Numerical Methods for Partial Differential Equations II - Einzelansicht

Many processes in science and engineering are governed by conservation laws. These conservation laws state that the temporal change of a conserved quantity in a given volume is equal to the integral of the flux of this quantity across the volume's boundary. When the flux is a function of the conserved quantities, these conservation laws lead to a system of first-order partial differential equations. Some examples among many others are the Euler equations in fluid dynamics or the modelling of traffic flow.

In this lecture we will be concerned with the analysis and numerical approximation of first-order partial differential equations arising from conservation laws. In particular we will study so-called Finite Volume methods, which are based on a discrete formulation of the conservation laws on a finite set of control volumes covering the computational domain. In the accompanying exercises, in addition to deepening our theoretical knowledge, we will gain practical experience with the implementation of finite volume schemes using the Python programming language.

• T. Barth und M. Ohlberger. Finite volume methods: foundation and analysis. In T.J.R. Hughes E. Stein, R. de Borst, editor, Encyclopedia of Computational Mechanics , volume 1, chapter 15. John Wiley & Sons, Ltd, 2004.
• R. Eymard, T. Galluoët und R. Herbin. Finite volume methods. In Handbook of numerical analysis, Vol. VII , pages 713-1020. North-Holland, Amsterdam, 2000.
• D. Kröner. Numerical schemes for conservation laws . Wiley-Teubner Series Advances in Numerical Mathematics. John Wiley & Sons Ltd., Chichester, 1997.

Some familiarity with partial differential equations and their numerical approximation (e.g. Finite Element method for elliptic equations) is helpful but not required. Basic knowledge of the Python programming language is required to follow the programming exercises.