= + +

Stability analysis of multipoint methods is a little more complicated than stability analysis of single-point methods. To illustrate the procedure, let's perform a stability analysis of the fourth-order Adams-Bashforth FDE, Eq. (7.229):

h Yn+l = Yn + 24 (55fn - 59fn_l + 37fn-2 - 9fn-3) (7.237)

Example 7.13. Stability analysis of the fourth-order Adams-Bashforth FDE

The amplification factor G is determined by applying the FDE to solve the model ODE, ji' + aji = 0, for whichf(t,ji) = -aji. Thus, Eq. (7.237) yields

h Yn+l = Yn + 24 [55(-aYn) - 59(-aYn-l) + 37(-aYn-2) - 9(-aYn-3)] (7.238)

For a multipoint method applied to a linear ODE with constant I'1t, the amplification factor G is the same for all time steps. Thus,

G =Yn+l =~ =Yn-l =Yn-2 (7.239) Yn Yn-l Yn-2 Yn-3

Solving Eq. (7.239) for Yn-" Yn-2' and Yn-3 gives Yn Yn Yn

Yn-l = G Yn-2 = G2 Yn-3 = G3 Substituting these values into Eq. (7.238) gives

(ah) (55 59Yn 37 Yn 9 Yn)Yn+l = Yn - 24 Yn - G + G2 - G3 Solving for G =Yn+l/Yn gives

G = I _ (ah) (55 _ 59 + 37 _~) 24 G G2 G3

Multiplying Eq. (7.242) by G3 and rearranging yields G4 (55(ah) _ I)G3 _ 59(ah) G2 37(ah) G _ 9(ah) = 0 (7.243)

+ 24 24 + 24 24

For each value of (ah), there are four values of G, G; (i = I, ... ,4). For stability, all four values of G; must satisfy IG;I s I. Solving Eq. (7.243) for the four roots by Newton's method gives the following results:

These results show that IGI .:::: 1 for (ex ,1t) .:::: 0.3.