 # ODE and numerical resolution

## Presentation

Programme (detailed contents):

1)    Basics of topology, open and closed set, compact sets, complete sets.

2)    Ordinary differential equations

i)     Introduction to the Cauchy problem

ii)    The Cauchy Lipschitz theorem, Gronwall lemma

iii)   Linear differential equations. Resolvente, Duhamel’s formula

iv)   Qualitative properties: stability of stationnary points, phase portraits (application to mechanic, biology, chemistry).

3) Numerical simulation of ODEs

i)     Euler schemes

ii)    Convergence, consistence and stability

iii)   Order of convergence and Runge Kutta Schemes

iv)   Stiff and conservative problem: design of adapted schemes.

Organisation:

Main difficulties for students:

Draw a phase portrait, apply the Cauchy Lipschitz theorem correctly. Notions of convergence, consistence and stability of numerical schemes.

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

i)     Define the Cauchy problem for a system of ODEs

ii)    Définition, existence and uniqueness of maximal solutions. Cauchy Lipschitz theorem.

iii)   Stability of stationnary solutions, stability criteria

iv)   Convergence, consistency, stability and order of convergence for a numerical scheme.

The student will be able to

i)     Solve classical scalar differential equations.

ii)    Give qualitative properties of the solutions to a system of ODEs: existence, uniqueness, regularity. Stability of steady states.

iii)   Propose and carry out an adapted numerical simulation of a system of ODEs.

## Needed prerequisite

Basic Differential Calculus. Linear algebra: reduction of matrices.

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...