# Contents

• Introduction to simulation
• Simulation of ordinary differential equations, including stiff problems
• Simulation of differential-algebraic equations
• Modelica and simulation of object-oriented models

# Examination

• Hand in of mandatory exercises in exercises (2022-01-27).
• For the DAE-part, mandatory participation in exercise solving classes, demonstrating your own solutions.
• Report of a mandatory mini-project in a topic related to the course. For examepl, choose a simulation model, possibly related to your research project, and demonstrate the correctness of the simulation.
• Oral examination: Exaplain and discuss theory and methods
• Extra work can merit extra credits

# Course plan

## Meeting 1 - Introduction/simulation of ordinary differential equations

Course meeting
Responsible: Lars E
Contents
Basic ODE:
Problem formulations (some classic problems from Hairer-Norsett-Wanner), existance and uniqueness.
Simple one-step methods, implicit and explicit.
Course material
Material covering existance and uniqueness from Hairer, Norsett and Wanner and from Dahlqvist och Björk.

## Meeting 2 - Introduction/simulation of ordinary differential equations

Course meeting
Responsible: Lars E, OH
Contents
Concepts: Convergence, consistency, 0-stability, absolute stability. Stiff decay.
Explicit one step methods: Runge-Kutta family.
Step length control, parameters för step length control.
Course material
First 4 chapters in Ascher-Petzold until page 95.

## Meeting 3 - Introduction/simulation of ordinary differential equations

Course meeting
Responsible: Lars E, OH
Contents
More on implicit and multi-step methods.
Steplength control, parameters for steplength control. Zero detection.
Start implementation of an explicit (or implicit) method with steplength control.
Continued work on implementation and exercises in the course.
Course material

## Meeting 4 - Introduction/simulation of ordinary differential equations

Course meeting
Responsible: Lars E, OH
Contents
A little on BVP.
Continued work on implementation and exercises.
Survey/review and analysis of Matlab's solvers.
Traps and pitfalls.
Course material

## Meeting 5 - Introduction/simulation of ordinary differential equations

Course meeting
Responsible: Lars E
Contents
Discussion about results from hand in assignments.
Course material

## Meeting 6 - Simulation of differential-algebraic equations

Responsible: Erik
Contents
• Motivating examples, some models
• Existence conditions for solutions to DAE:S, what do they look like?
• What separates an ODE from a DAER and when can an ODE solver be used to integrate DAE:s?
• Index, what is it and what does it mean? Complications and different definitions.
• Initial conditions.
• Why are index-1 easy and index larger than 1 difficult to simulate?
Course meeting
OH
Course material
Chapters 9 and 10 in Ascher/Petzold is good additional material. The paper S.L. Campbell and C.W. Gear, "The index of general nonlinear DAEs", Numerische Mathematik, Vol 72, No. 2, 173-196, 1995 gives a thorough description of different index for anyone interested.
Exercises: 2.1-3, 2.5, 2.7, 2.10a, 2.11

## Meeting 7 - Simulation of differential-algebraic equations

Responsible: Erik
Contents
• Introduction to method for semi-explicit index-1 DAE:s
• state-space
• epsilon-embedding + implicit Runge-Kutta
• BDF (DASSL)
• Pantelides algorithm for determination of consistent initial conditions
• Some methods for index reduction
• Problems with drift and possible solutions
• Baumgarte stabilization
• Projection methods
• Some about order and convergence with solvers, stiffly accurate methods
• ODE/DAE:er with invariants
• The unstructured problem F(y',y,x)=0 and overdetermined models
Course meeting
OH
Course material
Exercises: 2.26, 2.14-15, 2.18, 2.22, 2.33

## Meeting 8 - Modelica and simulation of object-oriented models

Responsible: Erik F
Contents
• Introduction to equation based models
• How to go from a Modelica model to simulation code in C
• Structural index
• How can you, in the model, give clues to the solver for choice of states and initial conditions.
• How to automatically generate crossing/event-functions
Course meeting
OH
Course material
Exercises: 2.9, 2.10b-d, 2.31

## Meeting 9 - Modelica and simulation of object-oriented models

Responsible: Erik F
Contents
• Pantelides algorithm
• Finding consistent initial conditions
• Computing structural index
• Index reduction using dummy-derivatives
Course meeting
OH
Course material
Exercises: 2.13, 2.29, 2.30

## Meeting 10 - DAE, exercise discussion 1

Responsible: Erik F
Contents
Presentations and discussion about mandatory exercises.
Course meeting

## Meeting 11 - DAE, exercise discussion 2

Responsible: Erik F
Contents
Presentations and discussion about mandatory exercises.
Course meeting

## Meeting 12 - DAE, exercise discussion 2

Responsible: Erik F
Contents
Presentations and discussion about mandatory exercises.
Course meeting

# Course material

• Huvudbok för delen om ordinära differentialekvationer är "Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations" av U. Ascher och L. Petzold.
• För delen om differential-algebraiska ekvationer kommer valda delar av "Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems" av E. Hairer och G. Wanner att användas. Avsnitt: VI.1, VII.1-VII.2.
• För delen om simulering av objekt-orienterade modeller kommer kapitel ur "Principles of Object-Oriented Modeling and Simulation with Modelica 2.1" av P. Fritzson att kopieras upp.

Utöver texterna ovan kommer nedanstående artiklar att ingå. De som ej finns i elektronisk version finns att tillgå som papperskopior.

Page responsible: Erik Frisk
Last updated: 2022-01-27