Numerical Analysis 1

Description : Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or 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 solution 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 exercises will illustrate the scientific computing tools seen in the course with simple examples.

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

Bachelor's degree in Mathematics in its entirety, with an emphasis on differential calculus and integration

Recommended prerequisites: It is recommended to have taken the numerical analysis modules of the Bachelor's degree covering the following topics: interpolation of functions, 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 unsteady problems, study of accuracy and stability.

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

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

Hourly volumes:

            CM : 21h

            TD :15h

            TP : 6h

            Land : 0