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 devoted to an introduction to EDPs, followed by their solution using well-known numerical methods such as the finite difference and finite volume methods. A section focused more on analysis—necessary for the introduction to the finite volume method—will be devoted to the analytical solution of scalar conservation laws. Four programming lab sessions will use simple examples to illustrate the scientific computing tools covered in class.

Introduce the numerical methods 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, integration by quadrature, and numerical methods for ODE. Programming experience is also desirable.

A sample 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 steady-state and then unsteady-state problems; analysis of accuracy and stability.

3) Analytical solutions to scalar conservation laws (SCLs): the characteristic method, weak solutions, the entropy inequality, and Riemann problems.

4) Finite Volume (FV) Methods: FV methods applied to LCS, the Godunov scheme, numerical fluxes, and TVD schemes.

Hourly volumes:

            CM: 9:00 p.m.

            TD: 3:00 p.m.

            Practical training: 6 hours

            Land: 0