401-3663-00L  Numerical Solution of Differential Equations

SemesterHerbstsemester 2011
DozierendeP. Grohs
Periodizitätjährlich wiederkehrende Veranstaltung
LehrspracheEnglisch
KommentarThis course unit is offered for the last time in this semester (HS 2011)



Lehrveranstaltungen

NummerTitelUmfangDozierende
401-3663-00 VNumerical Solution of Differential Equations (Numerik der Differentialgleichungen)
Wird im HS 2011 letztmalig in dieser Form durchgeführt.
4 Std.
Mo13:15-15:00HG D 5.2 »
Mi10:15-12:00HG D 5.2 »
P. Grohs
401-3663-00 UNumerical Solution of Differential Equations (Numerik der Differentialgleichungen)
Wird im HS 2011 letztmalig in dieser Form durchgeführt.
2 Std.
Mo08:15-10:00HG D 5.2 »
10:15-12:00HG D 5.2 »
P. Grohs

Katalogdaten

KurzbeschreibungDerivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in one and two spatial dimensions.
LernzielMain skills to be acquired in this course:
* Ability to implement advanced numerical methods for the solution of partial differential equations efficiently
* Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations
* Ability to select and assess numerical methods in light of the predictions of theory
* Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm
* Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations.

This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages.
Inhalt1 Case Study: A Two-point Boundary Value Problem
1.1 Introduction
1.2 A model problem
1.3 Variational approach
1.4 Simplified model
1.5 Discretization
1.5.1 Galerkin discretization
1.5.2 Collocation
1.5.3 Finite differences
1.6 Convergence
2 Second-order Scalar Elliptic Boundary Value Problems
2.1 Equilibrium models
2.1.1 Taut membrane
2.1.2 Electrostatic fields
2.1.3 Quadratic minimization problems
2.2 Sobolev spaces
2.3 Variational formulations
2.4 Equilibrium models: Boundary value problems
3 Finite Element Methods (FEM)
3.1 Galerkin discretization
3.2 Case study: Triangular linear FEM in two dimensions
3.3 Building blocks of general FEM
3.4 Lagrangian FEM
3.4.1 Simplicial Lagrangian FEM
3.4.2 Tensor-product Lagrangian FEM
3.5 Implementation of FEM
3.5.1 Mesh file format
3.5.2 Mesh data structures
3.5.3 Assembly
3.5.4 Local computations and quadrature
3.5.5 Incorporation of essential boundary conditions
3.6 Parametric finite elements
3.6.1 Affine equivalence
3.6.2 Example: Quadrilaterial Lagrangian finite elements
3.6.3 Transformation techniques
3.6.4 Boundary approximation
3.7 Linearization
4 Finite Differences (FD) and Finite Volume Methods (FV)
4.1 Finite differences
4.2 Finite volume methods (FVM)
5 Convergence and Accuracy
5.1 Galerkin error estimates
5.2 Empirical Convergence of FEM
5.3 Finite element error estimates
5.4 Elliptic regularity theory
5.5 Variational crimes
5.6 Duality techniques
5.7 Discrete maximum principle
6 2nd-Order Linear Evolution Problems
6.1 Parabolic initial-boundary value problems
6.1.1 Heat equation
6.1.2 Spatial variational formulation
6.1.3 Method of lines
6.1.4 Timestepping
6.1.5 Convergence
6.2 Wave equations
6.2.1 Vibrating membrane
6.2.2 Wave propagation
6.2.3 Method of lines
6.2.4 Timestepping
6.2.5 CFL-condition
7 Convection-Diffusion Problems
7.1 Heat conduction in a fluid
7.1.1 Modelling fluid flow
7.1.2 Heat convection and diffusion
7.1.3 Incompressible fluids
7.1.4 Transient heat conduction
7.2 Stationary convection-diffusion problems
7.2.1 Singular perturbation
7.2.2 Upwinding
7.3 Transient convection-diffusion BVP
7.3.1 Method of lines
7.3.2 Transport equation
7.3.3 Lagrangian split-step method
7.3.4 Semi-Lagrangian method
8 Numerical Methods for Conservation Laws
8.1 Conservation laws: Examples
8.2 Scalar conservation laws in 1D
8.3 Conservative finite volume discretization
8.3.1 Semi-discrete conservation form
8.3.2 Discrete conservation property
8.3.3 Numerical flux functions
8.3.4 Montone schemes
8.4 Timestepping
8.4.1 Linear stability
8.4.2 CFL-condition
8.4.3 Convergence
8.5 Higher order conservative schemes
8.5.1 Slope limiting
8.5.2 MUSCL scheme
9 Finite Elements for the Stokes Equations
9.1 Viscous fluid flow
9.2 The Stokes equations
9.3 Saddle point problems: Galerkin discretization
9.4 The Taylor-Hood element
SkriptLecture slides are available
at Link.
LiteraturChapters of the following books provide SUPPLEMENTARY reading
(Detailed references in course material):

• D. Braess. Finite Elements. Cambridge University Press, 2nd edition, 2001.
• S. Brenner and R. Scott. Mathematical theory of finite element methods. Texts in Applied
Mathematics. Springer–Verlag, New York, 1994.
• A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied
Mathematical Sciences. Springer, New York, 2004.
• Ch. Großmann and H.-G. Roos. Numerik partieller Differentialgleichungen. Teubner
Studienbücher Mathematik. Teubner, Stuttgart, 1992.
• W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of
Springer Series in Computational Mathematics. Springer, Berlin, 1992.
• P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential
Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
• S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of
Texts in Applied Mathematics. Springer, Heidelberg, 2003.
* R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2002.
Voraussetzungen / BesonderesHomework asssignments involve substantial coding, partly based on a finite element library. The examination will be computer based and will comprise coding tasks.

Leistungskontrolle

Information zur Leistungskontrolle (gültig bis die Lerneinheit neu gelesen wird)
Leistungskontrolle als Semesterkurs
Im Prüfungsblock fürBachelor-Studiengang Rechnergestützte Wissenschaften 2005; Ausgabe 17.06.2008 (Prüfungsblock Kernfächer)
Bachelor-Studiengang Rechnergestützte Wissenschaften 2008; Ausgabe 01.08.2014 (Prüfungsblock Kernfächer)
ECTS Kreditpunkte12 KP
PrüfendeP. Grohs
FormSessionsprüfung
PrüfungsspracheEnglisch
RepetitionDie Leistungskontrolle wird in jeder Session angeboten. Die Repetition ist ohne erneute Belegung der Lerneinheit möglich.
ZulassungsbedingungRegular attendance in tutorial classes.
Prüfungsmodusschriftlich 180 Minuten
Zusatzinformation zum PrüfungsmodusComputer based examination.
Hilfsmittel schriftlichLecture slides and software tools will be provided electronically. No other materials 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

Keine öffentlichen Lernmaterialien verfügbar.
Es werden nur die öffentlichen Lernmaterialien aufgeführt.

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Biomedical Engineering MasterEmpfohlene WahlfächerWInformation
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Physik BachelorAuswahl an Lehrveranstaltungen aus höheren SemesternWInformation
Physik MasterAuswahl: MathematikWInformation
Rechnergestützte Wissenschaften BachelorKernfächerOInformation