Finite-difference method for stationary PDEs

Observation of a phenomenon always leads the scientist to modeling, which in turn is accompanied by equationization of the problem studied; very often, the models obtained are constituted by differential equations or partial differential equations (PDE); unfortunately, the analytical methods for solving this type of mathematical problem only apply to a very limited class of equations. With the help of simplifying hypotheses, more or less justified according to the value of the parameters involved in the model, the scientist reduces himself to a type of problem that he knows how to solve, in a formal way; thus he uses highly simplified models to represent the observed phenomena, which are often complex.

Most of the time, the solutions of simplified equations only represent the phenomenon in the domain where the simplifying assumptions make sense; on the other hand, when the parameter values don't fit into this framework, the solution obtained doesn't always bear much relation to observation. To gain a more detailed understanding of the phenomenon under study, we need to include in the equations the terms that make it impossible to solve the problem analytically. This leads to an impasse, and we need to find a compromise that allows us to represent the observations as accurately as possible, while at the same time solving the equations describing the operating regime of the phenomenon.

However, before solving the problem of partial differential equations, an analytical study of the equations involved in the model must be carried out; at this stage, the scientist must ask questions about the existence and uniqueness of the solution(s), the sensitivity of the solution(s) to perturbations, the growth or decay of solutions as a function of time, the existence of bifurcation points, etc., This leads to the resolution of extremely complex mathematical problems, which nevertheless serve to validate the mathematical models developed by the scientist.

In addition, recent advances in automatic calculation have made it possible to implement calculation methods that were previously inconceivable. These numerical methods have made it possible to :

• the ability to perform a large number of calculations in a very short time;

• taking non-linearity into account in all kinds of equations, and solving them.

However, it's important to remember that the numerical resolution of a partial differential equation leads to an approximate solution. Indeed, as we shall see later, the principle of numerical methods is based on approximating the initial continuous problem by a discrete system of high-dimensional linear or non-linear (depending on the nature...