# MA8404 Numerical solution of time dependent differential equations

## Messages

• 12.09: The solutions to Exercise 1 is available at Dropbox. So is the Latex template used by the Cambridge group.
• 11.09: The additional lecture has been moved to 21.09, 9.15-11.00, in 738, SII (our regular room was occupied).
• 08.09: The lecture today is cancelled. To catch up, there will be an extra lecture Thursday 14.09, kl. 9:15-11.
• 23.08: The first lecture in the course will be given Wednesday August 30, 14:15-16:00. It will, if the technology is with us, possible to follow the lectures on Skype. The lectures will not be recorded.
• 03.08: There will be an information meeting at Monday 21. August, kl. 13:15-14:00, in 738, 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

### Lectures

• Wednesday 14:15-16:00,
• Friday 10:15-12:00,
• Optional: Thursday 9:15-11:00

All lectures will be in the video conference room 634, SBII.
The first lecture is Wednesday 30. August.

• Anne Kværnø
• Office: 1348

### Literature

• Main textbook
• Hairer, Lubich, Wanner, Geometric Numerical Integration (GNI), Springer (2006)
• Other sources
• Hairer, Nørsett, Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (SODE I) 2nd ed, Springer (1993)
• Hairer, Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations (SODE II) 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)
• Lecture notes from TMA4215 ++ odenote.pdf (LN). Updated 18.10.

All the books are available electronically from Springer Link

### Course Goals

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

• Select and implement numerical methods that are most suitable for a given application.
• Determine the stability, order of accuracy, and structure-preserving qualities of a given numerical method.
• Participate in scientific discussions and simple research related to the course topics.

Topics:

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

### Very Tentative Schedule

Week Topics Relevant Texts Suggested Exercises
34 Information meeting, Room 738, 13:15-15:00, 21. August.
35 Background on ODEs. Runge-Kutta methods.
Error analysis, convergence results.
LN: section 1-4.0
SODE1: p. 60-61 (Logarithmic norms) and Theorem 3.4 on page 160 (convergence) with proof.
GNI: Part I. (Examples of ODEs)
Local error analysis. B-series. Order conditions for RK methods. LN: section 4.1.
GNI: II 1.1-1.2 for an alternative derivation.
How to find order conditions from rooted trees.
Exercise 1 (already corrected twice, the last corrections in (2), (3) and (5))
36 Explicit Runge–Kutta methods, error estimation and stepsize selection
Collocation methods, discontinuous collocation
Simplifying assumptions
Lecture 8.9 cancelled
LN: 4.2-4.4
GNI: II.2
SODE1:p.208 (simplifying assumptions)
37 Preservation of invariants, symmetry
GNI II.3, IV.1-2, V.1-2.1
Partitioned, composition and extrapolation methods GNI II.2, II.4, IV.2.2-2.3, V.2.2 Exercise 2
Some corrections and clarifications, and they are marked in red.
38 Splitting methods
Hamiltonian problems, symplecticity
Start of project work (individual)
Additional lecture 21.09, 9:15-11:00, in room 738, SII
GNI II.5, III.4-5
GNI VI.1-4.
Exercise 3
39 No lectures
40 Backward error analysis and modified equations. GNI IX.1-3.1 and Theorem 3.3. Exercise 4
Backward error analysis for splitting methods GNI IX.4
41 Stiff ODEs and stability concepts.
Linear stability: stability function $R(z)$, region of absolute stability
A-, A($\alpha$)- and L-stability

Step-control stability
Nonlinear stability, B-stability, Algebraic stability
SODEII, p. 24-27.
LN 5.2 and SODEII IV.12, p. 180-184.
42 Differential-algebraic equations.
The differential index and Hessenberg forms.
RK-methods for index 1 DAEs.
LN 7.1-3 and SODEII: VI.1,3 Exercise 5
43 No lectures.
44 Linear multistep methods (self study)
Higher index DAEs.
BDF methods. Coordinate projection.
RK-methods for index 2 problems.
LN: 6 and 7.
SODEII p.470-474.
Exponential integrators. ETD and IF (Lawson) Note by Elena Celledoni, Chapter 2,
A review of exponential integrators for first order semi-linear problems by Minchev and Wright.
Exponential integrators and IMEX methods.
45-46 Numerical methods for stochastic differential equations
Brownian motions, Ito and Stratonovich integrals (the idea),
Ito's formula (the stochastic chain rule),
stochastic differential equations (SDEs),
Stochastic Taylor expansions (Wagner-Platen and B-series), some numerical methods,
Strong mean square and weak convergence
MS linear stability analysis
Higham: An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
Higham: Mean-Square and Asymptotic Stability of the Stochastic Theta method
Slides on stochastic B-series, Part II.
Exercise 6
47 Problem solving sessions and project presentations.

### Project

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. Projects must be submitted in written form, no longer than 10 pages, not including supplementary material such as a cover page or bibliography. Projects must also be presented orally to the class at the end of the semester.

• Choose a topic relevant for your research work. Discuss with you supervisor.
If you don't have a relevant topic, then contact me and we will work out something together.
• The focus of the project should be the time-integration algorithm: Choice of a method, important properties, theoretical considerations, ….
• And the method should be implemented and tested on some relevant problems.

Deadline for the project: November 21

Project presentations: November 24 (lecture hours).

### Exam

December 6.

Permitted aids:

• Hairer, Lubich, Wanner, Geometric Numerical Integration (GNI), Springer (2006)
A printout of the book is permitted.
• One yellow A4 sheet (both sides) with your own notes.