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Advanced Topics in Numer. Analysis 2 / Num. Math. III (Summer Semester 2007)

Lecture: Wednesday 8:00-9:30 Mittl. HS Raum 150 Geb. 10.91 Begin: 18.4.2007, End: 19.7.2007
Thursday 9:45-11:15 Mittl. HS Raum 150 Geb. 10.91

This course investigates the following advanced topics:

I. Numerical Treatment of Differential Equations: Ordinary/Partial Differential Equations (ODEs/PDEs), linear PDEs of order two (classification: elliptic, hyperbolic,parabolic type), model equations, examples, fundamental principles of numerical methods, finite differences, variational methods.

II. Boundary-Value and Eigenvalue Problems of ODEs: The boundary-value problem of Sturm, shooting method, multiple shooting, methods of finite differences, variational method, Ritz method, spline functions, comparison of the methods, eigenvalue problem of Sturm-Liouville, Euler's breaking load.

III. Boundary-Value Problems of Elliptic PDEs: Finite difference methods for Dirichlet problems, Laplace operator, Poisson equation, Dirichlet and Neumann problems of the Poisson equation, stability, solution of the finite difference equations, block SOR, cg-method, variational method and finite elements, Ritz method, ansatz functions.

IV. Initial- and Boundary-Value Problems of Hyperbolic and Prabolic PDEs: Finite difference methods for hyperbolic problems, wave equation, the CFL condition, conservation laws, Burgers equation, finite difference methods for parabolic problems, heat equation, order of convergence, Crank-Nickolson, method of lines, semi-discretization.


C. Großmann, H.-G. Roos: Numerische Behandlung partieller Differentialgleichungen (3rd ed.). Teubner, 2005.
W. Hackbusch: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1994.
M. Hanke-Bourgeois: Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens. Teubner, 2002.
P. Knabner, L. Angermann: Numerical methods for elliptic and parabolic partial differential equations. Translation from the German. Springer, 2003.
R.J. LeVeque: Numerical Methods for Conservation Laws. Birkhäuser, 1992.
K.W. Morton, D.F. Mayers: Numerical solution of partial differential equations. An introduction (2nd ed). Cambridge University Press, 2005.
A. Quarteroni, A. Valli: Numerical Approximation of Partial Differential Equations. Springer,1997.
H.R. Schwarz: Numerische Mathematik. Teubner, 1986.
J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (2nd ed.). Springer, 1996.
J.W. Thomas: Numerical partial differential equation. Conservation laws and elliptic equations.
Springer, 1999.

Scriptum appears after the progress in the class!