Numerical Analysis 1

Description : Partial differential equations (PDEs) are nowadays an essential mathematical tool for studying and understanding physical and biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, then to their resolution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming practical exercises will illustrate the scientific computing tools seen in class with simple examples.

Introduce the numerical schemes and numerical analysis tools needed to solve partial differential equations.

Comprehensive mathematics degree, with emphasis on differential calculus and integration

Recommended prerequisites: It is recommended to have taken the Bachelor's degree numerical analysis modules covering the following topics: function interpolation, quadrature integrals, and numerical methods for ODEs. Experience in programming is also desirable.

An indicative course schedule is as follows:

1) Introduction to PDEs: definition of PDEs, classification of PDEs (hyperbolic, elliptical, parabolic).

2) Finite difference (FD) methods: approximation of differential operators using FD methods, solution of stationary and then unsteady problems, accuracy and stability studies.

3) Analytical resolution of scalar conservation laws (LCS): method of characteristics, weak solutions, entropy inequality, Riemann problems.

4) Finite volume (FV) methods: FV methods applied to the LCS, Godunov scheme, numerical flows, TVD schemes.

Hourly volumes :

            CM: 21h

            TD :15h

            Practical work:6h

            Land: 0