Differential Algebraic Equations I & II

This lecture is designed for graduate and PhD students interested in the numerical time integration of stiff ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). These systems arise in various physical disciplines, including chemical reaction kinetics, electrical circuits, incompressible fluid dynamics, and constrained mechanical systems.

The first part of the course provides an introduction to DAEs, presenting a range of physical examples alongside benchmark problems with known analytical solutions. It then covers implicit Runge-Kutta methods and backward differentiation formulas (BDF) for numerical time integration. Programming exercises will guide students through simple implementations of both approaches.

The second part focuses on the implementation details of implicit Runge-Kutta methods, particularly the Radau IIA family, covering efficient solution strategies and practical considerations. This is followed by a survey of state-of-the-art open-source implementations of DAE solvers. The course concludes with a hands-on workshop, where students apply the learned concepts to research-oriented problems involving DAEs.

Note: It is possible to attend only Part I of the lecture (2 SWS / 3 ECTS). Alternatively, Part I & II must be attended (4 SWS / 6 ECTS).

Part I

• Differential algebraic equations: What are DAEs? The index concept and solvability. Index reduction and constraint stabilization. Consistent initial conditions.
• Examples from various physical disciplines and benchmark systems: Stiff ODEs, chemical reaction kinetics, electrical circuits, incompressible fluids and constrained mechanical systems.
• Implicit Runge-Kutta methods: Order conditions, simplifying assumptions, convergence, stability and Butcher tableaus.
• Backward differentiation formulas (BDF)

Part II

• Implementation of Radau IIA methods: Dense output, simplified Newton iterations, transformation of the linear system, error estimate and step-size control.
• Software for DAEs: DASSL.f, RADAU.f and Sundials IDA.
• Practical workshop: Application of the learned concepts on research topics on differential algebraic equations.

All course material will be posted on ILIAS.

• oral exam, duration 30 minutes
• time and date will be arranged individually
• closed book, no electronic devices (calculator/laptop/phone)

See "Stellung im Studienplan"