The goal of this course is to show some fundamental
elements of numerical approximation of partial difference
equations. It will be restricted to linear problems, like diffusion
and convection, since these phenomenons can be observed in most of
mechanical problems. Most of the course will be devoted to the method
of finite differences for one dimensional problems. Main notions like
accuracy, stability, convergence, numerical diffusion and dispersion,
will analyzed. The last part of the course will be an introduction to
the finite volume methods, which is well suited to multidimensional
problems, in particular for conservation laws of fluid mechanics.
The course is based on two books:
Finite Difference Methods for Ordinary and Partial Differential
Equations, Steady State and Time Dependent Problems, Randall
J. LeVeque, SIAM, 2007
Finite Volume Methods for Hyperbolic Problems, Randall J. LeVeque, Cambridge University Press, 2002.
The numerical methods presented in the course will be studied with
exercices and programming sessions.