16-dae-course-ovgu

Course DAEs - OvGU - 2016

The 5th exercise is up. We will discuss it in the last week of the course. That is Tuesday, July 5th.

There was a little mistake in the 5th exercise sheet (wrong initial conditions for the cars in the spring mass system) – it’s fixed now.

Here you find basic and current information and materials for the lecture DAEs at the OvGU in the summer term 2016.

Jump to the exercises section.

Course of the lecture (22+6=28)

1. Introductory considerations (1) [week 1]
   • DAEs in mathematical modelling
   • Applications areas and examples
   • Challenges in the numerical and analytical treatment of DAEs
   • Literature
2. General notions from DAE calculus (1+1)
   • Solutions and solvability
   • Consistency and regularity
   • Indices
3. Linear DAEs with constant coefficients (4+1)
   • Basic algebraic concepts [week 2]
   • Normal forms
   • Solvability and representations of solutions [week 3][week 4]
4. Linear time-varying and nonlinear DAEs (4+1)
   • Fundamental differences with the linear time-invariant case
   • Time-dependent equivalence transformations and canonical forms [week 5]
   • Derivative Arrays [week 6]
   • Differentiation-index and Strangeness-index
5. Numerical integration of DAEs (6+2)
   • Digression: Numerical integration of ODEs
   • Runge-Kutta methods (RKM) for DAEs with constant coefficients [week 7][week 8]
   • RKM methods for semi-explicit “index-1” DAEs [week 9]
   • RKM methods for implicit “index-1” DAEs
   • Backward Differencing for DAEs [week 10]
6. Numerical Methods for index reduction (1)
   • Derivative Arrays
   • Minimal Extension
7. DAEs with controls (1) [week 11]
   • Representation as Behavior
   • Index reduction through Feedback

Exercises

Week 1

Introductory considerations (1)

+++ DAEs are coupled differential and nondifferential (algebraic) equations +++ cf. the pendulum +++ which is naturally modelled as a DAE +++ as are electrical circuits, chemical reactions, and flows +++ in numerical schemes, equations are solved approximately - what does this mean for the pendulum? +++ back to overview

General notions from DAE calculus (1)

+++ we consider C1-solutions although there are many ways to define less regular solutions +++ existence of solutions depends on several factors +++ smoothness of right hand sides +++ consistency of initial values +++ hidden constraints and underlying ODE +++ many ways to classify DAEs <-> many indices +++ back to overview

Literature

Week 2

Linear DAEs with constant coefficients (1)

+++ variable transforms and scalings do not affect solvability +++ DAEs <-> (E, A) matrix pairs +++ canonical forms +++ Weierstrass canonical form +++ canonical form of a linear DAE with constant coefficients +++ back to overview

Linear DAEs with constant coefficients (2)

+++ splitting of DAEs into an ODE and a nilpotent DAE +++ explicit solution of the nilpotent DAE +++ index of a matrix pair (E,A) and its well-definedness +++ back to overview

Week 3

Linear DAEs with constant coefficients (3) - April 19th

+++ solvability solved +++ way to arrive at a explicit solution formula +++ definition of the Drazin inverse +++ properties of the Drazin inverse +++ back to overview

Course Exercise sheet I - April 20th

+++ multibody systems +++ separation of algebraic and differential parts +++ remodelling of the simple pendulum as ODE +++ Navier-Stokes equations +++ links to ode modelling of the pendulum and the overhead crane +++ back to overview

Week 4

Linear DAEs with constant coefficients (4) - April 26th

+++ DAE as superposition of a nilpotent DAE and an index-1 DAE +++ explicit formula for all solutions of the homogeneous equations +++ explicit form of a solution of the inhomogeneous equations +++ back to overview

Linear DAEs with time-varying coefficients (1) - April 27th

+++ regularity of matrix pairs does not say much about solvability of LTV DAEs +++ time-dependent state transformations +++ global and local equivalence of matrix function pairs +++ back to overview

Week 5

Linear DAEs with time-varying coefficients (2) - May 10th

+++ characteristic values +++ canonical form for local equivalence transformations +++ time-varying SVD +++ canonical form for global equivalence transformations +++ back to overview

Course Exercise sheet II - May 11th

+++ regularity and Kronecker form of 3x3 examples +++ index-1 condition +++ regularity and commutativity +++ Drazin inverse as group inverse +++ back to overview

Week 5

Linear DAEs with time-varying coefficients (3) - May 17th

Course Exercise sheet III - May 18th

Week 6

Linear DAEs with time-varying coefficients (4) - May 24th

+++ derivative arrays +++ strangeness free condensed form of linearized Navier-Stokes equations +++ derivative arrays for nonlinear DAEs +++ back to overview

Digression: Numerical Solutions of ODEs - May 25th

+++ basic definitions +++ implicit/explicit Euler +++ consistency and stability +++ Runge-Kutta schemes +++ BDF schemes +++ back to overview

Week 7

Numerical Solutions of DAEs (1) - May 31th

+++ basic notions and definitions +++ Kronecker product and perfect shuffle +++ Runge-Kutta methods +++ back to overview

Numerical Solutions of DAEs (2) - June 1st

+++ Numerical analysis of Runge-Kutta schemes for DAEs with constant coefficients +++ the local consistency error +++ back to overview

Week 8

Numerical Solutions of DAEs (3) - June 7th

+++ Numerical analysis of Runge-Kutta schemes for DAEs with constant coefficients +++ the global convergence error +++ back to overview

Course Exercise sheet IV - June 8th

+++ effect of rounding errors +++ consistency errors +++ two-stage Gauss method for ODEs and DAEs +++ Runge-Kutta method for linear DAEs +++ CODING: C1:Explicit Euler and rounding errors +++ C2:Implicit Euler for linear DAEs with time-varying coefficients +++ Resources: Matlab implementation by Jens Bremer – [zip file], Python implementation – [webview], [ipython notebook], [python file] +++ back to overview

Week 9

Numerical Solutions of DAEs (4) - June 20th

+++ Note on Runge-Kutta methods for linear time-varying DAEs +++ definition and analysis of Runge-Kutta schemes for semi-explicit “index-1” DAEs +++ back to overview

Numerical Solutions of DAEs (5) - June 21st

+++ stiffly accurate Runge-Kutta methods +++ definition and analysis of Runge-Kutta schemes for implicit “index-1” DAEs +++ back to overview

Week 10

Numerical Solutions of DAEs (6) - June 27th

+++ general remarks on collocation Runge-Kutta methods +++ Backward differencing schemes for DAEs +++ back to overview

Numerical Methods for Index Reduction - June 28th

+++ general concepts of index reduction +++ numerical approach to index reduction via derivative arrays +++ minimal extension +++ back to overview

Week 11

Course Exercise sheet V - July 5th

+++ mass-spring chain +++ minimal extension +++ 2-stage Radau IIa +++ CODING: Implicit Euler for the nonlinear pendulum equations — Radau IIa for the mass-spring manoeuvre — simulation of index reduced systems +++ Resources: Use the code from the previous exercise iv — check out the Oberwolfach snapshot on the mass-spring chain (in German) or the more verbose preprint (in English) +++ back to overview