LFU:online

Absolventinnen und Absolventen dieses Moduls haben einen Überblick über einige aktuelle Fragestellungen der höheren numerischen Mathematik und die Methoden zu deren Behandlung erworben. Weiters haben sie ein vertieftes Verständnis für das Gebiet der numerischen Mathematik erlangt und sind in der Lage, typische Probleme dieses Fachgebietes zu analysieren und zu lösen.

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++

Vortrag, Beurteilung aufgrund eines einzigen Prüfungsaktes am Ende der Lehrveranstaltung.

Lehrveranstaltungsprüfung gemäß § 7 Satzungsteil, Studienrechtliche Bestimmungen