Simulation of ordinary and differential-algebraic equations

Contents

1. Introduction to simulation
2. Simulation of ordinary differential equations, including stiff problems
3. Simulation of differential-algebraic equations
4. Modelica and simulation of object-oriented models

Schedule 2026

Examination

• Hand in of mandatory exercises in exercises.
• Report of a mandatory mini-project in a topic related to the course. For example, choose a simulation model, possibly related to your research project, and demonstrate the correctness of the simulation.
• Oral examination: Explain and discuss theory and methods

Course plan – ODE –

ODE 1 - Introduction/simulation of ordinary differential equations

• Responsible: Lars Eriksson
• 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.

ODE 2 - Introduction/simulation of ordinary differential equations

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

ODE 3 - Introduction/simulation of ordinary differential equations

• Responsible: Lars Eriksson
• Contents
  • More on implicit and multi-step methods.
  • Step length control, parameters for step length control.
  • Zero detection.
  • Start implementation of an explicit (or implicit) method with step length control.
  • Continued work on implementation and exercises in the course.
• Course material
  • Ascher och Petzold, remaining parts of Chapter 4 and Chapter 5.
  • Deuflhard och Bornemann, “Scientific Comupting with Ordinary Differential Equations”. Chapter 5 on step length control.
  • John Mathews, “Computer Derivations of Numerical Differentiation Formulae (Classroom Notes)”, Int. J. of Math. Education in Sci. and Tech., V 34, No 2 (March-April 2003), pp.280-287.

ODE 4 - Introduction/simulation of ordinary differential equations

• Responsible: Lars Eriksson
• Contents
  • Discussion about results from hand in assignments.
  • Stiff problems, A-stability, L-stability.
  • Implicit methods for stiff problems, BDF methods.
  • Implementation of implicit methods.
• Course material
  • Chapters 6-8 in Ascher and Petzold, with emphasis on Chapters 6-7.
  • More material on stiff differential-equations from Shampine och Gear “A User’s View of Solving Stiff Ordinary Differential Equations”.
  • Kjell Gustafsson, “Traps and Pitfalls in Simulantion”.

Course plan – DAE –

DAE 1 - Simulation of differential-algebraic equations

• Responsible: Erik Frisk
• Contents
  • Introduction to differential-algebraic equations (DAE:s)
  • Examples of DAE models
  • Existence and uniqueness of solutions to DAE:s
  • What is a DAE, compared to an ODE?
  • Index definitions for DAE:s
  • Initial conditions for DAE:s
  • Why are index-1 DAE:s easy and index>1 DAE:s difficult to simulate?
• Course material
  • C.W. Gear and L. Petzold, “ODE methods for the solution of differential/algebraic systems”, SIAM Journal on Numerical Analysis, Vol. 21, No. 4. (Aug., 1984), pp. 716-728.
  • Page 372-381 and 452-454 in Hairer part 2.
  • L. Petzold, “Differential/Algebraic Equations are not ODE’s”, SIAM journal on scientific and statistical computing, vol. 3, no. 3, 367-384, 1982.

  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

DAE 2 - Simulation of differential-algebraic equations

• Responsible: Erik Frisk
• 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 material
  • pages 455-480 in Hairer part 2
• Exercises: 2.26, 2.14-15, 2.18, 2.22, 2.33

DAE 3 - Modelica and simulation of object-oriented models

Responsible: Erik F

• Contents:
  • Adjoint sensitivity analysis
  • 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 material
  • If you want to known a lot on how DASSL/DASPK is implemented, then I recommend Chapter 5 in K.E. Brenan, S.L. Campbell and L.R. Petzold, “Numerical Solution of Initial-Value Problems in Differential.Algebraic Equations”, SIAM, 1996. Describes the solver in detail.

  • Cao, Yang, et al. “Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution”. SIAM journal on scientific computing 24.3 (2003): 1076-1089.

    Rather technical paper: I suggest to read and Section 1, 2.1, 4.1-4.2 in a first reading.

  • Chapter 17, 18 i “Principles of Object-Oriented Modeling and Simulation with Modelica 2.1”, P. Fritzson. For those not familiar with Modelica, recommended reading in Chapter 2.
  • G. Reissig, W.S. Martinson, P.I. Barton, “Differential–Algebraic Equations of Index 1 May Have an Arbitrarily High Structural Index”, SIAM Journal on Scientific Computing, Volume 21, Number 6, pp. 1987-1990, 2000.
  • For the interested, a brief, somewhat incomplete, but still interesting description of how OpenModelica goes from model to C code. Adrian Pop, OpenModelica Compiler Phases, Open Source Modelica Consortium, 2008-05-26.
  • H. Elmqvist and M. Otter, “Methods for tearing systems of equations in object-oriented modeling”, Proceedings of the European Simulation Multiconference, pp. 326-332, Barcelona, Spain, 1994.
  • Examples for Sundials – IDAS that compiles on charger (have a look at idasRoberts_ASAi_dns.c for adjoint sensitivity example). If you are interested, I recommend looking into the documentation, e.g., for the IDAS solver https://computing.llnl.gov/sites/default/files/public/idas_guide.pdf.
• Exercises: 2.9, 2.10b-d, 2.31

DAE4 - 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 material:
  • Kapitel 17,18 i “Principles of Object-Oriented Modeling and Simulation with Modelica 2.1”, P. Fritzson. For those not familiar with Modelica, recommended reading in Chapter 2.

  • C.C. pantelides “The consistent initialization of differential-algebraic systems”, SIAM Journal on scientific and statistical computing, Vol. 9, No. 2, pp. 213-231, March 1988.

    The description of the graph theoretical algorith to find all MSS:s can be skipped

  • Mattias Krysanders Matlab-implementation of Pantelides algoritm [zip, tar.gz]
  • S. Mattson and G. Söderlind, “Index reduction in differential-algebraic equations using dummy derivatives”, SIAM Journal on Scientific Computing, vol. 14, No. 3, pp. 677-692. 1993.
• Exercises: 2.13, 2.29, 2.35, 2.30