Introduction to the Modeling and Analysis of Complex Systems

13.5. SIMULATION OF CONTINUOUS FIELD MODELS 2550002020204040406060608080800 20 40 60 800 20 40 60 800 20 40 60 80Figure 13.14: Visual output of Code 13.6. Time flows from left to right.result shown in Fig. 13.15. This may look cool, but it is actually an invalid result. Youcan tell it is invalid if you understand the nature of the transport equation; it should onlytransport the surface and not change its shape. But even if you are unsure if the resultyou obtained is valid or not, you can try increasing the temporal/spatial resolutions (i.e.,making ∆t and ∆h smaller) to see if the result varies. If the result remains unaffected bythis, then you know that the original spatial/temporal resolutions were already sufficientand that the original result was thus valid. But if not, the original result probably containednumerical errors, so you need to keep increasing the resolutions until the result becomesconsistent.1.01.01.00.80.80.80.60.60.60.40.40.40.20.20.20.00.20.40.60.80.01.00.80.60.40.21.0 0.00.00.20.40.60.80.01.00.80.60.40.21.0 0.00.00.20.40.60.80.01.00.80.60.40.21.0 0.0Figure 13.15: Accumulation of numerical errors caused by increasing ∆t (Dt) to 0.1 inCode 13.5. Time flows from left to right.Before moving on to the next topic, I would like to point out that the discretization/simulationmethods discussed above can be extended to PDEs that involve multiple statevariables. Here is such an example: interactions between two spatially distributed popu-