Page 294 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 12 PARTIAL DIFFERENTIAL EOUATIONS 27 1

(1 2-23)

and the forward difference formula for dFldy

(1 2-24)

When these are substituted into equation 12-22, we obtain equation 12-25,
where r = Ay/(~l(hx)~). (Using forward and central differences simplifies the
expression.)
F,,/+I = f.(F,+l,/ + F,-l,/ ) + 0 - rk,/ (1 2-25)
Or, when i represents distance x andj represents time t,

Fx,t+l = r(Fx+,,, + Fx-l,, 1 + 0 - r)Fx,t (12-25a)
Equation 12-25a permits us to calculate the value of the function at time t+l
based on values at time t. This is illustrated graphically by the stencil of the
method.

-1 0 1
X

Figure 12-4. Stencil of the explicit method for the solution of a parabolic PDE.
The points shown as solid squares represent previously calculated values
of the function; the open square represents the value to be calculated.

An alternative to the use of equation 12-25 is to choose hx and Ay such that r
= 0.5 (e.g., for a given value of Ax, Ay = k(A~)~/2), that equation 12-25 is
so
simplified to

( 12-26)