# MA8404 - Numerical solution of time dependent differential equations

## Latest News

- The results have been handed in. There were 3 A's, 6 B's and 1 D. Candidates (10001, 10011, 10012) got A, 10005 got D, the rest got B.
- The location of the exam has been changed. It will take place in H3 (Hovedbygget), December 19, 15-19.

## Final exam

The final exam is written and will be held Monday December 19, 15-19 in room H3 (Hovedbygget). Aids: All the written material listed as lectured material, your course notes, a simple calculator, and the Rottmann formula book.

## Lectures

4 hours per week (2x2).

- Tuesday 10.15-12.00 in room 734, Sentralbygg 2.
- Thursday 08.15-10.00 in room 656, Sentralbygg 2.

First lecture, Thursday August 25, room 656, Sentralbygg 2.

## Lecturer

Brynjulf Owren, room 1350, SB2. Phone: (735)93518, email: bryn(at)math(dot)ntnu(dot)no

## Course description

The course is about numerical methods for solving ordinary and time dependent partial differential equations. We focus on modern, advanced methods. Emphasis will be put on equations which have particular features or structure for which numerical schemes should be be tailored. This may include equations with invariants, problems from mechanics, DAE problems, multiscale problems, highly oscillatory and stiff equations etc, the focus will also depend on the particular interests of the students. There will be a mandatory project which counts towards the final grade, this project will be individual and may to some extent be chosen by the student.

## Project

Everyone must complete a project which will count towards the final grade of the course. You may work alone or in groups, size of groups is preferably 1-2, but 3 may also be accepted. When marking the project, factors like group size and difficulty/size of project will be taken into account. You may come up with your own project proposal, if you are a PhD student I would like you to discuss the project with your supervisor. If you prefer, you can also do one of the enclosed projects in "Forslag.pdf".

Write a short report where you explain the methods you have used, and show a few carefully selected numerical experiments. We may decide to do also an informal oral presentation at the end of the term.

## Material lectured so far

##### Hairer, Lubich, and Wanner, Geometric Numerical Integration, Springer 2006, 2nd edition

The book is available in electronic form Springer

- Chapter I. Consider everything background or known material.
- Chapter II. Everything has been covered.
- Chapter III. Subsections 1.1 and 1.2.
- Chapter IV. Sections 1, 2.
- Chapter V. Sections 1, 2, 3 and 5.
- Chapter VI. Sections 1, 2, 3, 4, 6, 7.1 (p. 212-213 only)
- Chapter IX, Sections 1, 2, 3.1, 9.1, 9.2, 9.3 (sec 9, only cursory)

##### Hairer and Wanner, Solving Ordinary Differential Equations II, 2nd revised ed. 1996

The book is available in electronic form Springer

- Chapter IV.1 (background material)
- Chapter IV.2 (some material from here, but skip stiffess detection, and stepsize control)
- Chapter IV.3 Everything, but cursory
- Chapter IV.5 (only pp 71-77)
- Chapter IV.6
- Chapter IV.7 (pp 102-104), cursory on order conditions and stability. Nothing on implementation, "the Hump", and W-methods.
- Chapter IV.15 Here we have studied order behaviour the model problem (Prothero-Robinson) pages 225-227. Nothing on general B-convergence theory, but briefly on the last paragraph "Order reduction for Rosenbrock methods".

##### Exponential integrators

A brief introduction to the topic. I have used parts of the following articles

- Hochbruck and Ostermann,
*Exponential integrators,*Acta Numerica 2010, pp. 209–286. [link] - Minchev and Wright,
*A review of exponential integrators for first order semi-linear problems*, NTNU report 2/2005, Department of Mathematical Sciences. [link] - Berland, Owren, Skaflestad,
*B-series and Order Conditions for Exponential Integrators,*Siam Journal on Numerical Analysis**43**, pp 1715-1727 [link]