Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Michael Matthes
Universität zu Köln
Coupled systems as Abstract Differential-Algebraic Equations
We study so called Abstract Differential-Algebraic Equations (ADAEs) which are Differential-Algebraic
Equations (DAEs) with operators acting on infinite dimensional Hilbert or Banach spaces. In applications
these equations arise when coupling DAEs and partial differential equations (PDEs). Information is
shared between these two systems by certain coupling operators. For example specific models in circuit
simulation were already studied in the literature in the context of ADAEs or PDAEs.
While theory and numerics of DAEs and PDEs were well developed in the last 20 years there are just a
few results for coupled DAE-PDE systems concerning solvability and convergence. We study a general
prototype class of a coupled system where the infinite dimensional part of the system is governed by a
monotone operator. Investigating especially the coupling terms we give an existence and uniqueness
result of this prototype system. Also the Galerkin approach is discussed as it is important for numerical
applications.
Roland Pulch
Bergische Universität Wuppertal
Stochastic Galerkin Methods for Partial Differential Equations with Random Parameters
We consider partial differential equations (PDEs) of elliptic, parabolic and hyperbolic type. The PDEs include
parameters, which may be uncertain due to measurements or a lack of knowledge. We replace the
parameters by random variables to achieve an uncertainty quantification. The resulting stochastic model
can be resolved by a Monte-Carlo simulation, for example. Alternatively, we apply a spectral method involving
the expansion of the unknown random fields in the generalised polynomial chaos. This expansion
is based on orthogonal polynomials defined in the underlying random space. A stochastic Galerkin technique
yields a larger coupled system of PDEs satisfied by an approximation of the required coefficient
functions. We analyse the structure of the coupled systems in comparison to the original PDEs. The focus
is on hyperbolic systems of PDEs, where the coupled system of the stochastic Galerkin method is
not necessarily hyperbolic. Sufficient conditions are specified such that the coupled system inherits the
hyperbolicity. We present numerical simulations of corresponding test examples.
Literatur
Pulch, R.; van Emmerich, C. (2009). Polynomial chaos for simulating random volatilities. Math. Comput.
Simulat., 80, 245 - 255.
Pulch, R.; Xiu, D. (2011). Generalised polynomial chaos for a class of linear conservation laws. to appear
in: Journal of Scientific Computing, Springer.
Xiu, D. (2010). Numerical methods for stochastic computations: a spectral method approach. Princeton
University Press.
82