Figure 3 shows the results of the finite-difference method simulations. Three experiments were done with different values of Dt. A smooth line was fitted to the points to emphasize any trends in the data.  The exact solution is plotted as well for comparison. The solution which best matches the exact analytical solution is the experiment with the smallest Dt. The "worst" solution corresponds to the experiment with the largest Dt.

Figures 4 and 5 show the results of a Stella 5.0 simulation of the model system. As can be observed, the rate of change of the temperature is changing in a matter similar to the actual temperature of the cup.  A value of 1 minute was used for Dt.

Experiment 3 is the least computationally-expensive (only 5 points are calculated) but the least accurate compared to the exact solution. Experiment 1 gives a very good match to the exact solution, but 75 points are needed. All runs initially overestimate the actual rate of cooling (underestimating the actual temperature) but approach the real solution as time elapses. The degree of error is inversely proportional to the size of Dt.

In computer modeling, the modeler must be aware of the shortcomings of the numerical methods used in the model.  In this case, a computationally less-expensive method could be used if an accurate temperature wasn't needed until 40 minutes or a -10 ^oC (maximum) inaccuracy was acceptable.

This system was also notable because it was somewhat self-correcting so that no matter how poor a choice was made in regards to Dt, the numerical solution eventually approaches the exact solution. The self-correction was due to the relationship between rate of cooling and the coffee temperature. If the estimated temperature was low compared to the real solution, the rate of cooling is decreased (in the extreme case becoming negative). If the estimated temperature was high compared to the real solution, the cooling rate is apparently accelerated.  Both of these effects result in the simulated solution always approaching the exact solution.

Experimental Procedures

Feedback Mechanisms and Atmospheric Models