Modelling Physics with DAE

Master level course. Ecole Polytechnique 2017/18 (COMASIC)


Differential Algebraic Equations (DAE) arise as a natural modelling language for physical systems as they combine algebraic constraints and differential equations resulting from the laws of physics. This mathematical paradigm is at the basis of modelling languages like Modelica, a component-based language designed for the modeling of multi-physics systems. During this course we will be studying DAE from the computer scientist point of view, we will present the structural analysis at the heart of the state-of-the-art index reduction techniques. We will also present the main numerical integration schemes for DAE. The end of the course will be devoted to present the challenges reminiscent of supporting multi-mode DAE systems including the interactions with the nonsmooth behavior of some physical phenomena (e.g., impact laws), the multiple modes of operations, and the intrinsically discrete behavior of software components.

Keywords Modelling Physics, Differential Algebraic Equations, Structural Analysis, Numerical Integration, Compilation, Simulation


Graph Algorithms, Numerical Integration, Calculus, Abstract Algebra

Expected Outcome

The course aims at presenting the advantages, inherent difficulties, and future challenges of DAE-based programming languages like Modelica.


  • December 13th. Lecture 1: General Introduction
  • December 13th. Lecture 2: Structural Analysis: Index Reduction
  • December 20th. Lecture 3: Numerical Integration of DAEs
  • January 10th. Lecture 4: Algebraic Methods
  • January 17th. Lecture 5: Brief Introduction to Modelica [Online Tutorial]
  • January 24th. Lecture 6: Multi-mode DAE
  • February 9th. Presentations


The final score will be composed of two marks for the same project. Project proposals by January 10th.

  1. An oral presentation. Friday, February 9th 16th. 15-20mn Talk + 10-15mn Questions
  2. A Paper by Thursday, February 8th Latex Template
    • Expected Sections: Introduction, Model, Simulation, Results, Related Work, Conclusion



Internships (4 to 6 months) on related topics are available.
If interested, let me know as soon as possible.

Further Readings

cf. slides.

Office Hours

  • Wednesday 2–4 PM (Inria Saclay)
  • For any request/question, feel free to email me (see address below)