Numerical simulation and philosophy: Devils in the details and discrete Green´s function solution of a nonstandard finite difference model

We first discuss the philosophy of modeling and numerical simulation. Many researchers believe that the more accurate the model - i.e. the better it fits the data - the better. But this is not necessarily true. Besides accuracy, a good model must have the same essential mathematical characteristics as the phenomenon which it is supposed to describe. When this is not the case, a model can be very accurate but numerically unstable. Next we introduce high precision finite difference models of the wave equation and Maxwell´s equations based on what are called nonstandard (NS) finite differences (FD). Finally we introduce a new methodology to solve electromagnetic scattering and propagation problems based on solving NS-FD models with discrete Green´s functions.