# Milne method

A finite-difference method for the solution of the Cauchy problem for systems of first-order ordinary differential equations:

$$y ^ \prime = f ( x , y ) ,\ \ y ( a ) = b .$$

The method uses the finite-difference formula

$$y _ {i} - y _ {i-} 2 = \ 2 h f ( x _ {i-} 1 , y _ {i-} 1 ) ,$$

where

$$x _ {i} = a + i h ,\ \ i = 0 , 1 ,\dots .$$

In computations using this formula it is necessary, by some other means, to find an additional initial value $y _ {1} \approx y ( x _ {1} )$. The Milne method has second-order accuracy and is Dahlquist stable, that is, all solutions of the homogeneous difference equation $y _ {i} - y _ {i-} 2 = 0$ are bounded uniformly with respect to $h$ for $i = 0 \dots [ ( A - a ) / h ]$, for any fixed interval $[ a , A ]$. For Dahlquist stability it is sufficient that the simple roots of the characteristic polynomial of the left-hand side of the difference equation do not exceed one in modulus, and that the multiple roots are strictly less than one in modulus. In the case given, the characteristic polynomial $\rho ( \lambda ) = \lambda ^ {2} - 1$ has roots $\lambda = \pm 1$ and, consequently, the stability condition holds. However, in solving a system of equations $y ^ \prime = A y$ with a matrix $A$ having negative eigen values, a rapid growth in the calculation error occurs.

The predictor-corrector Milne method uses a pair of finite-difference formulas:

a predictor

$$\overline{y}\; _ {i} = y _ {i-} 4 + \frac{4h}{3} ( 2 f _ {i-} 3 + f _ {i-} 2 + 2 f _ {i-} 1 ) ,\ \ i = 4 , 5 \dots$$

and a corrector

$$\overline{ {\overline{y}\; }}\; _ {i} = \ y _ {i-} 2 + \frac{h}{3} ( f _ {i} + 4 f _ {i-} 1 + f _ {i-} 2 ) ,\ \ i = 4 , 5 \dots$$

where

$$f _ {i} = f ( x _ {i} , y _ {i} ) ,\ \ \overline{f}\; _ {i} = f ( x _ {i} , \overline{y}\; _ {i} ) .$$

An approximate expression for the error is given by the quantity

$$\epsilon _ {i} = \ \frac{1}{29} ( \overline{y}\; _ {i} - \overline{ {\overline{y}\; }}\; _ {i} ) .$$

Additional initial values $y _ {i} \approx y ( x _ {j} )$, $j = 1 , 2 , 3$, are calculated by some other means, for example, by the Runge–Kutta method, which has fourth-order accuracy. The method was proposed in .

How to Cite This Entry:
Milne method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Milne_method&oldid=47837
This article was adapted from an original article by V.V. Pospelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article