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A finite-difference method for solving Cauchy's problem for systems of first-order differential equations

In integration over a grid with a constant step the computational formulas may be based a) on extrapolation

or b) on interpolation

For a given , the latter formula gives more accurate results but a non-linear system of equations must be solved in order to find .

In practical work, an approximation by the former method is found and one or two corrections are performed according to the formula

which converge pointwise if . The initial conditions for Adams' method, which are needed to begin the calculations by formula a), must be found by some special method. The error in the solution can be written as

where is the solution of the system

when .

The structure of the term is such that, for small values of , it is usually uniformly small as compared to the main term on large intervals of integration. This means that Adams' method can be applied over a large integration interval in the case of an absolutely-stable solution of the differential problem. In particular, as distinct from the Milne method, it may be used for finding stable periodic solutions of differential equations. The standard Adams procedure for integration with automatic step selection is much more involved than the standard Runge–Kutta method, since the step-changing algorithm is much more involved and the selection of the initial values of is not standardized.

In the case of the equations , , the extrapolation formula a) has the form

Particular solutions of this equation are , where is a root of the equation

If , then one root of this equation is , and the rounding-off errors rapidly increase. When integrating with automatic step selection, this results in an unjustified diminution of the step size. However, Adams' method proves to be more economical than the Runge–Kutta method in most cases; it was first introduced by J.C. Adams in 1855.

#### References

 [1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , 2 , Pergamon (1973) (Translated from Russian) [2] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) [3] A.N. Tikhonov, A.D. Gorbunov, "Asymptotic expansions of the error in the difference method of solving Cauchy's problem for systems of differential equations" USSR Comput. Math. Math. Phys. , 2 : 4 (1962) pp. 565–586 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 4 (1962) pp. 537–548 [4] S.M. Lozinskii, "An estimate of the error of numerical integration of differential equations I" Izv. Vyssh. Uchebn. Zaved. Mat. , 5 : 6 (1958) pp. 52–90 (In Russian) [5] N.S. Bakhvalov, "On an estimate of the error at numerical integration of differential equations by Adams' extrapolation method" Dokl. Akad. Nauk SSSR , 104 : 5 (1955) pp. 683–686 (In Russian)