Asymptotic behavior of blow-up solutions to a degenerate parabolic equation

Anada Koichi

Waseda University, Tokyo, Japan

Abstract: We consider a degenerate parabolic equation with blow-up solutions which have faster rates than ones of self similar solutions. Our purpose is to investigate asymptotic behavior of $u/(\sup u)$ near the blow-up time where $u$ is the solution to our problem.

Mathematical and numerical modelling of forest fire front propagation

Balažovjech Martin and Mikula Karol

STU Bratislava, Slovakia

Abstract: We discuss mathematical model and numerical simulations of the forest fire front propagation. The model is based on tangentially stabilized curve shortening flow with a strong driving force which depends on non-homogeneous fuel field and includes the wind strength and direction and allows topological changes in the front evolution. We solve numerically an intrinsic partial differential equation for the front position vector by new higher order scheme which is a combination of the explicit forward Euler and the fully-implicit backward Euler schemes.

Numerical Simulation of Flow over a Rough Surface

Bauer Petr

Czech Technical University in Prague, Czech Republic

Abstract: We attempt to model a 2D rough surface by computing non-stationary Navier-Stokes flow over a periodic pattern. The solution is obtained by means of finite element method (FEM). We use non-conforming Crouzeix Raviart elements for velocity and piecewise constant elements for pressure. The resulting linear system is solved by multigrid method. We present computational studies of the problem.

Numerical simulation of transport of colloids in heterogeneous porous media

Beneš Pavel

Czech Technical University in Prague, Czech Republic

Abstract: This contribution deals with a numerical simulation of transport of colloids in heterogeneous porous media. The transport is described by the generalized convection-diffusion equation [Sun]. This equation is solved by means of the finite volume method using the operator splitting technique [Lev]: 1)the generalized convection-diffusion equation without the diffusion therm is solved explicitly using the finite volume method 2)the diffusion equation is solved implicitly by means of the finite volume method using solution from 1) as the initial condition. Some of our numerical simulations will be presented.

[Sun] N. Sun, M. Elimelech, N.-Z. Sun A novel two-dimensional model for colloid transport in physically and geochemically heterogeneous porous media. Journal of Contaminant Hydrology 49, (2001), 173-199

[Lev] R. LeVeque: The Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002

Numerical analysis of a two-scale model of acid attack on concrete

Chalupecký Vladimír

Faculty of Mathematics, Kyushu University, Fukuoka

Abstract: Concrete corrosion due to sulfuric acid attack is a phenomena occuring in sewer pipes that can eventually lead to their collapse. It is a complex process that includes numerous chemical reactions and interaction with anaerobic bacteria. We consider a semi-linear reaction-diffusion PDE/ODE system defined on two separate scales. It describes the evolution of gaseous $H_2S$ at the macro scale throughout the concrete matrix and the evolution of dissolved $H_2S$, $H_2SO_4$ as well as of moisture and gypsum at pore (micro) scale. We propose and analyse a semi-discrete numerical scheme based on finite difference discretization in space. We present an experimental order of convergence study and several numerical experiments demonstrating the general behavior of the system. We also study a fast micro-macro mass transfer limit as the Biot number tends to infinity.

Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems and Compressible Flow

Feistauer Miloslav

Charles University, Prague, Czech Republic

Abstract: In this paper we shall be concerned with two subjects. First we shall analyze the space-time discontinuous Galerkin method for the numerical solution of nonstationary convection-diffusion problems with nonlinear diffusion and nonlinear convection. The discontinuous Galerkin finite element method (DGFEM) is based on a piecewise polynomial approximation of the sought solution without any requirement on the continuity on interfaces between neighbouring elements. It is particularly convenient for the solution of conservation laws with discontinuous solutions, of singularly perturbed convection-diffusion problems with dominating convection, when solutions contain steep gradients, and of compressible flow. Here we apply the discountinuous approximation in space as well as in time with different polynomial degrees in the space discretization and the time discretization. An advantage of this technique is the possibility to use different meshes on different time levels, which can be used in practical computations for time dependent space adaptivity. We shall discuss the derivation of error estimates for this method.

In the second part, some applications of the DGFEM to the simulation of compressible flow, i.e. the solution of the compressible Euler and Navier-Stokes equations, will be presented. Our goal is to develop sufficiently accurate, efficient and robust numerical schemes allowing the solution of compressible flow for a wide range of Reynolds and Mach numbers, applicable to flow simulation in time dependent domains and to fluid-structure interaction. The applicability of the method will be demonstrated by computational results.

Higher-order numerical methods for simulation of two-phase flow in heterogeneous porous media

Fučík Radek

Czech Technical University in Prague, Czech Republic

Abstract: A higher-order numerical method is presented for simulating two-phase flow in heterogeneous porous media with sharp material interfaces. The numerical scheme uses the mixed-hybrid finite element method (MHFE) for the approximation of the phase-velocities. The MHFE method is combined with the discontinuous Galerkin (DG) approach that allows for representing the phase saturation as a piecewise continuous function. This approach is superior to the conventional finite difference (FD) or finite volume (FV) methods since it allows for using a higher order approximation of the saturation on complex unstructured meshes. We improve the MHFE-DG method proposed by Hoteit and Firoozabadi (2008) in order to model flow across sharp material discontinuities in situations where the capillary pressure is discontinuous preventing penetration of the non-wetting phase into the finer medium (the barrier effect). The capillary pressure at the material interface is treated using the extended capillary pressure condition. We show results of numerical simulations of several test problems and problems for which laboratory data are available.

Steady and Unsteady Turbulent Flow in External Aerodynamics

Furmánek Petr, Fürst Jiří and Kozel Karel

Aeronautical Research and Test Institute, Prague

Abstract: The aim of this work is to summarize results of numerical simulations of steady and unsteady transonic flow obtained by two different modern finite volume schemes in combination with Arbitrary Lagrangian-Eulerian method (computation on moving meshes) and various models of turbulence. The simulations were carried out both in 2D and 3D and the unsteady effects were presented by forced oscillations of the profile/wing around given reference point/axis. Implemented schemes were the so called Modified Causon’s scheme (based on TVD form of classical MacCormack scheme) and implicit WLSQR scheme (based on the WENO approach) combined with AUSMPW+ numerical flux in 2D and HLLC flux in 3D. As a 2D test case both inviscid and turbulent flow around the NACA 0012 profile wing have been simulated and the numerical results have been compared with experimental data. Both schemes were extended also for the 3D steady computations and tested on the transonic flow around the ONERA M6 wing. The computational area was discretized with two different types of finite volume meshes (H and C type). Comparison of the numerical results (both in-between and with experimental data) is satisfactory. Used turbulence models were: Spalart-Allmaras model, Kok’s TNT model and SST k-w model. The Modified Causon’s scheme in 3D form was also adapted for unsteady computation with the use of ALE method and was tested on inviscid transonic flow around the ONERA M6 wing (forced oscillation around a given axis). Experimental data for this case are unfortunately not available. However, the numerical results show all the characteristics as expected.
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Phase-field approach to epitaxial crystal growth under stress

Hoang Dieu Hung

Czech Technical University in Prague, Czech Republic

Abstract: This contribution deals with the numerical simulation of epitaxial growth with elastic effects. The numerical scheme, which was developed to solve this problem, is based on the finite difference method. The elastic equations were solved by the finite element method. In order to verify that the phase-field model validates the reported experimental observations a number of numerical tests was performed. We showed that elastic effects strongly influence the crystal surface.

Morphological Contour Decomposition and Reconstruction of Image.

Idogawa Tomoyuki and Lu Minhao

Shibaura Institute of Technology, Saitama, Japan

Abstract: A new algorithm to decompose and reconstruct a binary image will be proposed. It is based on mathematical morphology and uses two contours. These contours are defined as the edge and the second edge of the original image. Since these contours have informations of boundary and internal direction of image, the original image can be reconstructed exactly from them.

On time-parametrized bivariate copulas

Ishimura Naoyuki

Hitotsubashi University, Tokyo, Japan

Abstract: Copulas are known to be a basic tool of modelling the dependence structure among random variables. We here introduce a new type of bivariate copulas which are parametrized by time. Some applications are indicated.

Crystalline motion and eventual monotonicity of the shape

Ishiwata Tetsuya

Shibaura Institute of Technology, Saitama, Japan

Abstract: Crystalline motion is the one simple model of interface motion of crystals. In two dimensional case, we restrict the solution curves in certain class of polygonal curves in the plane. In this talk, we shall discuss the behavior of solution curves. During the time evolution, the solution curves may develop some singularities. We show the sufficient conditions to construct the solution beyond the singularity. We also investigate the deformation patterns of solution polygons and show eventual monotonicity property of the shape.

Finite difference scheme for the Ericksen-Leslie equation

Ishiwata Tetsuya

Shibaura Institute of Technology, Saitama, Japan

Abstract: Ericksen Leslie equation describes the time evolution of a spin vector and velocity in liquid crystals and has following property: (i) the length preserving of a spin vector, (ii) energy conservation or dissipation property, (iii) the incompressibility of a velocity vector. We will propose the new implicit scheme and show that the finite difference solutions satisfy the above three properties. We also show some numerical examples.

Parameterisations of properties of the turbulent flow above very rough surface

Jurčáková Klára

Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Prague

Abstract: Numerical methods solving Reynolds Average Navier-Stokes (RANS) equations require parameterisations of high order turbulent properties as turbulent transport of mass, heat and passive scalar. The most often used parameterisations are based on idea that turbulent transport of property is proportional to the gradient of the mean value of this property. This is plausible theory in the turbulent flow above smooth surface, however questionable in the turbulent flow above rough surface. Wind tunnel experiments provided direct measurement of the turbulent transport processes and allow comparing transport coefficient with widely used parameterisations as K-theory.

The best constant of discrete Sobolev inequality on regular polyhedron

Kametaka Yoshinori

Osaka University

Abstract: PDF version

Exponential decay of correlation functions in many-electron systems

Kashima Yohei

University of Heidelberg, Germany

Abstract: We show that for a class of tight-binding many-electron models the correlation functions decay exponentially at non-zero temperature in the thermodynamic limit if the interaction is sufficiently small depending on temperature. The decay bounds are valid in arbitrary space dimension. The proof is based on the U(1)-invariance property and perturbative bounds of the finite dimensional Grassmann integrals formulating the correlation functions.

Comparison of Wavelet Analyses Applied on Data From Very Rough Boundary Layer

Kellnerová Radka

Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Prague

Abstract: Continuous Wavelet analysis is applied on data from simulation of very rough boundary layer in wind-channel experiment. Morlet and Mexican hat functions are used as mother wavelet and mutually compared. Wavelet analysis is performed on velocity data from PIV and hot wire velocity measurements, both with high repetition rate. Results of wavelet power spectra are confronted with instantaneous PIV images in order to understand meaning of energetic structures detected by wavelet analysis and corresponding physical patterns in fully turbulent flow.

Numerical study of viscous flow of generalized Newtonian fluids

Keslerová Radka

Czech Technical University in Prague, Czech Republic

Abstract: In this paper the numerical results for steady and unsteady fluids flow are presented. The system of Navier-Stokes equations is considered as the governing system of the equations. We tested two models for the stress tensor in the right hand side of this system. First, Stokes model is used. In this model the power-law model as the viscosity function is used. Second, Jeffreys model is considered. For this model the constant viscosity is tested. The numerical results for Newtonian and for Oldroyd-B model are presented. For the unsteady computation we consider the dual-time stepping method. The high artificial compressibility coefficient is used in the artificial compressibility method applied in the dual time $\tau$ . The unsteady numerical results of fluids flow in the branching channel are presented.
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On some generalizations of polygonal motions

Kimura Masato

Kyushu University, Fukuoka, Japan

Abstract: We consider some extensions and generalizations of the polygonal motion (crystalline motion) proposed in Benes-Kimura-Yazaki (2009). Extensions to coupling with the external field, change of edge angles, and polyhedral motions will be discussed.

Local projection stabilization of convection-diffusion-reaction equations

Knobloch Petr

Charles University, Prague, Czech Republic

Abstract: We review classical approaches to local projection stabilization of finite element discretizations of convection-diffusion-reaction equations and present a generalized formulation allowing the use of local projection spaces defined on overlapping sets. This generalized formulation enables to increase the stability of the method and its robustness with respect to the choice of the stabilization parameter. Moreover, we present new error estimates with respect to a norm that is stronger than the usual local projection norm. The theoretical considerations will be illustrated by numerical results.

On the interpolation constants over triangular elements

Kobayashi Kenta

Kanazawa University, Japan

Abstract: In the finite element method, the local interpolation error is closely related to the a priori and a poteriori error estimaions. Therefore it is important to obtain sharp error bounds for interpolation. For this purpose, we will introduce the formulas which give very sharp upper bounds for the interpolation constants over triangular elements.

Mathematical Modeling of the Locomotion of Amoeba

Kobayashi Ryo

Hiroshima University, Japan

Abstract: Amoeboid motion is widely observed in a single cell movement of eucaryote. We concentrate on the locomotion of the naked amoebae which are crawling around on the substrate. An amoeba extends the part of its body to the direction of movement, which is called pseudopod. During the locomotion, contraction of the actomyosin fibers produce a power, a part of the cell is extended, and also sol-gel transformation is taking place. To guarantee the normal amoeboid locomotion, lots of processes are going simultaneously in the coordinated manner. A mathematical model is presented, in which we adopt the combination of the two models of different type, phase field model and smoothed particle hydro-dynamics.

A second order finite volume method for solving nonlinear parabolic equations in financial mathematics

Kutik Pavol, Mikula Karol

STU Bratislava, Slovakia

Abstract: Various generalizations of the classical linear Black-Scholes equation are made by adjusting the volatility to be a function of the option price itself. Such generalizations can be mathematically stated in the form of a nonlinear Black-Scholes equation, where the nonlinearity arises in the diffusion coefficient, the so-called Gamma equation. In this talk we introduce a new second order finite volume numerical scheme, based on the Crank-Nicolson type discretization, for solving such equations. We present numerical experiments showing properties of the new scheme and compare it with the well-known semi-implicit, fully-implicit and explicit schemes with respect to precision and computational efficiency.

Gene Expression Time Delays and Turing Pattern Formation

Lee Seirin

University of Tokyo, Japan

Abstract: There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self- organisation in biological systems via a Turing bifurcation has been explored, postulated or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. The investigations of our study demonstrate that the behaviour of Turing models profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression [1]. Furthermore, they also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing's mechanism in the presence of gene expression time delays [2, 3]. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics, and also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens.

[1] S. Seirin Lee, E. A. Gaffney, N. A. M. Monk, The influence of gene
expression time delays on Gierer-Meinhardt pattern formation system. Bulletin of Mathematical Biology (2010) DOI 10.1007/s11538-010-9532-5
[2] S. Seirin Lee, E.A. Gaffney, Aberrant behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays. Bulletin of Mathematical Biology (2010) DOI 10.1007/s11538-010-9533-4
[3] S. Seirin Lee, E.A. Gaffney, R.E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays. (Submitted in Bulletin of Mathematical Biology)

Graph Cuts in Segmentation of MRI Data

Loucký Jakub, Oberhuber Tomáš

Czech Technical University in Prague, Czech Republic

Abstract: We present applications of graph cuts in the image processing. An implementation of the basic algorithm performing segmentation of images from MRI is described. The original Ford-Fulkerson algorithm was modified for this particular application which led to major acceleration of the segmentation process.

Degenerate Diffusion Methods in Computer Image Processing and Applications

Máca Radek

Czech Technical University in Prague, Czech Republic

Abstract: This paper deals with segmentation of image data using a partial differential equation of the level set type. The level set formulation and modification of the level set equation is presented. The evolution process can be controlled by the segmented image data so that the edges of the objects can be found. The semi-implicit complementary volume numerical scheme to solve the level set equation is used. The essential part of the thesis is proposition and setting of the equation parameters which are used for the segmentation of the left heart ventricle from the cardiac MRI images.

Numerical Solution of the Gray-Scott Model

Mach Jan

Czech Technical University in Prague, Czech Republic

Abstract: This contribution deals with numerical solution of the Gray-Scott model [GS1983, GS1984]. We introduce numerical schemes for this model based on the method of lines. To perform spatial discretization we use FDM and FEM. Resulting systems of ODEs are solved using the modified Runge-Kutta method with adaptive time-stepping. We present some of our numerical simulations and perform comparison of these schemes from the qualitative point of view.

[GS1983] P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38:29-43 (1983)

[GS1984] P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B->3B, B->C. Chem. Eng. Sci. 39:1087-1097 (1984).

Finding an ideal path for a camera in virtual colonoscopy

Mikula Karol, Urban Jozef

STU Bratislava

Abstract: We develop a suitable mathematical models and computational methods for finding an ideal path in human colon represented by a visual information given by 3D computer tomography. This path will determine a trajectory of camera in virtual colonoscopy, medical technology dealing with colon diagnoses by computer. Physicians use this technology for searching polyps and tumours in colon. The first step in our approach is segmentation of the colon in medical data using a region-growing algorithm. Then we use a mathematical model for computing distance function inside the segmented volume. First we 6c compute this function as the shortest distance to a user's selected starting point. This function is used to create an initial guess for the curve, which will represent the searched path. In the next step we calculate distance function to the boundary of segmented volume. The gradient of this function determines the velocity vector field in which we insert the initial curve. Using projection of the vector field to the plane normal to evolving curve, a regularization of the motion by curvature and suitable tangential velocity, we end up with the smooth, asymptotically uniformly discretized curve representing optimal trajectory for the camera in virtual colonoscopy.

Numerical Simulation of Discrete Dislocation Dynamics

Minárik Vojtěch

Czech Technical University in Prague, Czech Republic

Abstract: The poster summarizes achievements in the field of numerical simulation of dislocation dynamics. A dislocation curve with a finite number of dipolar loops in a crystallic material is considered. Partial differential equations of degenerate parabolic type governing the dynamics of the system of dislocations are presented. The need for advanced numerical treatment of the model, e.g. special type of tangential redistribution of discrete nodes, is visualized in samples.

A numerical method for nonlinear cross-diffusion systems

Murakawa Hideki

University of Toyama, Japan

Abstract: This talk proposes a numerical scheme for general cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger, Brezis and Rogers [RAIRO Anal. Numer., 13 (1979), 297--312]. Analyses of numerical methods for general cross-diffusion systems are difficult. However, we can prove the convergence of the linear scheme by means of the theory of reaction-diffusion system approximation.

Homogenization limit of recurrent traveling waves in a two-dimensional cylinder with undulating boundary

Nakamura Ken-Ichi

University of Electro-Communications, Tokyo, Japan

Abstract: In this talk we will present a result on the speed of traveling waves in spatially heterogeneous media. More precisely, we consider a curvature-driven motion of plane curves in a two-dimensional cylinder with spatially undulating boundary. The function which determines the boundary undulation is assumed to be almost periodic, or more generally, recurrent. We study the homogenization problem as the boundary undulation becomes finer and finer. By constructing suitable upper and lower solutions, we give a sharp estimate of the average speed of the traveling wave for our problem. In the special case where the spatial undulation is periodic, the optimal rate of convergence to the homogenization limit of the average speed is derived from the estimate. This is a joint work with Bendong Lou (Tongji University) and Hiroshi Matano (University of Tokyo).

Numerical approach to the ODE system for the default risk model

Nakamura Masaaki

Nihon University, Tokyo, Japan

Abstract: We introduce the numerical study of the systems of ordinary differential equations (ODEs), which nonlinearly extend a looping default model of defaultable firms. Unknown functions are defined through a weighted integral of the tail distribution functions of the first jump time.

The splitted Do-Nothing Type of Boundary Condition for Navier-Stokes Equations in Cascade of Profiles, the Existence and Uniqueness of a Weak Solution

Neustupa Tomáš

Czech Technical University in Prague, Czech Republic

Abstract: We are concerned with the theoretical analysis of the model of incompressible, viscous, stationary flow through a plane cascade of profiles. The boundary value problem for the Navier-Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. More in PDF version.

A pressure-stabilized characteristics finite element scheme for the Navier-Stokes equations.

Notsu Hirofumi

Meiji University, Tokyo, Japan

Abstract: We have developed a characteristics finite element scheme for the Navier-Stokes equations. The P1/P1-element is employed with a pressure-stabilization technique. Advantages of the scheme are as follows. The resulting matrix is symmetric and it is useful for large scale computations. Two and three dimensional numerical results are shown.

Robust algorithms for singular values and eigenvalues

Ogita Takeshi

Tokyo Woman's Christian University, Japan

Abstract: In this talk, algorithms for accurately calculating singular values and eigenvalues of matrices is proposed. The proposed algorithms can treat the cases where the matrices are extremely ill-conditioned, i.e., their condition numbers are allowed to go far beyond the bounds of base precision such as IEEE 754 double precision. The algorithms require standard numerical algorithms, which are commonly implemented in several numerical libraries such as BLAS and LAPACK, and an algorithm for accurate matrix multiplication. Numerical results are presented for illustrating the performance of the proposed algorithms.

A rigorous derivation of mean-field models for diblock copolymer melts

Oshita Yoshihito

Okayama University, Japan

Abstract: We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241-1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615-632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.

Robustness Problem and Error-Free Determinant Transformation in Computational Geometry

Ozaki Katsuhisa

Shibaura Institute of Technology, Saitama, Japan

Abstract: Finite precision arithmetic like floating-point arithmetic may output an inexact result although rational arithmetic output an exact result. For example in computational geometry, if an algorithm outputting a convex full for a set of points is executed by floating-point arithmetic, then, the result may not be a convex, or some points may not be enclosed by the result. This problem is called a robustness problem. In this talk, main topic is how to overcome this problem by using only pure floating-point arithmetic.

Model of Topological Changes in Discrete Dislocation Dynamics

Pauš Petr

Czech Technical University in Prague, Czech Republic

Abstract: This contribution deals with the numerical simulation of dislocation dynamics, their interaction, merging and other changes in the dislocation topology. The glide dislocations are represented by parametrically described curves moving in gliding planes. The simulation model is based on the numerical solution of the dislocation motion law belonging to the class of curvature driven curve dynamics. Mutual forces between dislocations are incorporated in the model. We focus on the simulation of the cross-slip of two dislocation curves where each curve evolves in a different gliding plane and after applying certain stress, the curves may merge. The simulation of the Frank-Read source of dislocations which describes how new dislocations are created is also presented. Merging and splitting of multiple (more than two) dislocation curves in persistent slip bands and their interactions in channels of the bands are also simulated.

Numerical Simulation of Multicomponent Compressible Flow in Porous Medium

Polívka Ondřej and Jiří Mikyška

Czech Technical University in Prague, Czech Republic

Abstract: The article focuses on the numerical modelling of the compressible single-phase ow of a mixture composed of several components in a porous medium. Firstly, the physical-chemical properties of mixtures with the constant number of components are described and the equations describing the single-phase ow are introduced. On the basis of the equations and established initial and boundary conditions the mathematical model of the solved problem is formulated. Further, the composed model is solved numerically using a combination of the mixed-hybrid nite element method for the Darcy's law discretization and the nite volume method for the discretization of the transport equations. For the time discretization the Euler method is used. The combination of the numerical schemes leads to a large system of nonlinear algebraic equations which is solved using the Newton-Raphson iterative method. The dimensions of the resulting matrices of the system of linear algebraic equations are signicantly reduced using the hybridization technique so that they do not depend on the number of mixture components. Finally, the results of simulations, which have been computed by written programme, are introduced. The convergence of the numerical scheme is veried on two testing problems in a homogeneous medium. Moreover, we present several results of methane injection simulations into a reservoir lled with propane and spreading of the mixture in the heterogeneous reservoir containing the blocks of fractured medium.
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Numerical modeling of Generalized Newtonian Flows in Channels

Prokop Vladimír

Czech Technical University in Prague, Czech Republic

Abstract: In this work are considered generalized Newtonian fluids described by the system of conservation laws and viscosity function. The system consists of equation of continuity and three momentum equations in 3D case. Energy conservation is not taken into account because temperature variations are in our case negligible. Numerical solution of steady system is in our case based on artificial compressibility method. There are different approaches to solve unsteady system of Navier-Stokes equations. One of them is the artificial compressibility that allows to use time marching method after the continuity equation is changed by addition of pressure time derivative divided by $\beta^2\rightarrow\infty$. The other possible approach is the stepping method based on the addition of the derivative of W in fictitious dual time. The time derivative in the real time can be discretized by three–point backward formula. The scheme is implicit in the real time. Space derivatives are discretized using finite volume method in the cell centered formulation. Numerical solution of incompressible generalized Newtonian flow is sought in the geometry of channel and bypass or in geometry of branched channels.
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Mathematical Modelling of Pulverized Coal Combustion

Straka Robert

Czech Technical University in Prague, Czech Republic

Abstract: We describe behavior of the air-coal mixture using the Navier-Stokes equations for the mixture of air and coal particles, accompanied by a turbulence model. The undergoing chemical reactions are described by the Arrhenian kinetics (reaction rate proportional to $e^{-\frac{E}{RT}}$). We also consider the heat transfer via conduction and radiation. The system of PDEs is discretized using the finite volume method (FVM) and an advection upstream splitting method as the Riemann solver. The resulting ODEs are solved using the 4th-order Runge-Kutta method. Simulation results for typical power production level are presented together with the air-staging impact on NO production.

MEGIDDO: Numerical issues behind the MR-DTI visualization algorithm

Strachota Pavel

Czech Technical University in Prague, Czech Republic

Abstract: For the purpose of MR-DTI data visualization, we have developed a numerical algorithm based on a mathematical model of texture diffusion. Accompanied by data preprocessing and postprocessing procedures, this algorithm forms the cornerstone of the MEGIDDO (Medical Employment of Generating Images by Degenerate Diffusion Operator) software tool, which is briefly introduced in this contribution. Afterwards, we focus on investigating the properties of the numerical solution methods. Emphasis is put on the assessment of several numerical schemes with respect to artificial diffusion. Both visual and quantitative methods for scheme comparison are discussed.

Finite element solver for modern parallel computer with GPU

Suzuki Atsushi

École Nationale des Ponts et des Chaussées, Champs sur Marne, France

Abstract: Three dimensional computation of flow problem is still challenging with modern parallel computers. GPU (graphics processing unit) is now extended to have capability of huge floating arithmetic for scientific computation and is regarded as a successor of vector processor. CUDA is provided as a programming environment of GPU parallel and thread computing from nVIDIA. To obtain high performance of GPU, some technical issues need to be considered, i.e. multi-thread computing and memory hierarchy. Finite element computation for flow problem has an advantage in dealing with complex mesh structure, and iterative solver (CG or GMRES) consists of matrix-vector multiplication is efficient. However, direct implementation of sparse matrix-vector multiplication can not utilize computational capability of GPU, due to random memory access from sparseness of the matrix. On the other hand, simple LU factorization with BLAS3 operation can enjoy high performance of GPU. A hybrid algorithm with direct and iterative solver based on iterative substructuring method is a candidate for GPU computation. A numerical results by using GPU cluster in French supercomputer center, CCRT will be presented.

On the free boundary problem arising from pricing path-dependent options and its numerical approximation

Ševčovič Daniel

Comenius University Bratislava, Slovakia

Abstract: In this talk, we analyze the model for pricing American-style Asian options. The problem can be formulated in terms of a solution to a free boundary problem for a parabolic partial differential equation. We focus on the qualitative and quantitative analysis of the early exercise boundary position. The first order Taylor series expansion of the early exercise boundary close to expiry is constructed. We furthermore propose an efficient numerical algorithm for determining the early exercise boundary position based on the front fixing method. Construction of the algorithm is based on a solution to a non-local parabolic partial differential equation for the transformed variable representing the synthesized portfolio. Numerical results and comparisons of our method and the projected successive over-relaxation method studied by (Dai and Kwok 2006) are presented. This is a joint work with Tomas Bokes (arXiv:0912.1321).

Simulation of the 2D and 3D Stratified Flows in Atmospheric Boundary Layer

Šimonek Jiří, Kozel Karel, Tauer M. and Jaňour Zbyněk

Czech Technical University in Prague, Czech Republic

Abstract: The work deals with the numerical solution of the 2D and 3D turbulent stratified flows in atmospheric boundary layer over the simplified hill. Reynolds averaged Navier-Stokes equations (RANS) for incompressible turbulent flow with addition of the equation of density change (Bousinesq model) have been used as a mathematical model.
The density is changing linearly from $\rho_{0}$ (near the Earth's surface) to $\rho_{h}$ (in the maximum height above the Earth's surface). The hydrostatic pressure function has been used to evaluate pressure, which decreases with the increasing height.
The artificial compressibility method together with the finite volume method has been used as discretization method in all solved cases. The Lax-Wendroff scheme (MacCormack form) has been used with Cebecci-Smith algebraic turbulence model.

Acknowledgments
This work was partially supported by Research Plan MSM 6840070010 and MSM 6840770003 and grant GAČR no. 201/08/0012.

Theoretical and computational aspects of global constraints in evolutionary problems

Švadlenka Karel

Kanazawa University, Japan

Abstract: We present partial theory and applications for a globally constrained evolutionary free boundary problem. Approximate solutions in the existence proof are constructed via a minimization method and the same method proves to be efficient also in numerical realization. We show a few examples of computational results for coupled problems, which are stable in spite of using an explicit sc

The best constant of Sobolev inequality corresponding to clamped-free boundary value problem for (−1)^M (d/dx)^2M

Takemura Kazuo

Nihon University

Abstract: PDF version

Reaction-diffusion equations in combustion

Tomica Vít

Czech Technical University in Prague, Czech Republic

Abstract: Combustion is a chemical process during which a fuel is turned into heat energy. Its course is strongly influenced by physical phenomena such as viscosity, heat conduction and diffusion. Reaction-diffusion equations are usually used to the modelling of physical-chemical processes. During combustion of preximed gases under certain conditions a patterns such as circles and spirals were observed. This phenomenon is described by Scott-Wang-Showalter model which includes a system of two partial differential equations. The equations are discretized by finite difference or finite volume method and semi-implicit time-stepping scheme is used. Circular patterns and also an influence of the advection were simulated using schemes mentioned above.

Numerically repeated support splitting and merging phenomena in a porous media equation with strong absorption

Tomoeda Kenji

Osaka Institute of Technology, Japan

Abstract: Numerical experiments to nonlinear diffusion equations suggest several interesting phenomena. One of them is the occurrence of numerical support splitting phenomena caused by strong absorption [1]. The most remarkable property is that the interaction between diffusion and absorption causes numerically repeated support splitting and merging phenomena. More in PDF version.

[1] T.Nakaki and K.Tomoeda, A finite difference scheme for some nonlinear diffusion equations in absorbing medium: support splitting phenomena, SIAM J. Numer. Anal., 40(2002), 945–964.

Numerical solution of 2D and 3D inviscid and viscous transonic flows

Trefilík Jiří, Huml Jaroslav and Kozel Karel

Czech Technical University in Prague, Czech Republic

Abstract: The contribution will present some results for solution of 2D inviscid transonic flows in the GAMM channel and through the 8% DCA cascade. Also the results of viscous turbulent flow problems using turbulence models (Baldwin­Lomax and Wilcox k­ω model) will be presented. 3D flows (inviscid) over a swept wing will be also presented and compared to other numerical results. Transonic flows through the DCA 8% cascade will be compared to experimental results of IT AS CZ.

Stationary patterns of a reaction-diffusion-advection system

Tsujikawa Tohru

University of Miyazaki, Japan

Abstract: We are concerned with a reaction-diffusion-advection system which describes phase transition phenomena on Platinum surf For this system, several stationary patterns have been shown by numerical simulations. Here, we discuss the sufficient solutions. Next, we introduce a shadow system in the limiting case that diffusion coefficients tend to infinity and sho dimensional domain.

Synchronized Flux Corrected Remapping of Mass, Momentum and Energy for ALE Simulations

Váchal Pavel, Liska Richard, Shashkov Mikhail and Wendroff Burton

Czech Technical University in Prague, Czech Republic

Abstract: We present a new method for synchronized flux-corrected remapping (conservative interpolation) of variables between two meshes, primarily designed for the Arbitrary Lagrangian-Eulerian (ALE) simulations in fluid dynamics and plasma physics. Our recently proposed method has been extended also for energy, so that now it simultaneously remaps conserved variables - mass, momentum and total energy - while preserving local bounds in density, velocity and internal energy. This allows more general application and prevents over-correction of fluxes which arises in existing methods that treat the variables separately or sequentially.

Hierarchical structure of Green functions and the best constant of Sobolev inequality corresponding to a bending problem of a beam

Yamagishi Hiroyuki

Tokyo Metropolitan College of Industrial Technology, Japan

Abstract: PDF version

Numerical study on Hele-Shaw flows in a time-dependent gap

Yazaki Shigetoshi

University of Miyazaki, Japan

Abstract: The Hele-Shaw flow describes motion of a viscous fluid in almost two dimensional space between two parallel plates. The boundary of the Hele-Shaw-flow-region moves in time dependeing on the gradient of pressure of liquid. If the distance of two plates changes in time, then the moving boundary depends on not only the pressure, but also speed of the distance. In the talk, model equation, numerical methods and simulation will be shown. Numerical methods include boundary elemet method, uniformly tangential redistribution method, and the flowing control volume method.

Image segmentation using CUDA implementations of the Runge-Kutta-Merson and GMRES methods

Žabka Vítězslav, Oberhuber Tomáš

Czech Technical University in Prague, Czech Republic

Abstract: Modern GPUs are well suited for performing image processing tasks. We utilize their high computational horsepower and memory bandwidth for image segmentation purposes. We use segmentation technique based on numerical solution of a partial differential equation of the Allen-Cahn type. We implement two different algorithms for solving the equation on the CUDA architecture. One of them is based on the Runge-Kutta-Merson method for the approximation of solutions of ordinary differential equations, the other uses the GMRES method for the numerical solution of systems of linear equations. The CUDA implementations of both algorithms are much faster than corresponding single threaded CPU implementations.