401-2654-00L Numerical Analysis II

Semester	Frühjahrssemester 2020
Dozierende	H. Ammari
Periodizität	jährlich wiederkehrende Veranstaltung
Lehrsprache	Englisch

Lehrveranstaltungen

Nummer

Titel

Umfang

Dozierende

401-2654-00 V

Numerical Analysis II

3 Std.

Mo	13:15-15:00	HG G 5 »
Fr	13:15-14:00	HG G 5 »

401-2654-00 U

Numerical 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.

Katalogdaten

Kurzbeschreibung	The 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.
Lernziel	The 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.
Inhalt	Chapter 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
Skript	Lecture notes including supplements will be provided electronically. Please find the lecture homepage here: https://www.sam.math.ethz.ch/~grsam/SS20/NAII/ All assignments and some previous exam problems will be available for download on lecture homepage.
Literatur	Note: 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 / Besonderes	Homework problems involve Python implementation of numerical algorithms.

Leistungskontrolle

Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
Im Prüfungsblock für	Bachelor-Studiengang Mathematik 2010; Ausgabe 24.02.2016 (Prüfungsblock 2) Bachelor-Studiengang Mathematik 2016; Ausgabe 25.02.2020 (Prüfungsblock 2)
ECTS Kreditpunkte	6 KP
Prüfende	H. Ammari
Form	Sessionsprüfung
Prüfungssprache	Englisch
Repetition	Die Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich.
Prüfungsmodus	schriftlich 180 Minuten
Zusatzinformation zum Prüfungsmodus	A 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 schriftlich	Excerpts 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.

Lernmaterialien


Hauptlink	Lecture homepage
Es werden nur die öffentlichen Lernmaterialien aufgeführt.

Gruppen

401-2654-00 U

Numerical Analysis II

Gruppen

G-01

G-02

G-03

G-04

Do	10:15-12:00	CHN D 46 »
20.02.	10:15-12:00	ML J 37.1 »
27.02.	10:15-12:00	ML J 37.1 »
05.03.	10:15-12:00	ML J 37.1 »
12.03.	10:15-13:00	ML J 37.1 »

G-05

G-06

Einschränkungen

Keine zusätzlichen Belegungseinschränkungen vorhanden.

Angeboten in

Studiengang	Bereich	Typ
Mathematik Bachelor	Prüfungsblock II	O