### Scientific Computing II: Introduction to Dynamical Systems

Fall/Winter 2019, Otto-von-Guericke Universität Magdeburg

Details:

Vorlesungsverzeichnis

Lecturer:

Christian Lessig

Lectures: Tuesdays, 15:00-17:00, G05-300

Tutorials: Thursdays, 15:00-17:00; G29-426

*Content:* The course provides the fundamentals to describe the time evolution of ordinary and partial differential equations, e.g. time integrators, the finite element method, and spectral methods.

- No tutorial on 21/11/2019.
- See the Vorlesungsverzeichnis for details. Note that the course is cross listed and you can take it as a B.Sc. and M.Sc. student (again see the LSF for the details of the offering). For further questions please get in touch<./li>

Week 1:

Recap linear algebra; dual spaces, dual bases and Riesz representation theorem

Week 2:

Motivation: description of transport by advection equation; analytic solution

Week 3:

Finite difference methods for advection equation

Week 4:

Courant-Friedrichs-Lewy stability condition

Upwind scheme code
Courant-Friedrichs-Lewy's original paper

Week 5:

Consistency, stability, and convergence: Lax-Richtmyer theorem; Galerkin projection; spectral methods

Week 6:

Von Neumann stability analysis; Galerkin projection; spectral methods

Week 7:

Finite element method: fundamentals

Week 8:

Wave equation, theory and finite element formulation

Week 9:

Wave equation, dispersion relationship and stability

Week 10:

Wwave equation, boundary conditions; heat equation, spectral ansatz

Week 1:

Linear algebra recap; implementation of biorthogonal bases

Week 2:

Finite dimensional function spaces and L_2 inner product

Week 3:

[holiday]

Week 4:

Implicit time stepping schemes

Skeleton code

Week 5:

CFL condition and order for Lax-Richtmyer scheme

Week 6:

[no tutorial]

Week 7:

Von Neumann stability analysis; finite element method

Week 8:

Triangular finite elements

Assignment 1:

Task
Skeleton code

Assignment 2:

Task
Skeleton code

We will be working with python and the

Numpy library in this course. On Linux you can install it using your package manager. On other operating systems it is convenient to use the

Anaconda distribution which contains all necessary packages. An introduction to python and Numpy can be found

here.

- Discrete exterior calculus course notes
- L. N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, 1996
- J. Kirkwood, Mathematical physics with partial differential equations. 2018.
- G. H. Golub and J. M. Ortega, Scientific computing and differential equations: an introduction to numerical methods. 2014.
- G. Evans, J. M. Blackledge, and P. Yardley, Numerical methods for partial differential equations. Springer, 2000.
- D. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations. London: Springer London, 2010.