702722 Introduction to Higher Numerical Mathematics: Numerical methods for partial differential equations

winter semester 2014/2015 | Last update: 09.01.2015 Place course on memo list
702722
Introduction to Higher Numerical Mathematics: Numerical methods for partial differential equations
VO 2
4
weekly
annually
English

Graduates of this module have an overview of some current issues in higher numerical analysis and acquired the methods for their treatment. Furthermore, they gained deeper understanding of the field of numerical analysis and are able to study and solve problems of this field.

In this lecture we will introduce a number of partial differential equations (PDEs) that are important in applications. Based on these equations numerical methods are introduced, explained, and analyzed which enable the efficient numerical solution of the PDE under consideration. We consider methods for both space and time discretization. Special emphasize is devoted to the the selection of a suitable numerical method for a given class of PDEs. A more detailed outline of the lecture and the methods covered can be found below.

Syllabus

07.10 Transport equation, finite differences space discretization (stability & consistency, uniform boundedness principle, Lax equivalence theorem)

14.10 Discontinuous solutions for transport problems (method of characteristics, upwind scheme, modified equation), Burgers' equation (shock waves)

21.10 Burgers' equation (weak solutions, entropy condition, conservative schemes, Godunov's method, Godunov's theorem)

28.10 Poisson's equation (weak solution, Finite element method with hat functions in one dimension)

04.11 Poisson's equation (finite element method, weak solution, Lax-Milgram theorem, conforming elements, L^{2} optimality)

11.11 Poisson's equation (finite element method of order 1 on triangles in 2D, boundary conditions)

18.11 Heat equation (Fourier series, smoothing property, semigroups)

25.11 Heat equation with non-constant diffusivity (Runge-Kutta finite difference method, CFL condition, implicit methods, collocation methods, Radau methods, Backward differentiation formula)

02.12 Stability of numerical schemes (stability region, A-stability, error analysis for implicit Euler)

09.12 Vlasov-Poisson equation (splitting time discretization, semi-Lagrange space discretization)

16.12 Semilinear problems in chemical kinetics (variations of constants formula, exponential Euler & Rosenbrock methods)

12.01 Overview of the activities of the work group (instead of the PS)

13.01 Semilinear problems including the Laplacian (Fast Fourier transform, splitting time discretization)

20.01 Maxwell's equation (physical motivation, theory, staggered grid space discretization)

27.01 Maxwell's equation (staggered grid space discretization, Gauss RK methods)

03.02 Elements of GPU programming in C++

Lecture, assessment is based on a single oral examination at the end of the course.

Course examination according to § 7, statute section on "study-law regulations"

07.10.2014
Group 0
Date Time Location
Tue 2014-10-07
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-10-14
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-10-21
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-10-28
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-11-04
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-11-11
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-11-18
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-11-25
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-12-02
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-12-09
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2014-12-16
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2015-01-13
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2015-01-20
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2015-01-27
14.15 - 16.00 HS 10 HS 10 Barrier-free
Tue 2015-02-03
14.15 - 16.00 HS 10 HS 10 Barrier-free