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.

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News

• 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>

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Lectures

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

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Tutorials

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

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Assignments

Assignment 1: Task   Skeleton code Assignment 2: Task   Skeleton code

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Software

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.

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Literature

• 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.