Numerical Solution of Differential Equations (Numerik der Differentialgleichungen)

Grading Policy:

There will be 12 homework assignments consisting of theoretical and programming problems. A passing grade will depend on attendance in the exercise sessions. Solutions should be submitted during the exercise sessions together with a note about which problems you solved.

Problem sheets

Lecture notes

Aims of the course

Introduce students of Applied Mathematics, Computational Science and Engineering and Computer Science to the most widely used numerical solution methods for ordinary and partial differential equations, their mathematical properties, and their computer implementation.

Students should be able to:

• identify types of differential equations,
• know the main discretization techniques, their scope and limitations,
• identify sources of simulation error: Modelling, Data and discretization errors,
• diagnose numerical instability
• be in a position to code Finite Element and Finite Volume solvers for stationary and transient problems in general two-dimensional domains.

Programming Language: All implementations will be done in MATLAB.

Content of the course

• The finite element method (FEM)
• Special elliptic boundary value problems
• Parabolic boundary value problems
• Numerical methods for conservation laws
• Adaptive finite element schemes

Related lectures

• Num. PDE FS 2010
  
• Num. Meth. CSE 2009
• Num. Meth. CSE 2008
• Num. PDE 1         2009
  

Matlab links

Students of ETH can download Matlab via Stud-IDES for free (product name 'Matlab free').

• Matlab online documentation from MathWorks
• Matlab primer

Literature

• D. Braess, Finite Elements, 2nd Edition, Cambridge University Press, 2001.
• T. Hughes, The Finite Element Method, Dover Publications, 2000.
• C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.
• R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
• W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2003.
• A. Quarteroni, R. Sacco und F. Salieri, Numerische Mathematik, Springer, 2002.

Further literature (beyond the scope of the lecture)

• C. Canuto, M.Y. Hussaini, A. Quarteroni und T.A. Zang, Spectral Methods, Springer, 2006.
• M. Feistauer, J. Felcman, and I. Straskraba, Mathematical and Computational Methods for Compressible Flow, Oxford: Clarendon Press, 2003.
• A. Tveito und R. Winther, Einführung in Partielle Differentialgleichungen, Springer, 2002.