Mathematics for electricians - Numerical methods

Numerical methods useful to the engineer are numerous. However, the basic algorithms are already described in the treatise Fundamental Sciences. We have therefore chosen to present here the methods for discretizing the equations of physics, concentrating on the conduction model.

Mathematics makes common use of the notions of infinity and continuity. The exact solution to a differential or partial differential equation problem is a continuous function. Computers know only the finite and the discrete. Approximate solutions will ultimately be calculated as collections of discrete values in the form of components of a solution vector of a matrix problem.

For the transition from an exact (continuous) problem to an approximate (discrete) one, several competing techniques are available: finite differences, finite elements and finite volumes. Each of these three methods corresponds to a different formulation of the equations of physics:

• balance of forces at each point for finite differences ;

• energy minimization or virtual work principle for finite elements ;

• conservation law and flow calculation for the finite volume method.