• • •

Consider element (i) in Figure 12.19c. Node i in element (i) corresponds to node i in the general element illustrated in Figure 12.19a. Thus, the element equation corresponding to node i in element (i) is Eq. (12.205). Thus,

+ (12.209)

Multiplying Eq. (12.208) by 6/fu:_ and Eq. (12.209) by 6/fu:+ and adding yields the nodal equation for node i:

( +4( + ( + 6rx(fi - fi-I) _ 6rx(fi+l - fi) _ Q-U-I)( ( +2() Ji-I 'Ji Ji+1 fu:2 ~ Ji-I:li

- +

- QU)(2fi +fi+l) + 3(ji"U-I) + ji"U) = 0 (12.210)

Next, let's develop a finite difference approximation for j. Several possibilities exist. For example,

·11 fll+1 - f" f = I1t

f ll+1 f"/"+1/2 = - M

(12.211)

The first expression is a first-order forward-time approximation, the second expression is a first-order backward-time approximation, and the third expression is a second-order centered-time approximation. When using any of these finite difference approximations, the function values in Eq. (12.210) must be evaluated at the corresponding time level.