Search result: Catalogue data in Spring Semester 2016
|Computational Science and Engineering Bachelor|
|401-0674-00L||Numerical Methods for Partial Differential Equations|
Not meant for BSc/MSc students of mathematics.
|O||8 credits||4V + 2U + 1A||R. Hiptmair|
|Abstract||Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in Python in one dimension and in C++ in 2D.|
|Objective||Main 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.
* Skills in the efficient implementation of finite element methods on unstructured meshes.
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.
|Content||1 Case Study: A Two-point Boundary Value Problem|
1.2 A model problem
1.3 Variational approach
1.4 Simplified model
1.5.1 Galerkin discretization
1.5.2 Collocation [optional]
1.5.3 Finite differences
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 in C++
3.5.1 Mesh file format (Gmsh)
3.5.2 Mesh data structures (DUNE)
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 [optional]
4 Finite Differences (FD) and Finite Volume Methods (FV) [optional]
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 [optional]
5.7 Discrete maximum principle [optional]
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
6.2 Wave equations [optional]
6.2.1 Vibrating membrane
6.2.2 Wave propagation
6.2.3 Method of lines
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
7.3 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
8.4.1 Linear stability
8.5 Higher order conservative schemes [optional]
8.5.1 Slope limiting
8.5.2 MUSCL scheme
8.6. FV-schemes for systems of conservation laws [optional]
|Lecture notes||Lecture documents and classroom notes will be made available to the audience as PDF.|
|Literature||Chapters of the following books provide SUPPLEMENTARY reading|
(Detailed references in course material):
* D. Braess: Finite Elemente,
Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer 2007 (available online)
* S. Brenner and R. Scott. Mathematical theory of finite element methods, Springer 2008 (available online)
* 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: Numerical Treatment of Partial Differential Equations, Springer 2007
* 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.
However, study of supplementary literature is not important for for following the course.
|Prerequisites / Notice||Mastery of basic calculus and linear algebra is taken for granted.|
Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential.
Important: Coding skills in MATLAB and C++ are essential.
Homework asssignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.
|529-0431-00L||Physical Chemistry III: Molecular Quantum Mechanics||O||4 credits||4G||B. H. Meier, M. Ernst|
|Abstract||Postulates of quantum mechanics, operator algebra, Schrödinger's equation, state functions and expectation values, matrix representation of operators, particle in a box, tunneling, harmonic oscillator, molecular vibrations, angular momentum and spin, generalised Pauli principle, perturbation theory, electronic structure of atoms and molecules, Born-Oppenheimer approximation.|
|Objective||This is an introductory course in quantum mechanics. The course starts with an overview of the fundamental concepts of quantum mechanics and introduces the mathematical formalism. The postulates and theorems of quantum mechanics are discussed in the context of experimental and numerical determination of physical quantities. The course develops the tools necessary for the understanding and calculation of elementary quantum phenomena in atoms and molecules.|
|Content||Postulates and theorems of quantum mechanics: operator algebra, Schrödinger's equation, state functions and expectation values. Linear motions: free particles, particle in a box, quantum mechanical tunneling, the harmonic oscillator and molecular vibrations. Angular momentum: electronic spin and orbital motion, molecular rotations. Electronic structure of atoms and molecules: the Pauli principle, angular momentum coupling, the Born-Oppenheimer approximation. Variational principle and perturbation theory. Discussion of bigger systems (solids, nano-structures).|
|Lecture notes||A script written in German will be distributed. The script is, however, no replacement for personal notes during the lecture and does not cover all aspects discussed.|
|227-0014-00L||Computer Engineering II||O||4 credits||2V + 2U||R. Wattenhofer|
|Abstract||We learn the important functions of operating systems. Networking: IP, routing, transport, flows, applications, sockets, link and physical layer, Markov chains, PageRank, security. Storage: memory hierarchy, file systems, caching, hashing, data bases. Computation: virtualization, processes, threads, concurrency, scheduling, locking, synchronization, mutual exclusion, deadlocks, consistency.|
|Content||Computers come in all shapes and sizes: servers, laptops, tablets, smartphones, smartwatches, all the way down to that tiny microcontroller in a washing machine. People buy a computer because (i) it gives them access to the Internet, (ii) it provides storage, and probably also because (iii) it computes. While having network access seems to be vital, advanced storage and computing capabilities more and more move to designated servers ("the cloud"). In this lecture, we learn how computers provide networking, storage, and computation by means of an operating system.|
We start out with networking, and discuss the internet protocol, addressing, routing, transport layer protocols, flows, some representative application layer protocols, and how to implement these with sockets. We also discuss the link and physical layer, Markov chains and PageRank, and selected topics in security. Regarding storage, we talk about the memory hierarchy, file systems, caching, efficient data structures such as hashing, and data base principles. Concerning computation, we discuss the virtualization of the processing units with processes and threads. We focus on concurrency and examine scheduling, locking, synchronization, mutual exclusion, deadlocks, and consistency.
The lecture will use various teaching paradigms. The majority of the lecture will be based on blackboard discussions, supported by a script. Where appropriate we will also use slides or demonstrations. A few lectures will be flipped classroom style. The lecture will feature weekly paper exercises.
However, some of the course material is best learned in front of an actual computer. In addition to the lecture we offer exciting hands-on exercises in a lab environment.
- Page 1 of 1