MA8404 Numerical solution of time dependent differential equations, fall 2019


  • 29.08: Handwritten notes and other working documents will be made available at Blackboard. If you still not have access, I can send them by mail on request.
  • 06.08: There will be an information meeting at Tuesday 20. August, kl. 14:15-15:00, in 734, SBII.
    If you want to follow the course, but can not attend the information meeting, please send me an email beforehand.

    The course is suitable for both PhD and Master students.

Course Information


  • Tuesday 08:15-10:00
  • Thursday 12:15-14:00


  • Friday 10:15-12:00 (irregular)

The lectures and the exercises will be in room 656, 6th floor, SBII, with one exception: The lecture 03.09 will be in 734.



We will not rely on one specific textbook. Some useful resources are listed below:

  • Hairer, Lubich, Wanner, Geometric Numerical Integration (GNI), Springer (2006)
  • Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (HNW) 2nd ed, Springer (1993)
  • Hairer, Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations (HW) 2nd ed, Springer (1996)
  • Kloeden, Platen, Numerical Solution of Stochastic Differential Equations (KP) 2.ed. Springer,
  • Hundsdorfer, Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. (HV), Springer (2003)
  • Butcher, Numerical Methods for Ordinary Differential Equations (B) 3.ed. Wiley.

All the books, except from the one by Butcher, are available electronically from Springer Link

Course Goals

The aim of this course is to teach students how to derive, analyse and implement numerical methods for solving time dependent differential equations. To achieve these aims, students will numerically solve mathematical problems and mathematically analyse the methods used for numerical solution. This course will cover the basic topics of stability, accuracy, and efficiency for various numerical algorithms. At the end of the course, the student will be able to:

  • Select and implement numerical methods that are most suitable for a given application.
  • Analyse the methods, using suitable techniques.
  • Participate in scientific discussions and simple research related to the course topics.


  • Classical methods for ordinary differential equations
  • Structure preserving integrators
  • Integration methods for PDEs
  • Numerical solution of stochastic differential equations (SDEs)

It is possible to do minor adjustments, depending on the students interests.

Tentative Schedule

Week Topics Suggested Exercises
34 Information meeting, room 734, 14:15-15:00, 20. August.
35 Introduction to the course.
Example of problems to be solved, and simple methods for solving them.
Runge-Kutta methods
Definition of the method (LN. sec. 4, GNI II.1.1)
Global error analysis of one-step methods (LN. Theorem 2.2+proof, HNW II.3)
36 Local error analysis, B-series and rootet trees (LN 4.1, GNI III.1.1-2.)
Be aware that the construction of the series differ in the two references, but the result is the same.
Exercise 1
with solution.
Error estimation and stepsize control, dense output (LN, 4.2-3, HNW II.4)
37 Collocation methods (LN 4-4, GNI II.1.2, HNW II.7)
38 Stiff differential equations. A- and B-stability (LN, section 5, HW IV.3, IV.12),
Step-control stability (HW IV.2, p.24)
Differential-algebraic equations (DAEs), index, Hessenberg forms (LN, 7.1-7.2)
Runge-Kutta methods applied to DAEs (HW VI.1)
Index reduction and projection methods (HW VII.2).
Exercise 2 with solutions (The last exercise to appear), sol2.m,
39 Linear Multistep Methods
Zero-stability, consistency, convergence
Adams and BDF methods (HNW III.1-4)
Linear stability of LMM (HW, V1.1)
Exercise 3 with solution and with accompanying Jupyter notebook.
40 Geometric integration
Preservation of invariants (GNI IV.1, 2.1, 2.2 and 3.1)
Hamiltonian problems and symplectic methods (GNI VI.1-4)
41 Variational integrators (GNI VI.6) Exercise 4 with solution and accompanying Jupyter notebook.
42 Time integration of PDEs
Splitting methods and the BCH formula (GNI, III.4-5, HV IV.1)
Exponential and integrating factor methods. Note by Elena Celledoni, section 2.
43 Partitioned methods (GNI II.2, III.2, HV IV.4) See also Ascher
Stochastic differential equations
Brownian motions
Euler-Maruyama and Milstein methods (at least)
See e.g. Higham (2001)
44 SDE: Itô and Stratonovich integrals (briefly), and some of their properties.
Wanger-Platen series, Stochastic Runge-Kutta methods
Strong and Weak convergence,
45 SDE: More on weak convergence. Linear stability analysis (see e.g. Higham 2000.
Multilevel Monte-Carlo methods are not a part of the curriculum, but if someone is interested, you can find it here Giles (2015).
Backward error analysis and modified equations
(GNI IX 1-2)
Application to asymptotic global error expansions and from there to extrapolation methods.
Exercise 5 with solutions
46 Modified equations for splitting methods and Hamiltonian problems. (GNI IX 3-4) Exercise 6 with solutions.
Generating functions (GNI VI 5.1, 5.3-4).
47 Project presentations.


Choose a topic related to your own Master or PhD studies and use the methods/techniques developed in this course in conjunction with your own research.


Exam: December 9. The exam is oral.

2019-11-28, Anne Kværnø