Modeling and numerical solution of the Laplace equation in 2D by the finite difference method case of the heat equation - Study of stability

The objective of this paper is to show the procedure to follow to analyze a physical phenomenon, with a simple and effective method. Mathematical stability examination was performed to define which resolution is stable and physically feasible. We have seen the approach and the steps that must be taken to go from a mathematical model to a numerical model. To do this, we used the Poisson equation as an example, and to show the results, we used the heat transfer equation. In real life, the most difficult step is not the numerical processing and analysis, but to understand the physical phenomenon and to translate it into a mathematical formulation. The finite difference approach is employed to resolving those equations numerically. The resulting schemes were resolved by the iterative GAUSS Seidel technique. These differential schemes have an approximate order 0(h 2 ) , and are absolute stable. Differential schemes are a linear algebraic equation system which solution can be solved by the Gaussian technique of elimination. This paper provides a good basis for the analysis of physical phenomena that deal with partial differential equations in 2D. In several sectors and branches, whether in the industrial framework.