# Forschungsgruppe ''Numerische Mathematik und Wissenschaftliches Rechnen''

## Seminar Numerische Mathematik / Numerical mathematics seminars*

aktuelles Programm / current program

**Matheon - Special Guest Lecture 2010:**

Prof. Dr. A. Ern (CERMICS, France) Discontinuous Galerkin methods

Prof. Dr. A. Ern (CERMICS, France) Discontinuous Galerkin methods

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Donnerstag, 16. 12. 2010, 14:00 Uhr (ESH)
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A. Kulikovsky(Forschungszentrum Juelich GmbH)

Analytical modeling of fuel cells, cell components and stacks

Abstract:

We report an overview of recent work on analytical modeling of catalyst layers, fuel cells and stacks. A generalized Perry-Newman-Cairns model for the catalyst layer (CL) with solid electrolyte enables to understand the regimes of CL operation and gives the general polarization curve of the CL. A model of CL degradation based on the PNC-model and the features of CL operation in PEMFC, DMFC and SOFC will be discussed. A quasi-2D fuel cell model based on the PNV-model predicts a number of interesting effects: the instability of PEMFC operation due to oxygen starvation, the degradation wave along the oxygen channel and formation of electrolytic domain in DMFCs. Similar approach leads to a simple model of repeating element in a fuel cell stack. Asymptotic solution to the steady-state heat transport problem for such an element will be discussed. Dynamic version of this model reveals thermal instability of SOFC stack operation. Finally, we discuss a fast parallel model of a fuel cell stack based on the model of repeating element. The model predicts the effect of anomalous thermal transparency of a planar SOFC stack.

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Donnerstag, 11. 11. 2010, 14:00 Uhr (ESH)
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Dr. E. Hoarau(Agence Nationale pour la Gestion des D{\'e}chets Radioactifs)

Finite-volume simulation of carbon steel corrosion in a nuclear waste repository

Abstract:

In France, the concept explored to manage the high-level and intermediate-level long-lived wastes is to store them in a deep geological repository. The waste overpacks and the cell disposal liners, which will receive the packages, are made of carbon steel. ANDRA, the French national agency for radioactive waste management, performs studies dedicated to the assessment of the long-term degradation of carbon steel. In particular, this material is subject to generalized corrosion, which leads to the development of a passive layer consisting of a dense oxide inner layer and a porous hydroxide outer layer.

The corrosion processes are under the influence of the geochemical features of the groundwater, under anaerobic conditions. Therefore the groundwater is assumed to be anoxic. The processes depend also on the temperature, which is assumed to decrease from its initial value of 90grad celsius to its steady state value of 40grad celsius.

In this context, the Diffusion Poisson Coupled Model (DPCM) has been developed which describes the electrochemical behavior of the dense oxide layer, which covers the metal and is in contact with the solution in the pore space of the Callovo-Oxfordian claystones (repository host formation). The model is governed by unsteady electro-migration equations, describing the mass and charge transport, coupled with a Poisson equation, describing the electrostatic potential. In the literature, most existing models assume that the electrostatic potential is derived from a homogeneous electric field, and steady charge transport.

The boundary conditions of the DPCM are Robin type. The electrochemical surface reactions, which at the interfaces with the adjacent layers, are governed by Butler-Volmer laws. The potential drops at the interfaces are given by Gauss-Helmholtz laws. The model also considers moving boundaries and an additional equation for the free corrosion potential. According to the electrochemical mode performed, either the potential is constant (potentiostatic mode) or the current density is constant (galvanostatic mode). In the case of a zero current density, the galvanostatic mode gives us the free corrosion potential.

The model is similar to semi-conductor models. Numerical methods have been developed which ensure robustness and efficiency of the solution. In particular, all unknowns of the system, including boundary positions, are calculated using an implicit Euler method. Newtons method is used to solve the nonlinear systems occurring in each time step ensuring fast and accurate solutions. The linearized system is solved using the Schur complement, which takes into account the coupling between the drift-diffusion-Poisson system and the system describing the moving boundaries and the electrochemical potential.

Results of Numerical experiments will be presented showing the dependence on the electrochemical potential and the pH value for the steady state solution, as well as for the transient unsteady corrosion processes.

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Donnerstag, 4. 11. 2010, 14:00 Uhr (ESH)
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Prof. J. Novo(Universidad Autonoma de Madrid, Instituto de Ciencias Matematicas)

Adaptive schemes for evolutionary convection dominated problems

Abstract:

In the first part of this talk we study a procedure to stabilize Galerkin finite element approximations to linear evolutionary convection-reaction-diffusion equations in the convection dominated regime. It is well known that standard finite element approximations to this kind of equations develop spurious oscillations when convection dominates diffusion. We propose a procedure to considerably reduce the oscillations. The idea is the following. One first compute the standard Galerkin approximation at a given time and then solve a steady convection-reaction-diffusion problem with data based on the previously computed Galerkin approximation over the same finite element space but using the SUPG stabilized method. We then propose an adaptive algorithm based on this idea that is able to compute an oscillation-free Galerkin approximation over an automatically adapted mesh. In the second part of the talk we study the numerical approximation of the same equations using ENO (essentially non oscillatory) schemes in space together with TVD (total variation diminishing) Runge-Kutta methods in time. We will show that the methods are able to obtain non-oscillatory approximations both inside the domain and at the boundary even when the solution develops steep boundary layers due to the imposed boundary conditions. An adaptive scheme is introduce to improve the performance of the methods.

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Donnerstag, 21. 10. 2010, 14:00 Uhr (ESH)
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Prof. V. Springel(Heidelberger Institut für Theoretische Studien)

Tessellating the universe: Astrophysical fluid dynamics on a moving Voronoi mesh

Abstract:

Hydrodynamic cosmological simulations are a powerful tool to study structure formation in the Universe. At present, they usually either employ the Lagrangian smoothed particle hydrodynamics (SPH) technique, or Eulerian hydrodynamics on a Cartesian mesh with adaptive mesh refinement (AMR). However, both of these methods have a number of disadvantages that can negatively impact their accuracy in certain situations. In my talk I describe a novel scheme to calculate hydrodynamical flows which can largely eliminate these weaknesses. It is based on a moving unstructured mesh that is defined as the Voronoi tessellation of a set of discrete points. The new scheme adjusts its spatial resolution to the local clustering of the flow automatically and continuously, and hence retains a principle advantage of SPH for simulations of cosmological structure growth. At the same time it drastically reduces advection errros present in tradiation Eulerian schemes. I discuss the accuracy of the method in a set of illustrative examples, and also talk about technical implementation aspects on massively parallel supercomputers.

Prof. Dr. A. Ern(CERMICS, France) Montag,11. 10. 2010,13:00 Uhr (ESH)

Discontinuous Galerkin methods Part IMittwoch,13. 10. 2010,13:00 Uhr (ESH)

Discontinuous Galerkin methods Part IIFreitag,15. 10. 2010,13:00 Uhr (ESH)

Discontinuous Galerkin methods Part IIIMittwoch,20. 10. 2010,13:00 Uhr (405/405)

Discontinuous Galerkin methods Part IV

Abstract:

Discontinuous Galerkin (DG) methods are a modern approach for the discretization of partial differential equations. Prof. Ern is a world-leading expert in this field. The lectures will give an introduction into DG methods and they will discuss their application to various kinds of problems.

prerequisite: basic knowledge on numerical methods for partial differential equations

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Donnerstag, 14. 10. 2010, 14:00 Uhr (ESH)
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Prof. E. Burman(University of Sussex, Department of Mathematics)

Nitsche's method and unfitted FEM

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Donnerstag, 2. 9. 2010, 14:00 Uhr (ESH)
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J. Heiland(TU Berlin)

Control of drop size distributions in liquid/liquid dispersions

Abstract:

The project presented is dedicated to the control of drop size distributions in stirred liquid/liquid dispersions which are of major importance in chemical industries. The design of reactors usually requires expensive experimental investigations. Mathematical modelling, model based simulation and control are often the only feasible approach to achieve the technological goals. For the complex process of formation of drop size distributions (DSD) in a stirred tank such mathematical methods are currently not available. The global long-term perspective of the project is to develop model based methods for the prediction and the control of the complex process of DSD formation in stirred liquid/liquid dispersions in a stirred tank. The technological vision is to be able to achieve a desired average drop size and a defined size distribution, using control parameters such as the stirrer speed. Therefore a specific stirrer setup is investigated analytically, numerically and by means of experiments. In my talk I will present the concrete physical setup, introduce the underlying mathematical model and address solution and implementation approaches. The model is given by a coupling of the RANS equations for the mixture, treated as a single flow, and a population balance equation (PBE) describing the dispersed phase. To soften the high computational loads of a direct numerical solution of the PBE the method of moments (MOM) will be used for the simulations. In view of control or system identification an interface between the flow/PBE solver and Matlab is implemented.

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Donnerstag, 26. 8. 2010, 14:00 Uhr (ESH)
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G. R. Barrenechea(University of Strathclyde, Scotland)

Residual local projection finite element methods

Abstract:

In this talk we review the Local Projection Stabilized (LPS) finite element technique to solve problems in incompressible fluid mechanics. Old and new methods are presented, and a way to derive new methods from a Variational Multiscale strategy is presented and analyzed. The methodology is applied to both the Stokes and Navier-Stokes equations, and the resulting methods are analyzed and numerically tested.

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Donnerstag, 12. 8. 2010, 14:00 Uhr (ESH)
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Dr. P. Knobloch(Charles University, Institute of Numerical Mathematics, Czech Republic)

Local projection stabilization for convection-diffusion-reaction equations

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Donnerstag, 22. 4. 2010, 14:00 Uhr (ESH)
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Dr. J. Rang(TU Braunschweig, Institut für Wissenschaftliches Rechnen)

Diagonally implicit Runge-Kutta methods and order reduction

Abstract:

It is well-known that diagonally implicit Runge-Kutta methods (DIRK-methods) have order reduction if they are applied on stiff ODEs or on the semi-discretised incompressible Navier-Stokes equations. Fully implicit DIRK-methods converge only with order one and semi-implicit DIRK-methods (as the Crank-Nicolson scheme) converge with order 2 if they are applied on very stiff problems as the Prothero-Robinson example. In this talk the order reduction is discussed and the problem of Prothero-Robinson is considered. It will be shown that some further order conditions should be satisfied to reduce the order reduction. New DIRK-methods satisfying the newconditions are applied to the incompressible Navier-Stokes equations.

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Donnerstag, 18. 3. 2010, 14:00 Uhr (ESH)
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Prof. F. Schieweck(Otto-von-Guericke Universität Magdeburg)

A stable discontinuous Galerkin--Petrov time discretization of higher order

Abstract:

We present a time discretization for evolution equations in a Hilbert space which is based on a discontinuous Galerkin-Petrov technique. The discrete solution space is constructed with continuous piecewise polynomial functions of degree k and the discrete test space with discontinuous polynomial functions of degree k-1. The method is A-stable and of optimal order k+1 in the maximum norm. Moreover, numerical experiments show that the method dGP(2), where k=2, is super-convergent of order 4 in the end points of the discrete time intervals. Applied to a gradient flow of an energy functional, the method guarantees that the discrete energy approximation decays monotonously for each time step.

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Donnerstag, 4. 3. 2010, 15:00 Uhr (ESH)
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D. Naumov(Helmholtz-Zentrum Potsdam, Deutsches GeoForschungsZentrum)

Fluid flow in sandstones

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Donnerstag, 4. 3. 2010, 14:00 Uhr (ESH)
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Dr. G. Blöcher and Dr. M. Cacace(Helmholtz-Zentrum Potsdam, Deutsches GeoForschungsZentrum)

Meshing and numerical solvers - Requirements for geosciences