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-
Guy CHAVENT: Professor of Mathematics at Paris-Dauphine University - Scientific Director at the French National Institute for Research in Computer Science and Control (INRIA-Rocquencourt)
INTRODUCTION
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 x
1
... 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
and homogeneous from a known initial temperature (u
0
(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 (
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