Research Group "Numerical Mathematics and Scientific Computing"

Publications

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Articles in Refereed Journals

R. Bordás, V. John, E. Schmeyer, D. Thévenin, Numerical methods for the simulation of a coalescence-driven droplet size distribution, Theor. Comput. Fluid Dyn., 27 (2013) pp. 253--271.
Abstract

A droplet size distribution in a turbulent flow field is considered and modeled by means of a population balance system. This paper studies different numerical methods for the 4D population balance equation and their impact on an output of interest, the time-space-averaged droplet size distribution at the outlet which is known from experiments. These methods include different interpolations of the experimental data at the inlet, various discretizations in time and space, and different schemes for computing the aggregation integrals. It will be shown that notable changes in the output of interest might occur. In addition, the efficiency of the studied methods is discussed.
C. Bertoglio, A. Caiazzo, M.A. Fernàndez, Fractional-step schemes for the coupling of distributed and lumped models in hemodynamics, SIAM J. Sci. Comput., 35 (2013) pp. B551--B575.
A. Bradji, J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Appl. Math., 58 (2013) pp. 1--38.
Abstract

A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives.
A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, Th. Koprucki, K. Gärtner, A. Glitzky, Self-heating, bistability, and thermal switching in organic semiconductors, Phys. Rev. Lett., 110 (2013) pp. 126601/1--126601/5.
Abstract

We demonstrate electric bistability induced by the positive feedback of self-heating onto the thermally activated conductivity in a two-terminal device based on the organic semiconductor C60. The central undoped layer with a thickness of 200 nm is embedded between thinner n-doped layers adjacent to the contacts minimizing injection barriers. The observed current-voltage characteristics follow the general theory for thermistors described by an Arrhenius-like conductivity law. Our findings including hysteresis phenomena are of general relevance for the entire material class since most organic semiconductors can be described by a thermally activated conductivity.
V. John, J. Novo, A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 255 (2013) pp. 289--305.
Abstract

A robust residual-based a posteriori estimator is proposed for the SUPG finite element method applied to stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of this paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypotheses. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided.
CH. Batallion, F. Bouchon, C. Chainais-Hillairet, J. Fuhrmann, E. Hoarau, R. Touzani, Numerical methods for the simulation of a corrosion model in a nuclear waste deep repository, J. Comput. Phys., 231 (2012) pp. 6213--6231.
Abstract

In this paper, we design numerical methods for a PDE system arising in corrosion modelling. This system describes the evolution of a dense oxide layer. It is based on a drift-diffusion system and includes moving boundary equations. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance. Finally, numerical experiments with real-life data shows the efficiency of the developed methods.
R. Bordás, V. John, E. Schmeyer, D. Thévenin, Measurement and simulation of a droplet population in a turbulent flow field, Comput. Fluids, 66 (2012) pp. 52--62.
Abstract

The interaction of a disperse droplet population (spray) in a turbulent flow field is studied by combining wind tunnel experiments with simulations based on the model of a population balance system. The behavior of the droplets is modeled numerically by a population balance equation. Velocities of the air and of the droplets are determined by non-intrusive measurements. A direct discretization of the 4D equation for the droplet size distribution is used in the simulations. Important components of the numerical algorithm are a variational multiscale method for turbulence modeling, an upwind scheme for the 4D equation and a pre-processing approach to evaluate the aggretation integrals. The simulations of this system accurately predict the modifications of the droplet size distribution from the inlet to the outlet of the measurement section. Since the employed configuration is simple and considering that all measurement data are freely available thanks to an Internet-based repository, the considered experiment is proposed as a benchmark problem for the simulation of disperse two-phase turbulent flows.
M. Natale, A. Caiazzo, E.M. Bucci, E. Ficarra, A novel Gaussian extrapolation approach for 2D gel electrophoresis saturated protein spots, Genomics Proteomics Bioinformatics, 10 (2012) pp. 336--344.
R. Eymard, Th. Gallouët, R. Herbin, A. Linke, Finite volume schemes for the biharmonic problem on general meshes, Math. Comp., 81 (2012) pp. 2019--2048.
Abstract

We propose a finite volume scheme for the approximation of a biharmonic problem with Dirichlet boundary conditions. We prove that the piece-wise constant approximate solution converges to the exact solution, as well as the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. These results are confirmed by numerical results.
A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, Th. Koprucki, J. Fuhrmann, K. Gärtner, A. Glitzky, Self-heating effects in organic semiconductor crossbar structures with small active area, Organic Electronics, 13 (2012) pp. 2461--2468.
Abstract

We studied the influence of heating effects in an organic device containing a layer sequence of n-doped / intrinsic / n-doped C₆₀ between crossbar metal electrodes. A strong positive feedback between current and temperature occurs at high current densities beyond 100 A/cm², as predicted by the extended Gaussian disorder model (EGDM) applicable to organic semiconductors. These devices give a perfect setting for studying the heat transport at high power densities because C₆₀ can withstand temperatures above 200° C. Infrared images of the device and detailed numerical simulations of the heat transport demonstrate that the electrical circuit produces a superposition of a homogeneous power dissipation in the active volume and strong heat sources localized at the contact edges. Hence, close to the contact edges, the current density is significantly enhanced with respect to the central region of the device, demonstrating that three-dimensional effects have a strong impact on a device with seemingly one-dimensional transport.
K. Galvin, A. Linke, L. Rebholz, N. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg., 237--240 (2012) pp. 166--176.
Abstract

We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.
W. Hackbusch, V. John, A. Khachatryan, C. Suciu, A numerical method for the simulation of an aggregation-driven population balance system, Internat. J. Numer. Methods Fluids, 69 (2012) pp. 1646--1660.
Abstract

A population balance system which models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain and the equation for the particle size distribution (PSD) in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel.
V. John, J. Novo, On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations, J. Comput. Phys., 231 (2012) pp. 1570--1586.
Abstract

Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.
V. John, F. Thein, On the efficiency and robustness of the core routine of the quadrature method of moments (QMOM), Chem. Engng. Sci., 75 (2012) pp. 327--333.
Abstract

Three methods are reviewed for computing optimal weights and abscissas which can be used in the Quadrature Method of Moments (QMOM): the Product-Difference Algorithm (PDA), the Long Quotient-Modified Difference Algorithm (LQMDA, variants are also called Wheeler algorithm or Chebyshev algorithm), and the Golub--Welsch Algorithm (GWA). The PDA is traditionally used in applications. It is discussed that the PDA fails in certain situations whereas the LQMDA and the GWA are successful. Numerical studies reveal that the LQMDA is also more efficient than the PDA.
T. Streckenbach, Parameter determination of an evolution model for phase transformations in steel, ZAMM Z. Angew. Math. Mech., 92 (2012) pp. 217--235.