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Adams' Method

Adams' method is a numerical method for solving first-order ordinary differential equations of the form

(dy)/(dx)=f(x,y).
(1)

Let

h=x_(n+1)-x_n
(2)

be the constant step interval, and consider the Taylor series of an analytic solution y about x_n for sufficiently small h,

y_(n+1)=y_n+((dy)/(dx))_nh+1/2((d^2y)/(dx^2))_nh^2+...
(3)
((dy)/(dx))_(n+1)=((dy)/(dx))_n+((d^2y)/(dx^2))_nh+....
(4)

Define the function values and their backward differences by

q_n=f(x_n,y_n)
(5)
del q_n=q_n-q_(n-1)
(6)
del ^2q_n=del q_n-del q_(n-1),
(7)

and similarly for higher backward differences.

Using constant interpolation gives the first-order method

y_(n+1)=y_n+q_nh.
(8)

For an s-step method, let q(x) be the polynomial of degree at most s-1 interpolating q_n, q_(n-1), ..., q_(n-s+1) at x_n, x_(n-1), ..., x_(n-s+1). The finite difference integration formula (Beyer 1987, p. 455) is

int_0^1q(x_n+ph)dp=(1+1/2del +5/(12)del ^2+3/8del ^3+(251)/(720)del ^4+(95)/(288)del ^5+(19087)/(60480)del ^6+...)q_n
(9)

where terms through del ^(s-1) are retained. Integrating this polynomial over the step gives the update

y_(n+1)-y_n=h(q_n+1/2del q_n+5/(12)del ^2q_n+3/8del ^3q_n +(251)/(720)del ^4q_n+(95)/(288)del ^5q_n+...).
(10)

Note that von Kármán and Biot (1940) confusingly use the symbol normally used for forward differences delta to denote backward differences del .

See also

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

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 896, 1972.Bashforth, F. and Adams, J. C. Theories of Capillary Action. London, England: Cambridge University Press, 1883.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 455, 1987.Jeffreys, H. and Jeffreys, B. S. "The Adams-Bashforth Method." §9.11 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 292-293, 1988.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.Whittaker, E. T. and Robinson, G. "The Numerical Solution of Differential Equations." Ch. 14 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 363-367, 1967.

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Adams' Method

Cite this as:

Weisstein, Eric W. "Adams' Method." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AdamsMethod.html

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