Adams' Method

Adams' method is a numerical Method for solving linear First-Order Ordinary Differential Equations of the form

$${dy\over dx}=f(x,y).$$ (1)

Let

$$h=x_{n+1}-x_n$$ (2)

be the step interval, and consider the Maclaurin Series of $y$ about $x_n$,

$$y_{n+1}=y_n+\left({dy\over dx}\right)_n(x-x_n)+{1\over 2}\left({d^2y\over dx^2}\right)_n(x-x_n)^2+\ldots$$ (3)

$$\left({dy\over dx}\right)_{n+1} = \left({dy\over dx}\right)_n+\left({d^2y\over dx^2}\right)_n(x-x_n)^2+\ldots.$$ (4)

Here, the Derivatives of $y$ are given by the Backward Differences

$$q_n \equiv \left({dy\over dx}\right)_n={\Delta y_n\over x_{n+1}-x_n} = {y_{n+1}-y_n\over h}$$ (5)

$$\nabla q_n \equiv \left({d^2y\over dx^2}\right)_n = q_n-q_{n-1}$$ (6)

$$\nabla^2 q_n \equiv \left({d^3y\over dx^3}\right)_n = \nabla q_n-\nabla q_{n-1},$$ (7)

etc. Note that by (1), $q_n$ is just the value of $f(x_n,y_n)$.

For first-order interpolation, the method proceeds by iterating the expression

$$y_{n+1}=y_n+q_n h$$ (8)

where $q_n\equiv f(x_n,y_n)$. The method can then be extended to arbitrary order using the finite difference integration formula from Beyer (1987)

$$\int_0^1 f_p\,dp=(1+{\textstyle{1\over 2}}\nabla+{\textstyle... ...88}}\nabla^5+{\textstyle{19087\over 60480}}\nabla^6+\ldots)f_p$$ (9)

to obtain

$y_{n+1}-y_n=h(q_n+{\textstyle{1\over 2}}\nabla q_{n-1}+{\textstyle{5\over 12}} \nabla^2q_{n-2}+{\textstyle{3\over 8}}\nabla^3 q_{n-3}$
$+{\textstyle{251\over 720}}\nabla^4 q_{n-4}+{\textstyle{95\over 288}}\nabla^5 q_{n-5}+\ldots).\quad$	(10)

Note that von Kármán and Biot (1940) confusingly use the symbol normally used for Forward Differences $\Delta$ to denote Backward Differences $\nabla$.

See also Gill's Method, Milne's Method, Predictor-Corrector Methods, Runge-Kutta Method

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 455, 1987.

Kármán, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems. New York: McGraw-Hill, pp. 14-20, 1940.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 741, 1992.