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 [3].

References

[1]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[2]	B.P. Demidovich, I.A. Maron, E.Z. Shuvalova, "Numerische Methoden der Analysis" , Deutsch. Verlag Wissenschaft. (1968) (Translated from Russian)
[3]	W.E. Milne, "Numerical solution of differential equations" , Dover, reprint (1970)

Comments

In the Western literature, the method here called "Milne method" is called the (explicit) midpoint rule. Instead, the corrector appearing in the "predictor-corrector Milne method" is called the Milne method or a Milne device. This method is direct generalization of the Simpson quadrature rule to differential equations. The terminology "Dahlquist stability" is nowadays seldom used in the English literature. More frequently, one uses "zero-stabilityzero-stability" , "root-stabilityroot-stability" , or just "stability" .

References

[a1]	P. Henrici, "Discrete variable methods in ordinary differential equations" , Wiley (1962)