Approximation of partial differential equations

In this article, we focus on the discretization of partial differential equations, in which the unknown is a function u (temperature, for example) depending on several space variables x₁ ... x_n (abbreviated to x) and time t. We'll call Ω the domain of space and [0, T] the time interval where we're trying to find out the temperature. Thus, the evolution of the temperature (u (x, t )) in an infinite bar $\left(\Omega =ℝ\right)$ and homogeneous from a known initial temperature (u₀ (x )) is given by :

where c is the heat capacity and a is the thermal conductivity of the bar. The finite-difference method was historically the first known method for calculating an approximate solution of (1) on a computer. The idea was to replace the search for the function u (x, t ) by that of a vector ($u_i^n$, i = ...– 2, – 1,0,1,2... ; n = 0,1,2...) whose...