Fall 2017
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, 9.1511.00, 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. 9:1511.
 23.08: The first lecture in the course will be given Wednesday August 30, 14:1516:00. 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. 13:1514:00, 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 14:1516:00,
 Friday 10:1512:00,
 Optional: Thursday 9:1511:00
All lectures will be in the video conference room 634, SBII.
The first lecture is Wednesday 30. August.
Lecturer
 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 DifferentialAlgebraic Equations (SODE II) 2nd ed, Springer (1996)
 Kloeden, Platen, Numerical Solution of Stochastic Differential Equations (KP) 2.ed. Springer,
 Hundsdorfer, Verwer, Numerical Solution of TimeDependent AdvectionDiffusionReaction Equations. (HV), Springer (2003)
 Lecture notes from TMA4215 ++ odenote.pdf (LN). Updated 18.10.
 Additional material (to appear)
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 structurepreserving 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, 13:1515:00, 21. August.  
35  Background on ODEs. RungeKutta methods. Error analysis, convergence results.  LN: section 14.0 SODE1: p. 6061 (Logarithmic norms) and Theorem 3.4 on page 160 (convergence) with proof. GNI: Part I. (Examples of ODEs)  
Local error analysis. Bseries. Order conditions for RK methods.  LN: section 4.1. GNI: II 1.11.2 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: 4.24.4 GNI: II.2 SODE1:p.208 (simplifying assumptions)  
37  Preservation of invariants, symmetry The adjoint of a method  GNI II.3, IV.12, V.12.1  
Partitioned, composition and extrapolation methods  GNI II.2, II.4, IV.2.22.3, 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, 9:1511:00, in room 738, SII  GNI II.5, III.45 GNI VI.14.  Exercise 3 
39  No lectures  
40  Backward error analysis and modified equations.  GNI IX.13.1 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 Lstability  LN: section 55.1, see also SODEII IV.2 and IV3  
Stepcontrol stability Nonlinear stability, Bstability, Algebraic stability  SODEII, p. 2427. LN 5.2 and SODEII IV.12, p. 180184.  
42  Differentialalgebraic equations. The differential index and Hessenberg forms. RKmethods for index 1 DAEs.  LN 7.13 and SODEII: VI.1,3  Exercise 5 
43  No lectures.  
44  Linear multistep methods (self study) Higher index DAEs. BDF methods. Coordinate projection. RKmethods for index 2 problems.  LN: 6 and 7. SODEII p.470474.  
Exponential integrators. ETD and IF (Lawson)  Note by Elena Celledoni, Chapter 2, A review of exponential integrators for first order semilinear problems by Minchev and Wright.  
Exponential integrators and IMEX methods.  
4546  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 (WagnerPlaten and Bseries), 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: MeanSquare and Asymptotic Stability of the Stochastic Theta method Slides on stochastic Bseries, 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 timeintegration 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.