## 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.

Day | Time | Place |
---|---|---|

Tuesday | 17:00-18:30 | G05-313 |

Wednesday | 09:30-11:00 | G05-313 |

Jump to the exercises section.

### Course of the lecture (22+6=28)

- Introductory considerations (1) [week 1]
- DAEs in mathematical modelling
- Applications areas and examples
- Challenges in the numerical and analytical treatment of DAEs
- Literature

- General notions from DAE calculus (1+1)
- Solutions and solvability
- Consistency and regularity
- Indices

- Linear DAEs with constant coefficients (4+1)
- Linear time-varying and nonlinear DAEs (4+1)
- Numerical integration of DAEs (6+2)
- Numerical Methods for index reduction (1)
- Derivative Arrays
- Minimal Extension

- DAEs with controls (1) [week 11]
- Representation as
*Behavior* - Index reduction through
*Feedback*

- Representation as

### Exercises

Date | Topic | Sheet |
---|---|---|

April 20th | I - Introductory Considerations and Basic Notions | ueb1.pdf |

May 11th | II - Linear DAEs with constant coefficients | ueb2.pdf |

May 18th | III - linear daes with time-varying coefficients | ueb3.pdf |

June 8th | IV - one step methods | ueb4.pdf |

July 5th | V - higher index and nonlinear equations | ueb5.pdf |

### 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

Author | Title | comments |
---|---|---|

Kunkel, Mehrmann | Differential-Algebraic Equations | Main reference, very concise, sometimes hard to read |

Hairer, Wanner | Solving ODEs. (Stiff and DAEs) | standard reference for solving ODEs (the first volume), intuitive and practical approach to numerical analysis of certain DAEs |

### 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