For example,

A model of personal savings that assumes a fixed yearly growth rate, r, in savings (S) implies that time rate of change in saving d(S)/dt is given by,

d(S)/dt= r (S) ** eqn. 1 **

(this example is also used to discuss analytical models so that numerical and analytical models can be compared and contrasted more easily).

An example of a numerical solution to this fundamental differential equation is given shown in Table 1 along with the corresponding values from the analytical solution, S=SoEXP(rt).

The numerical values in the Table 1 are generated by using the difference equation,

S(t+dt) = S(t) + d(S) = S(t)+ r S(t) dt = S(t) [1+ r dt] ** eqn. 2**

and assuming r=0.10 (1/yr) and a time step, dt, of one month (0.083 yr) for illustration. Since the change in savings, d(S), is rather small each time step this numerical solution agrees fairly well with the analytical solution.

Table 2 compares numerical and analytical results for r=2.0 (1/yr) and dt=0.083 yr). After 1 year there is a significant discrepancy between the numerical solution and the analytical (exact) solution. A smaller time step would be required to get better agreement between the numerical solution and the analytical solution. Using a time step of 0.01 yr gave a savings value of $724.46 with 100 numerical calculations compared with the exact result of $738.91.

This highlights a drawback of numerical solutions to model equations; to get good results many iterative calculations may be required. With fast computer speeds available today this is not really an issue for most model equations that one would explore in an introductory geoscience course. In addition, the precision can be greatly improved for a given time step by using a numerical procedure which is more sophisticated than the rather simple Euler's method described by eqn 2.

Numerical solutions have several advantages over analytical solutions. First, the equations are much more intuitive. Students can clearly understand the meaning of eqn 2 and can generate Table 1 by hand or by using Excel. The exponential form of the analytical solution is clear to those with strong mathematics skills but not so clear to others. Second, the basic procedure S(t+dt) = S(t) + d(S) is the same regardless of how complicated the formulas are which describe d(S). This is not true of analytical solutions as it is relatively easy to get into mathematics which is much too complicated to obtain analytical solutions. Thus more realistic models of greater complexity can be investigated using numerical techniques.

For introductory geoscience courses regardless of whether one uses analytical solutions or numerical solutions to model equations students should still use graphical output, animations, and tabular data to interpret, understand, and explain model behavior. There are three primary venues for introductory geoscience models Excel, Stella II, and JAVA type interactive web based activities. Excel is great for graphing and exploring analytical models and can be used quite successfully for numerical solutions (this may involve copying your basic formulas down to hundreds of rows). Stella only uses numerical procedures and it has some options for which numerical procedure to use. JAVA type interactive Web activities use graphical output, animations, and tabular data to display the results of solving the model equations and hence the question of whether the programmer used an analytical or numerical solution becomes academic.

**Other Information**
This site about Euler's Method (more info) uses an exponential growth model similar to ours. It has an interactive JAVA type graph that lets one explore the effects of time step variations on the solution.