Scientific Computing II: Introduction to Dynamical Systems
Fall/Winter 2019, Otto-von-Guericke Universität Magdeburg
Lecturer: Christian Lessig
Lectures: Tuesdays, 15:00-17:00, G05-300
Tutorials: Thursdays, 15:00-17:00; G29-426
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>
Recap linear algebra; dual spaces, dual bases and Riesz representation theorem
Motivation: description of transport by advection equation; analytic solution
Finite difference methods for advection equation
Courant-Friedrichs-Lewy stability condition
Upwind scheme code
Courant-Friedrichs-Lewy's original paper
Consistency, stability, and convergence: Lax-Richtmyer theorem; Galerkin projection; spectral methods
Von Neumann stability analysis; Galerkin projection; spectral methods
Finite element method: fundamentals
Wave equation, theory and finite element formulation
Wave equation, dispersion relationship and stability
Wwave equation, boundary conditions; heat equation, spectral ansatz
Linear algebra recap; implementation of biorthogonal bases
Finite dimensional function spaces and L_2 inner product
Implicit time stepping schemes
CFL condition and order for Lax-Richtmyer scheme
Von Neumann stability analysis; finite element method
Triangular finite elements
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.