Objectifs :
The course aims to make the student aware of the cutting-edge algorithms used to numerically perform time integration and their properties in order to optimally choose the proper time integrator according to the considered phyisical model.

Programme :
• Theory of ODEs: examples, classification, existence and uniqueness of solutions.
• Simple methods: explicit and implicit Euler, convergence, stability analysis, properties.
• High order classical methods.
◦ Runge--Kutta methods: construction, explicit, implicit, IMEX, error and stability analysis, properties, the Butcher tableau.
◦ Multistep methods: explicit, implicit methods, error and stability analysis, convergence.
• Iterative explicit high order methods: Deferred Correction (DeC), Arbitrary Derivative (ADER) methods, properties, stability and convergence analysis.
• Unconditionally positivity preserving schemes: implicit Euler, high order schemes, modified Patankar schemes and their stability and convergence analysis.
• Entropy conservative high order schemes: relaxation Runge--Kutta methods.

Pré-requis :
Basics of numerical analysis

Compétences acquises à l'issue de la formation :
The student will be able to study time integration methods, perform analysis on them, code them and choose between a variety of schemes according to considered physical problem.