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Guy CHAVENT: Professor of Mathematics at ParisDauphine University  Scientific Director at the French National Institute for Research in Computer Science and Control (INRIARocquencourt)
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 finitedifference 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 ( , i = ...– 2, – 1,0,1,2... ; n = 0,1,2...) whose component represented an approximation of u (x, t) at the point (x _{...}
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Approximation of partial differential equations
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