Numerics II

News

• The results of the make-up exam can be found here.
• The results of the exam can be found here.
• The morning lecture on February 7 is skipped in light of the written exam on the same day.
• Section 5.4.3 Generalized minimal residual method (GMRes) has been revised.
• The 12th problem set is online.
• The exam will take place on Monday, 7 February at 14:15. There will be no lecture in the morning.
• The 11th problem set is online.
• The 10th problem set is online.
• The nineth problem set is online.
• Section 1.9 Extrapolation Methods has been revised (special thanks to Vincent Dallmer).
• New reference added: P. Kunkel and V. Mehrmann: Differential-algebraic equations
• The eigth problem set is online.
• The seventh problem set is online.
• The sixth problem set is online.
• Due to the rapidly increasing number of Covid-19 infections, we came to the conclusion that switching to a non-virtual environment is not responsible at this point. Therefore, all lectures and tutorials will remain online for the time being.
• The fifth problem set is online.
• The fourth problem set is online.
• The third problem set is online.
• The second problem set is online.
• The first problem set is online.
• The introductory notes from the first lectures can be found here
• Starting from December, we plan to have lectures and the tutorial sessions as regular (non-online) meetings at the institute. The usual measures will apply (Proof of vaccination, proof of recovery, or recent negative test).

Dates

General Information

Description

Extending basic knowledge on odes from Numerics I, we first concentrate on one-step methods for stiff and differential-algebraic systems and then discuss Hamiltonian systems. In the second part of the lecture we consider the iterative solution of large linear systems.

Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS.

Prerequisites

Basics of calculus (Analysis I, II), linear algebra (Lineare Algebra I, II) and numerical analysis (Numerik I).

Registration

All participants should register at the Whiteboard. In addition, depending on your program of study, you have to register in the Campus Management.

Exercises and Criteria for a Certificate

Tutorial & Exercises

• The tutorial offers the possibility to discuss and better understand the presented material and exercises. Furthermore solutions of exercises are presented by the students.
• Each week a sheet with exercises will be made available electronically on this web page (see below).
• The exercises are intended to be solved by teams of three members.
• The exercises consist of theoretical and numerical problems. The latter should be solved using Python 3 (or Matlab, if you must). Both types of exercises are rated seperately.
• The solutions have to be finished before the tutorial a week after they were handed out.
• The solutions of the numerical problems should be delivered by email to l.hinrichsen@fu-berlin.de. Note that a complete solution for a numerical experiment consists of a running code, a program executing the required test runs, and protocols of the execution of these test runs. Delivering a correct and running code without knowing what is going on in the code will be regarded and rated as attempt of deception.
• The solutions of the theoretical problems have to be presented in the tutorial on request (without regard to preference and presence) by a member of a given group.

Criteria for the Certificate

• Passing the exam.
• Regular participation in the tutorials, i.e. 85% attendance at the tutorials (not checked).
• Active participation: at least as much success as failure in the presentation of the solution of theoretical problems and 50% of the maximal number of points for numerical problems.
• The grade of this course is based only on the result of the exam.

Exercises

Here you'll find problem sets which will help you to understand the material.

• 1st problem set
• 2nd problem set
• 3rd problem set
• 4th problem set
• 5th problem set
• 6th problem set
• 7th problem set
• 8th problem set
• 9th problem set
• 10th problem set (Note that you can do the programming part in Python, of course).
• 11th problem set
• 12th problem set (Fixed due date)

Accompanying Material

Lecture Notes

The lecture notes are available here. Lecture notes for Numerik I are also available here (only in German).

Papers

• P. Kaps, P. Rentrop: Generalized Runge-Kutta Methods of Order Four with Stepsize Control for Stiff Ordinary Differential Equations Numer. Math. 33(1), pp. 55-68 (1979).
• E. Hairer, Ch. Lubich: Asymptotic Expansions of the Global Error of Fixed-Stepsize Methods Numer. Math. 45(3), pp. 345-360 (1984)

Python

• Think Python 2nd Ed. (Free introduction into programming in general and Python in particular).
• Numpy for Matlab users (Helps you to transit from Matlab to Python (or rather, Numpy).

Literature

The following selection of textbooks can be found in the library.

1. Deuflhard, Peter: Newton Methods for Nonlinear Problems. Springer, Berlin, 2004.
2. Deuflhard, Peter und Folkmar Bornemann: Numerische Mathematik II - Gewöhnliche Differentialgleichungen. Walter de Gruyter, Berlin, 2002.
3. Deuflhard, Peter und Folkmar Bornemann: Scientific computing with ordinary differential equations. Springer, Berlin, 2002.
4. Deuflhard, Peter und Andreas Hohmann: Numerische Mathematik I - Eine algorithmisch orientierte Einführung. Walter de Gruyter, Berlin, 2002.
5. Golub, Gene und Charles Van Loan: Matrix computations. Johns-Hopkins-University Press, Baltimore, 1993.
6. Hairer, Ernst, Syvert Paul Nørsett und Gerhard Wanner: Solving Ordinary Differential Equations I - Nonstiff Problems, Band 8 der Reihe Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, New York, 1987.
7. Hairer, Ernst und Gerhard Wanner: Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems, Band 14 der Reihe Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, New York, 1991.
8. Kunkel, Peter, and Mehrmann, Volker. Differential-algebraic equations: analysis and numerical solution. Vol. 2. European Mathematical Society, 2006.
9. Meister, Andreas: Numerik linearer Gleichungssysteme. Vieweg, Braunschweig, 1999.
10. Ortega, James und Werner Rheinboldt: Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1972.
11. Quarteroni, Alfio, Riccardo Sacco und Fausto Saleri: Numerische Mathematik 1. Springer, Berlin, 2002.
12. Quarteroni, Alfio, Riccardo Sacco und Fausto Saleri: Numerische Mathematik 2. Springer, Berlin, 2002.
13. Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 1. Springer, Berlin, 2005. aus dem FU-Netz auch Online verfügbar: Springer Link
14. Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 2. Springer, Berlin, 2005. aus dem FU-Netz auch Online verfügbar: Springer Link
15. Walter, Wolfgang: Gewöhnliche Differentialgleichungen - eine Einführung. Springer, Berlin, 1996.
16. Werner, Dirk: Funktionalanalysis. Springer, Berlin, 2000.

Further literature on numerics is deposited under the shelf marks H.1.0 and H.1.1.

Introductory literature in computer science can be found under XA.1, introductions in Unix operating system under XD.4.0.

Contact