Université Paris 13 MACS 1 - Modeling and case studies: differential equations Professor : Caroline Japhet

Many physical phenomena can be formulated as differential equations, for example a simple pendulum (see also here), a predator-prey model, ... Most of these equations however can not be solved exactly by analytical tools. Their solution has to be approximated using numerical methods. Among these methods we will study :

• explicit and implicit Euler, Crank-Nicolson, and BDF schemes
• first and second orders linear differential equations
• nonlinear differential equations, requiring the resolution of a nonlinear algebraic equation at each time step by fixed-point or Newton methods
• differential systems
• finite difference method
  
• how to write the corresponding codes in C language, using separate compilation and Makefile, dynamic memory allocation, and function pointers
• how to visualize the numerical solutions as functions of time, and the error schemes, using gnuplot.
  

After the course you will be able, from a given problem and method, to write the corresponding codes in C languageto recover the theoretical results (consistency, stability,...) studied in class, compare differentnumerical methods and choose the "best one" to get an accurate solution, close to the physics with a minimum computational time andstorage., and to validate these codes by finding the theoretical results (convergence to the exact solution if known, scheme error, stability, ...) and the physics studied in class.

Handouts

• Schémas

• TP1 (EDO linéaire du 1er ordre à coefficients constants : schémas d’Euler explicite et implicite, visualisation des solution)
• TP2 (EDO linéaire du 1er ordre à coefficients constants : schémas de Crank-Nicolson et BDF, visualisation de l'erreur des schémas),
  
• TP3 (EDO linéaire du 2nd ordre à coefficients constants : allocation dynamique),
  
• TP4 (EDO linéaire du 2nd ordre à coefficients constants avec conditions aux limites : méthode de tir),
• TP5 (EDO non linéaire du 2nd ordre : pendule simple, résolution numérique utilisant une méthode de point fiwe ou de Newton),
• TP6 (Résolution d’équations non-linéaires en langage C, passage de fonction en paramètre d'entrée),
• TP7 (Résolution d’un problème aux limites par différences finies sur maillage 1D non régulier),
• TP8 (Résolution d'un système linéaire - stockage des matrices creuses),
• TP9 (Système de Lotka - Volterra),

TPs corrigés : voir le site où a lieu le cours