|Stochastic Models for Nonlinear Convection-Dominated Flows|
|Project manager:||Prof. Dr. Christian Rohde|
|Deputy:||Prof. (jun.) Dr.-Ing. Wolfgang Nowak, M.Sc.|
|Duration:||1.9.2008 - 31.8.2011|
This project is part of the research area:
Stochastic modelling of subsurface flow and transport processes
Abstract:For many application problems in the natural and technical sciences the behavior of macroscopic
quantities strongly depends on small time-dependent influences and the structure of the underlying
medium. The large variability of heterogeneities on different spatial and temporal scales seems
to be hard to capture using purely deterministic mathematical models. Specific examples of
such situations are the solute transport in complex geological formations, macroscopic transport
phenomena in tumors, phase transformation processes in materials science, or the dynamics of
atmospheric flows. From the viewpoint of modeling a promising alternative is the use of stochastic
evolution equations. In the last two decades there has been substantial progress in theory and
numerics for probabilistic partial differential equations, in particular for linear elliptic equations.
However, for nonlinear and/or convection dominated problems the field is still in its infancy.
In this project we plan to develop, implement, and analyze higher-order Discontinuous-Galerkin methods
(DG) for a class of nonlinear convection-dominated evolution equations including random
processes in time and space. Time-discretizations for Wiener processes appropriate for high-order
space discretization will be constructed. To reduce the extreme computational expenses
associated with spatial noise when employing classical Monte-Carlo-methods (MC) we shall use
the Karhunen-Loeve based moment-equation approach (KLME) for the spatial noise. This has to
be integrated in the framework of DG-methods for the evolution equation itself. Technically we
want to construct a dynamical a-posteriori error control that governs the mesh parameter, the
order of the DG-method, and the finite dimension of the stochastic parameterization.
The overall goal on the numerical side is to apply the algorithm to a number of realistic subsurface
flow problems in a heterogeneous randomly given geological formation. Concerning analysis we
seek to obtain well-posedness results for simple model problems and first a-priori/a-posteriori error
estimates for the used approximation methods.