Solving Parabolic Partial Differential Equations The Explicit Method

Using equation 12-21 as an example and writing it in the form

82 F dF

dx2 dy

we can replace derivatives by finite differences, using the central difference formula for <^FI3c2

and the forward difference formula for dFldy dF

ij+1

dy Ay

When these are substituted into equation 12-22, we obtain equation 12-25, where r = Ay/(k(Ax)2). (Using forward and central differences simplifies the expression.)

Or, when i represents distance x and j represents time t,

Equation 12-25a permits us to calculate the value of the function at time t+\ based on values at time t. This is illustrated graphically by the stencil of the method.

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.

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 Ax and Ay such that r = 0.5 (e.g., for a given value of Ax, Ay = k{Ax)2/!), so that equation 12-25 is simplified to fmj +Fi-\J

+1 0

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