# Computational Methods in Ordinary Differential Equations by J. D. Lambert

By J. D. Lambert

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Oo (~/n = 0 if and only if 1',1"; 1, ' it is clear that the method will not be convergent if any of the roots (, has modulus greater than one. Also, consider the case when e, is a reaCroot of pm with mUltiplicity m > 1. 3n(n - 1) + ... mn(n - 1)". (n - m + 2)](~. Since for q :;;" 1, lim hnq(: = x lim nq-Ie~ = 0 if and only if h ..... O nh=x n-oo 1',1 < 1, it is clear that the method will not be convergent if p(() has a multiple root of modulus greater than or equal to one. The argument extends to the case when the roots of p(() are complex, and motivates our next definition.

J h2A2 = [exp (nhA)] { exp (hA) - I - hA - ""2 - thA[exp(hA) - I - hAJ} = [exp(nhA)][(1 - 1hA) exp (hA) - (I Also Y•• I - Y. = 1hA(y•• I + yJ + 1hA)]. , / 30 Computational methods in ordinary differential equations + thA) exp (nhA), or (1 - 4hA)y = (1. • "+ 1 Y. + 1] = by the localizing assumption (l - thA) exp [In = [exp(nhA)][(! '. = + I)hA]- that + thA) exp (nhA) thA)exp(hA) - (1 + 1M)] (1 -2'[y{x,); h], verifying (24). 8 Consistency and zero-stability Definition The linear multistep method (2) is said to be consistent order p ;;.

1). l is not critical. l -:- 0 since this causes two coefficients, namely ct: 1 and 11:3, to vaDlsh. Another sunphfylOg choice is Jl = 1\. which causes P2 to-vanish; the resulting method turns out to be Quade's method, defined in exercise 5 (page 27). Exercise 11. Find the range of a. (Y,+2 - y,+,) - Y. +,) is zero-stable. Show that there exists a value of a. t if the method is to be zero~stable, its order cannot exceed 2. 10 Specification oflinear multistep methods In the days of desk computation, it was common practice to write the right-hand side of a linear multistep method in terms of a power series in a difference operator.