Modelling and applied mathematics
Numerical simulation involves using a computer to reproduce the functioning of a system, once it has been described by a set of models. It relies on specific mathematical and computational methods.
At any point in the object being studied, several physical values (velocity, temperature, etc.) describe the state and evolution of the system. These are not independent. Rather they are linked and governed by equations which generally include partial derivatives. These equations represent the mathematical translation of the physical laws used to model the object's behaviour. Simulating the state of this object involves determining the numerical values of its parameters, ideally at all points. Since there are an infinite number of points, and hence an infinite number of values to calculate, this goal can only be achieved in special cases. A natural approximation is to only consider a finite number of points. There will then be only a finite number of values to calculate for the parameters and the required operations become accessible using a computer. The effective number of points considered, depends on the power of this computer: the greater the power, the better will be the final description of the object.
Underlying the calculation of these parameters and underlying numerical simulation therefore is the concept of reducing the infinite to the finite. This is known as discretisation.
