Next: The explicit heat-flow equation Up: FINITE DIFFERENCING Previous: First derivatives, explicit method

## First derivatives, implicit method

Let us solve the equation
 (24)
by numerical methods. The most obvious (but not the only) approach is the basic definition of elementary calculus. For the time derivative, this is
 (25)
Using this in equation (24) yields the the inflation-of-money equations (22) and (23), where .Thus in the inflation-of-money equation the expression of dq/dt is centered at ,whereas the expression of q by itself is at time t. There is no reason the q on the right side of equation (24) cannot be averaged at time t with time ,thus centering the whole equation at .When writing difference equations, it is customary to write more simply as qt+1. (Formally one should say and write qn+1 instead of qt+1, but helpful mnemonic information is carried by using t as the subscript instead of some integer like n.) Thus, a centered approximation of (24) is
 (26)
Letting , this becomes
 (27)
which is representable as the difference star

For a fixed this star gives a more accurate solution to the differential equation (24) than does the star for the inflation of money. The reasons for the names explicit method'' and implicit method'' above will become clear only after we study a more complicated equation such as the heat-flow equation.

Next: The explicit heat-flow equation Up: FINITE DIFFERENCING Previous: First derivatives, explicit method
Stanford Exploration Project
10/31/1997