## Course DAEs - OvGU - 2018

Major typo in the exercise sheet. We want to analyze the BDF for the DAE `Nx'=x+f` rather than the ODE `x'=Nx+f`. I have updated the sheet.

I have added the Butcher scheme of `RadauIIa-2` to the script to compute the `kappa`s in exercise IV.

Here is the paperhive lecture. Looks like you have to join this channel to see the document.

Here you find basic and current information and materials for the lecture DAEs at the OvGU in the winter term 2018.

Day Time Place
Thursday 09:00-11:00 G12-129
Friday 11:00-13:00 G05-308

Consultation hours: Thursday 11:00-12:00 – please make an appointment by email. Contact data is on my webpage.

Here is my write-up of some selected topics.

### 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
• Normal forms [week 2]
• Solvability and representations of solutions
4. Linear time-varying and nonlinear DAEs (4+1)
• Fundamental differences with the linear time-invariant case
• Time-dependent equivalence transformations and canonical forms
• Derivative Arrays
• 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
• RKM methods for semi-explicit “index-1” DAEs
• RKM methods for implicit “index-1” DAEs
• Backward Differencing for DAEs
6. Numerical Methods for index reduction (1)
• Derivative Arrays
• Minimal Extension
7. DAEs with controls (1)
• Representation as Behavior
• Index reduction through Feedback

### Exercises

Date Topic Sheet
October 25th I - Introductory Considerations and Basic Notions ueb1.pdf
November 2nd II - Linear DAEs with constant coefficients ueb2.pdf
November 16th III - linear daes with time-varying coefficients ueb3.pdf
December 6th IV - one step methods ueb4.pdf
January 17th 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 +++ 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

### 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

#### Course Exercise sheet I - Oct 25th

+++ 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

#### Linear DAEs with constant coefficients (3)

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

### Week 4

#### Linear DAEs with constant coefficients (4)

+++ 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

#### Course Exercise sheet II - Nov 2nd

+++ 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 (1)

+++ 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

#### Linear DAEs with time-varying coefficients (2)

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

### Week 6

#### Linear DAEs with time-varying coefficients (3)

+++ global equivalence ctd. +++ strangeness index +++ derivative arrays +++ back to overview

#### Course Exercise sheet III - Nov 16th

+++ global/local equivalence of matrix pairs +++ time-varying Drazin inverse +++ back to overview

### Week 7

#### Linear DAEs with time-varying coefficients (4)

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

#### Index-Reduction

general ideas and notions +++ local computation of the strangeness-free reformulation +++ minimal extension for NSE +++ see Kunkel/Mehrmann book 6.4 or this original paper on minimal extension +++ back to overview

### Week 8

#### Homework – Numerical Methods for ODEs

How does Euler’s method work. +++ What is the consistency error? +++ What is stability? +++ What is meant by convergence? +++ What are Runge-Kutta methods +++ Here is a scan of the lecture I had on this by Hans-Görg Roos back then. +++ back to overview

#### Numerical Solutions of DAEs (1)

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

### Week 9

#### Course Exercise sheet IV

+++ 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] – [py-script to compute κs] +++ back to overview

#### Numerical Solutions of DAEs (3)

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

### Week 10

#### Numerical Solutions of DAEs (4)

+++ Numerical analysis of Runge-Kutta schemes for DAEs with constant coefficients +++ the global consistency error +++ stiffly accurate RKM +++ back to overview

#### Numerical Solutions of DAEs (5)

+++ Numerical analysis of Runge-Kutta schemes for DAEs with constant coefficients +++ BDF schemes +++ formulation and warnings concerning RKM for DAEs with time-varying coefficients +++ back to overview

### Week 11

#### Numerical Solutions of DAEs (6-7)

+++ online lecture +++ RKM for nonlinear DAEs +++ semi-explicit strangeness-free +++ collocation and RKM +++ back to overview

### Week 12

#### Numerical Solutions of DAEs (8)

RKM for nonlinear DAEs +++ semi-explicit index-2 +++ the Hairer-Wanner analysis approach +++ recap: collocation and RKM +++ back to overview

#### Overview on Results and Software

+++ see the script +++ back to overview

### Week 13

#### 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