Numerical Analysis 1

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

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

Introduce numerical schemes and numerical analysis tools necessary for solving partial differential equations.

Bachelor's degree in Mathematics as a whole, with an emphasis on differential calculus and integration

Recommended prerequisites: It is recommended that students have completed the undergraduate numerical analysis modules covering the following topics: function interpolation, quadrature of integrals, and numerical methods for ODEs. Programming experience is also desirable.

An indicative course schedule is as follows:

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

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

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

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

Hourly volumes:

            CM: 9 p.m.

            Tutorial: 3:00 p.m.

            Practical work: 6 hours

            Land: 0