401-2654-00L  Numerical Analysis II

SemesterFrühjahrssemester 2020
DozierendeH. Ammari
Periodizitätjährlich wiederkehrende Veranstaltung


401-2654-00 VNumerical Analysis II3 Std.
Mo13:15-15:00HG G 5 »
Fr13:15-14:00HG G 5 »
H. Ammari
401-2654-00 UNumerical Analysis II
Gruppeneinteilung erfolgt über myStudies.
Thu 10-12 or Thu 13-15 as allocated.
Students who registered for MMP II take the slot Thu 13-15
2 Std.
Do10:15-12:00CHN D 46 »
10:15-12:00HG G 26.1 »
10:15-12:00HG G 26.5 »
10:15-12:00LFW B 3 »
13:15-15:00HG F 26.5 »
13:15-15:00ML H 34.3 »
20.02.10:15-12:00ML J 37.1 »
27.02.10:15-12:00ML J 37.1 »
05.03.10:15-12:00ML J 37.1 »
12.03.10:15-13:00ML J 37.1 »
H. Ammari


KurzbeschreibungThe central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation.
LernzielThe course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments.
InhaltChapter 1. Some basics
1.1. What is a differential equation?
1.2. Some methods of resolution
1.3. Important examples of ODEs
Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case
2.1. Banach fixed point theorem
2.2. Gronwall’s lemma
2.3. Cauchy-Lipschitz theorem
2.4. Stability
2.5. Regularity
Chapter 3. Linear systems
3.1. Exponential of a matrix
3.2. Linear systems with constant coefficients
3.3. Linear system with non-constant real coefficients
3.4. Second order linear equations
3.5. Linearization and stability for autonomous systems
3.6 Periodic Linear Systems
Chapter 4. Numerical solution of ordinary differential equations
4.1. Introduction
4.2. The general explicit one-step method
4.3. Example of linear systems
4.4. Runge-Kutta methods
4.5. Multi-step methods
4.6. Stiff equations and systems
4.7. Perturbation theories for differential equations
Chapter 5. Geometrical numerical integration methods for differential equation
5.1. Introduction
5.2. Structure preserving methods for Hamiltonian systems
5.3. Runge-Kutta methods
5.4. Long-time behaviour of numerical solutions
Chapter 6. Finite difference methods
6.1. Introduction
6.2. Numerical algorithms for the heat equation
6.3. Numerical algorithms for the wave equation
6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension
Chapter 7. Stochastic differential equations
7.1. Introduction
7.2. Langevin equation
7.3. Ornstein-Uhlenbeck equation
7.4. Existence and uniqueness of solutions in dimension one
7.5. Numerical solution of stochastic differential equations
SkriptLecture notes including supplements will be provided electronically.

Please find the lecture homepage here:


All assignments and some previous exam problems will be available for download on lecture homepage.
LiteraturNote: Extra reading is not considered important for understanding the
course subjects.

Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994.

Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996.

Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002.

L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009.

Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993.

Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972.

Walter: Ordinary differential equations, Springer-Verlag, New York, 1998.
Voraussetzungen / BesonderesHomework problems involve Python implementation of numerical algorithms.


Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
Im Prüfungsblock fürBachelor-Studiengang Mathematik 2010; Ausgabe 24.02.2016 (Prüfungsblock 2)
Bachelor-Studiengang Mathematik 2016; Ausgabe 25.02.2020 (Prüfungsblock 2)
ECTS Kreditpunkte6 KP
PrüfendeH. Ammari
RepetitionDie Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich.
Prüfungsmodusschriftlich 180 Minuten
Zusatzinformation zum PrüfungsmodusA mid-term (60 points) and an end-term exam (60 points) will be held during the
teaching period on dates specified in the beginning of the semester. Both
non-mandatory exams are without aids and test only standard knowledge and
routine skills. Neither for the mid-term nor for the end-term exam will a
repetition be offered. These two exams will be considered as learning tasks.
The points gained in the mid-term (say x points) and end-term (say y points)
exams give a bonus of 0.25 if x+y≥80 and 0 otherwise.
The session exam comprises theoretical problems (to be solved on paper) as well
as implementation problems (Python, to be executed on a computer that is
provided by ETH).
Hilfsmittel schriftlichExcerpts from the lecture notes are provided electronically. No additional aids are allowed.
Falls die Lerneinheit innerhalb eines Prüfungsblockes geprüft wird, werden die Kreditpunkte für den gesamten bestandenen Block erteilt.
Diese Angaben können noch zu Semesterbeginn aktualisiert werden; verbindlich sind die Angaben auf dem Prüfungsplan.


HauptlinkLecture homepage
Es werden nur die öffentlichen Lernmaterialien aufgeführt.


401-2654-00 UNumerical Analysis II
Do10:15-12:00HG G 26.1 »
Do10:15-12:00HG G 26.5 »
Do10:15-12:00LFW B 3 »
Do10:15-12:00CHN D 46 »
20.02.10:15-12:00ML J 37.1 »
27.02.10:15-12:00ML J 37.1 »
05.03.10:15-12:00ML J 37.1 »
12.03.10:15-13:00ML J 37.1 »
Do13:15-15:00HG F 26.5 »
Do13:15-15:00ML H 34.3 »


Keine zusätzlichen Belegungseinschränkungen vorhanden.

Angeboten in

Mathematik BachelorPrüfungsblock IIOInformation