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# Cours scientifiques - MAA312 : Numerical Methods for ODEs

## Descriptif

Welcome to MAA312! The topic of this course is the numerical simulation of ODE's.

Let's get one question out of the way immediately: your final grade will be calculated as follows:
• Quizz after each class: 20%.
• One assignement from TD (in the form of a Jupyter notebook): 20%
• Test: 20%
• Final exam: 40%.
• Attendance and participation: bonus points between 0 and 2/20 given by the TAs.
Quizzes are open the day of the lecture and must be completed before they close the following Monday at noon.

The test will take place during the second hour of the class on 11/05.

Students must pair to complete together the assignment. Pairs can be composed of students of two different groups.
The first assignment will be given on 16/05. It must be turned back before 23/05 1pm on Moodle.

You can find this information as well as the description of the course in the Syllabus.

The course MAA207 "Series of functions, differential equations" is a pre-requisite.

And now, on to the science !

We will use two textbooks:

-Numerical Mathematics, Chapter 11, by A. Quarteroni, R. Sacco, F. Saleri, for the numerical aspects;
-Differential Equations, dynamical systems and an introduction to Chaos, Chapters 3, 8 & 9, by M. Hirsch, S. Smale and R. Devaney, for the qualitative aspects.

Both books are available at the BCX (both electronic and hard copies)

## Objectifs pédagogiques

The aim of the class is to give a large overview on numerical methods for ODEs exploring both the theory and their implementation in Python. A complementary aspect of the course is the study of qualitative properties of ODEs.

réparties en:
• Contrôle Final : 1
• Contrôle : 1
• Travaux pratiques : 6
• Cours magistral : 9
• Travaux dirigés : 3

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE MAA207

Calculus, Basics of Python, Series of functions, differential equations

Numérique sur 20

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 11)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 4 ECTS

La note obtenue rentre dans le calcul de votre GPA.

### Programme détaillé

A brief review of the theory of ordinary differential equations (Cauchy-Lipschitz theorem). Presentation of typical examples.

One-Step methods:

-Explicit and implicit Euler schemes

-Stability, consistency, convergence of the numerical schemes

-Implementation in Python and numerical examples

Runge-Kutta methods

Multi-step methods

Phase portraits for planar systems:

-Linear case

- Nonlinear case : stability of hyperbolic points, a glimpse at bifurcation theory

Global Nonlinear techniques:

-Stability of equilibria: Liapounov functions