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First derivatives, implicit method

Let us solve the equation  
 \begin{displaymath}
{ dq \over dt } \eq 2\ r\ q\end{displaymath} (19)
by numerical methods. The most obvious (but not the only) approach is the basic definition of elementary calculus. For the time derivative, this is  
 \begin{displaymath}
{ dq \over dt } \ \ \ \approx \ \ \ 
{ q(t+ \Delta t )\ -\ q ( t ) 
\over \Delta t }\end{displaymath} (20)
Using this in equation (19) yields the the inflation-of-money equations (17) and (18), where $2\,r = .1$.Thus in the inflation-of-money equation the expression of dq/dt is centered at $ t+ \Delta t / 2 $,whereas the expression of q by itself is at time t. There is no reason the q on the right side of equation (19) cannot be averaged at time t with time $t+\Delta t$,thus centering the whole equation at $ t+ \Delta t / 2 $.When writing difference equations, it is customary to write $q(t+\Delta t)$ more simply as qt+1. (Formally one should say $t=n\Delta t$ 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 (19) is  
 \begin{displaymath}
q_{{t+1}}\ -\ q_t \eq 2\,r \,\Delta t \ \ { q_{{t+1}}\ +\ q_t \over 2 }\end{displaymath} (21)
Letting $\alpha = r \Delta t $, this becomes  
 \begin{displaymath}
( 1- \alpha )\ q_{{t+1}}\ \ - \ ( 1+ \alpha )\ q_t \eq 0\end{displaymath} (22)
which is representable as the difference star

\begin{displaymath}
\begin{tabular}
{cc}
 &\begin{tabular}
{\vert c\vert} \hline...
 ...r}
{c}
 t \\  $\downarrow$ \\  \end{tabular} & \\ \end{tabular}\end{displaymath}

For a fixed $ \Delta t $ this star gives a more accurate solution to the differential equation (19) 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.


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Next: Explicit heat-flow equation Up: FINITE DIFFERENCING IN (omega,x)-SPACE Previous: First derivatives, explicit method
Stanford Exploration Project
12/26/2000