x - {x_{n - 1}}}}{{ - h}} \end{equation}
Upon integrating, we get,
\begin{equation} \label{wt8} {y_{n + 1}} = {y_n} + h\left( {\frac{3}{2}{f_n} - \frac{1}{2}{f_{n - 1}}} \right) \end{equation}
The higher schemes require more algebra but calculated in a similar way as shown in the example above. Furthermore, implicit methods are derived by the same formula but includes $f_{n+1}$ information while approximating the polynomial, $P\left( x \right)$.
More information here.
