# Mathematical Modelling and Numerical Simulation of Pollution Transport in the Boundary Layer of Atmosphere

## P. Bauer

### Czech Technical University in Prague, Czech Republic

Abstract: Air pollution is one the serious problems in almost all countries, especially those with high population density and large industrial centers. We create a mathematical model based on Navier-Stokes equation for media flow and diffusion-convection equation describing pollution transport, and solve the model using finite element methods (FEMs). The FEMs allows us to use different terrain shapes and examine the terrain influence. Essential part of the work is numerical analysis, which examines the properties of algorithms with respect to numerical parameters, and analyzes selected cases of media flow in boundary layer of atmosphere, and pollutant adsorption on the surface in the case of instant, steady or periodicaly interupted source of pollution. The most significant results are obtained in direct application to real-based problems like the tranport of pollution caused by a stack with time dependent intensity or the model of ecological accident.

# Computational Model of Fluid Flow Induced by a Temperature Profile

## M. Bačák

### Czech Technical University in Prague, Czech Republic

Abstract: The main goal of our work is to develop a mathematical model of convection in liquids induced by temperature difference. The solved problem is described as the Benard convection. We consider the Boussinesq approximation which proceeds from the system of Navier-Stokes equations after vanishing inconsiderable terms. To find a numerical solution of our problem we use finite element method applied in numerical software FEMLAB. We attempt to modify some procedures in the program at present.

# Generalization of the Benchmark Solution for the Two-Phase Porous- Media Flow

## R. Fučík

### Czech Technical University in Prague, Czech Republic

Abstract: The contribution presents a discussion of one-dimensional models of the two-phase flow through porous media used in verification of more complex numerical models. We give information how a particular solution of the simplified model based on the McWorther-Sunada formula can be generalized. To enlarge the class of admissible boundary and initial conditions, we offer a numerical algorithm solving the transport equation for saturation, which is based on the Finite-Difference Method in space and time and yields values of the solution at given time levels and on a spatial grid of positions. The use of the algorithm is demonstrated on a series of computations in one-dimensional spatial domain.

# Numerical Solution of Inviscid and Viscous Flows Using Modern Schemes and Quadrilateral or Triangular Mesh

## J. Fürst and K. Kozel

### Czech Technical University in Prague, Czech Republic

Abstract: The work deals with the numerical solution of inviscid and viscous problems, mainly directed to transonic flows computation. The problem of inviscid flows is described by the system of Euler equation for compressible medium and one mentions new schemes suitable for computation of transonic flows without oscillations near shock waves like TVD, ENO and composite schemes on quadrilateral or triangular meshes for 2D flows. The other problem mentioned here is a problem of transonic viscous flows computation and relation between physical and artificial viscosity for high Re. Some numerical examples of transonic flows through 2D and 3D cascades or in a channel are also presented.

# Conley Index and Numerical Verification

## Y. Hiraoka

### Osaka University, Osaka, Japan

Abstract: The purpose of my work is to develop a numerical technique to prove the existence of stationary solutions and to detect connecting orbits among them in dissipative PDEs. The Conley index, topological quantities defined on invariant sets in dynamical systems, is applied to these problems. We consider the cubic Swift-Hohenberg equation as an example and study how to rigorously verify stationary solutions and connecting orbits among them. An effective algorithm by using FFT for this verification technique is also shown. This algorithm is applied to study the snaky bifurcation structure appearing in the quintic Swift- Hohenberg equation.

# Interfacial Phenomena in Fluid Dynamics: Linking atomistic and macroscopic properties

## R.F. Holub

### Colorado School of Mines, Golden, USA

Abstract: Many fields, like solid state physics and material sciences, are trying to introduce multiscale modeling of complex materials in order to obtain better agreement between theory and experiment and eventually to be able, for instance, to predict a failure in a given material. Using as an example the transport of Rn through porous media, and also of the so called "surface diffusion", in dependence on temperature, which involves both molecular theory diffusion (Fick's equation) and viscous flow (Navier-Stokes equation), we will attempt to show that the atoms of Rn behave in ways that cannot be described by a mere macroscopic continuum approach. We will show specifically interplay of some quantum effects, surface structure and interface morphology. It is possible that even the intractable problem of "fingering" might be clarified by employing such microscopic approach. The biggest practical problem for experimentalists and engineers is to have models that "marry" mathematically the microscopic (atomic) aspects to to the macroscopic (continuum) ones (also called "handshaking"), and to come up with practical models that can predict the behavior of transporting fluids. We think this is an opportunity for applied mathematicians to help solve a problem that has been been too long ignored (due to its difficulty?), and which is of great practical importance.

# Flow Over a Given Profile in a Channel with Dynamical Effects

## R. Honzátko, J. Horáček and K. Kozel

### Czech Technical University in Prague, Czech Republic

Abstract: The work deals with a numerical solution of 2D inviscid incompressible flow over profile NACA 0012 in a channel. The finite-volume method in a form of cell-centered scheme at quadrilateral C-mesh is used. Governing system of equations is the system of Euler equations. Some numerical results are compared with experimental data. Steady state solutions of the flow and also unsteady flows caused by the prescribed oscillating behaviour of the profile were computed. The method of artificial compressibility and the time dependent method are used for a computation of the steady state solution.

# Numerical Studies of Cahn-Hilliard Equation and Applications in Image Processing

## V. Chalupecký

### Czech Technical University in Prague, Czech Republic

Abstract: In our talk we present a numerical scheme for solving the Cahn-Hilliard equation which models the phase separation in binary alloys. We consider the Cahn- Hilliard equation both with constant mobility which corresponds to Mullins- Sekerka problem, and with degenerate mobility which approximates the motion by surface diffusion.

The presented numerical scheme based on the method of lines consists of the finite-difference discretization in space leading to a system of ordinary differential equations. This system is solved by the Cash-Karp modification of the Runge-Kutta method which enables us to adaptively change the time step. Care has to be taken when treating the case with degenerate mobility. We present results concerning the experimental order of convergence of our numerical scheme. Finally, we also mention a parallelization technique for our numerical scheme and show some results demonstrating possible applications of the Cahn- Hilliard equation in image processing.

# Propagation Speeds of Disturbances are Bounded for a Class of Reaction-Diffusion Systems

## R. Ikota

### Department of Mathematics, Kyushu University Fukuoka, Japan

Abstract: The aim of our study is to show that propagation speeds of disturbances are bounded for a class of reaction-diffusion systems. It turns out that solutions for various initial states are confined by traveling waves. Thanks to the operator-splitting methodology, which is originally meant for numerical computations, we can make a simple proof.

# A Variational Approach to Very Singular Gradient Flow Equations

## Y. Kashima

### University of Sussex, Brighton, England

Abstract: Some models describing a motion of crystal have strong singularity, therefore they do not make sense mathematically in their original forms. We would like to introduce a method to formulate such singular gradient flow equations and consider the behaviour of their solutions defined properly. The singular gradient of crystalline energy can be formulated by a subdifferential of the convex functional in a suitable functional space. By characterizing the subdifferential operator, we observe the speed of the solution solving the initial boundary value problem. Especially we apply this method to 4th order singular gradient flow equations and investigate the motion of a crystalline surface.

# Variational Mathematical Model to Quasi-Stationary Phase Transition with the Gibbs-Thomson Effect

## M. Kimura and S.Luckhaus

### Kyushu University, Fukuoka, Japan Max Planck Society, Leipzig, Germany

Abstract: We consider a variational mathematical model to quasi-stationary phase transition with the Gibbs-Thomson effect. First, we start from a two-phase quasi-stationary Stefan-Gibbs-Thomson problem, and we consider its natural implicit time discretization. We show that a solution of the discretized problem exists and is given by a variational problem. Next, we consider a new mathematical modeling approach by means of energy minimizing sequences to our phase transition problem. The obtained time discretized model is shown to be equivalent to the previous model. We also give some mathematical estimates for the variational model.

# On Korn's Inequality for Nonconforming Finite Elements

## P. Knobloch

### Charles University Prague, Czech Republic

Abstract: We investigate the Korn first inequality for low order nonconforming finite elements and clarify the dependence of the constant in this inequality on the discretization parameter. Then we use the nonconforming elements for approximating the velocity in a discretization of the Stokes equations with boundary conditions involving surface forces. We show that, in some cases, convergence results can be proved even if the Korn inequality does not hold uniformly with respect to the discretization parameter.

# Arbitrary Lagrangian Eulerian Code for Plasma Simulations

## M. Kuchařík and R. Liska

### Czech Technical University in Prague, Czech Republic

Abstract: Lagrangian codes with mesh moving with the fluid suffer in many cases by mesh distorsion leading to tangled mesh which in principle stops the computation as basic assumptions of the numerical method as e.g. positive volume of cells are not valid any more. Arbitrary Lagrangian Eulerian (ALE) method overcomes this difficulty by untangling or smoothing the mesh and conservative remapping of conserved quantities from the old mesh to the new one. After this Lagrangian computation can continue. We will report on the development of ALE code for simulation of problems from plasma physics.

# Mathematical Model of a Heat Exchanger

## J. Makovička

### Czech Technical University in Prague, Czech Republic

Abstract: We describe a transformation of the Euler equations from the conservative form to the variables of pressure, temperature and mass flow, which are preferred in the applications of the system control. This model is used to describe steam and flue gas flow in two pipes coupled by a wall with finite thickness. Then, we deal with the numerical solution of the system and suggest a variant of a finite-volume scheme which is cell-centered in pressure and temperature and vertex-centered in mass flow rate. A note on a model of the wall and an injection cooler is also included. Finally, we present a comparison with a theoretically computed temperature profile for a stationary state.

# Irreversible Thermodynamics - Application to Fluids and Solids

## F. Maršík

### Czech Academy of Sciences, Institute of Thermomechanics, Czech Republic

Abstract: Thermodynamics of open systems offers a new concept for description of real material objects including the living systems. The II. Law of Thermodynamics can be interpreted as an evolution law of all material systems, which are in interaction with their surroundings. The most important quantity is the entropy, which is defined by the balance law of entropy. The production of the entropy gives an information about the processes into the systems. The convexity of the entropy informs us about the stability of the system states. Under the appropriate outer conditions the fluctuations can to drive the systems to an instability. The consequence is the creation or decay of dissipative structures. When the new dissipative structure appears the system is going further from the thermodynamic equilibrium to the new stable state. However, if the dissipative structure disappears the systems tends to the thermodynamic equilibrium, which from the biological point of view equals to death. In this lecture the application to the stability of fluid flow and solid is shown.

# Modelling Multiphase Flow in Heterogeneous Porous Media

## J. Mikyška

### Czech Technical University in Prague, Czech Republic

Abstract: Multiphase models that simulate the behavior of non-aqueous phase liquids in porous media can be used to obtain fundamental understanding of the complex behavior and predict the fate of waste chemicals in the subsurface. Existing models have limitations in simulating highly heterogeneous systems to be able to represent realistic field conditions. The presentation reports development of a new multiphase flow code called VODA. It starts with a brief introduction of the mathematical model of the multiphase flow in porous media. Then, the Control Volume Finite Element (CVFE) discretization is described and finally, several numerical experiments are presented concerning validation of the developped code. Experimental convergence analysis is carried out for two well-known onedimensional two-phase flow problems. Examples of several further two-phase flow computations in a heterogeneous medium will be also given.

# Numerical Simulation of Dislocation Dynamics

## V. Minárik, J. Kratochvíl, K. Mikula and M. Beneš

### Czech Technical University in Prague, Czech Republic, Slovak Technical University, Bratislava

Abstract: The aim of this contribution is to present the current state of our research in the field of numerical simulation of dislocations moving in crystalline materials. The simulation is based on recent theory treating interactions of dislocation curves and dipolar loops, both occurring in the material and interacting by means of forces of elastic nature. The mathematical model describes the motion and interaction laws for one dislocation curve and finite number of dipolar loops placed in 3D space. The interactions occur not only between the dislocation curve and dipolar loops but also between dipolar loops themselves, which makes the model more complex. Equations of motion for a parametrically described dislocation curve are discretized by the flowing finite volume method in space. The interaction force is computed for each dipolar loop and along the discretized curve. The resulting system of ordinary differential equations is solved by a higher order time solver.

# Numerical Computations to Moving Boundaries in Multi-Dimensional Porous Medium Equations

## T. Nakaki and H. Murakawa

### Kyushu University, Fukuoka, Japan

Abstract: We are concerned with the numerical scheme to multi-dimensional porous medium equations. The basic idea of our scheme is to use a singular limit solution of a reaction-diffusion system, and the advantage is that moving boundaries can be easily and clearly captured at low computational cost. We note that, by using similar idea, we already propose the scheme to Stefan problems, and some mathematical justifications are made. The aim in this contribution is to review the results on Stefan problems, and to propose a numerical scheme to the porous medium equations. Some numerical simulations shall be also demonstrated.

# A Structure of Self-Excitation Movement of a One-Dimensional Free Boundary Problem of Hyperbolic Type

## K. Nakane

### Osaka Institute of Technology, Osaka, Japan

Abstract: The phenomena of trying to remove a thin tape from a table can be seen in ordinary life. However, because the length of the peeled tape changes with time, Fourier analysis can not be applied. Therefore, it is difficult to represent this phenomena mathematically. We treat this phenomena as a variational problem. By supposing that the tape is not distorted, the action integral is a hyperbolic type. To find a stationary point of this functional, we solved Euler-Lagrange equation, and it consists a free boundary problem of hyperbolic type.

A sufficient condition for global existence and a well-posedness of this problem have been given. On the other hand, there are physical experiments which are concerned with this problem, and its free boundary carries out periodic behavior. In this talk, we will show the existence of a periodic solution, and explain the structure of periodic movement.

# Numerical Recovery of the Signed Distance Function

## T. Oberhuber

### Czech Technical University in Prague, Czech Republic

Abstract: In our presentation, we discuss a new method for recovering the signed distance function used in the levelset methods. Out algorithm is efficient in case when the initial guess is close to the desired signed distance function as well as in case when the initial guess only defines the levelset of our interest. The approach is based on an evolution PDE of eikonal type. We derive new schemes to overcome the problems with discontinuities of derivatives. As the theoretical background for the equation is not zet developed, we emphasiye features of the numerical algorithm and give several computational examples.

# $L^1$ Conservative Finite-Element Scheme to a Parabolic-Elliptic System Modelling Chemotaxis

## N. Saito

### Toyama University, Japan

Abstract: The aggregation of slime molds caused by their chemotactic features is described as a nonlinear parabolic-elliptic system called a simplified Keller-Segel system. A remarkable property of its solution is the conservation of the $L^1$ norm, and it brings us various information on the behavior of the solution. Especially, if the $L^1$ norm of an initial datum is large enough, the solution may blow up. One of the main purpose of this paper is to make a finite-element scheme which preserves the conservation of a discrete $L^1$ norm. To accomplish the purpose, we apply a conservative upwind finite-element approximation proposed by K. Baba and M. Tabata. Stability and error estimate of the scheme are also established.

# Numerical Simulation of Reaction-Diffusion Dynamics

## R. Straka

### Czech Technical University in Prague, Czech Republic

Abstract: Reaction-diffusion system Brusselator is theoretical model of nonlinear chemical reaction in stirred reactor tank. In the reaction scheme the initial components A and B are transformed into products D and E via the reaction intermediates X and Y. It is well known that this model exhibit various types of solution depending on the characteristic length of reactor tank L. Keeping L small $L\in\langle 0,0.513)$ we obtain trivial invariant set - stable fixed point. If L increases we obtain stable symmetic periodic solution - we crossed first Hopf bifurcation point $L^*=0.513$. The branch of stable antisymmetric solutions bifurcate in neighbourhood of $L^+=1.225$ and further develop to invariant torus. Development of this torus were studied using methods of Poincare mapping and it's fractal dimension and Lyapunov characteristic exponents. Symmetry of solutions and period of periodic solution were studied as well.

# Parallel Finite Element Computation of the Stokes Equations in a Spherical Shell Domain

## A. Suzuki

### Kyushu University, Fukuoka, Japan

Abstract: The Stokes equations in a spherical shell domain with slip boundary conditions plays a key role in a mathematical model of the Earth's mantle convection problem. The other equation concerned in the problem is the convection-diffusion equation. In the simulation of the Earth's mantle convection problem, almost all the time is consumed in solving the Stokes problem. Consequently a fast Stokes solver is required. We have developed a computation code for shared memory-type parallel computers using P1/P1 finite element approximation with a stabilization technique for discretization, an orthogonal projection onto a solution space for treatment of slip boundary conditions, and a preconditioned conjugate gradient method for the indefinite symmetric stiffness matrix. We show efficiencies of our code in numerical simulation and also show an iterative substructuring method for distributed memory-type parallel computers.

# Evolution of Curves on a Surface Driven by the Geodesic Curvature and External Force

## D. Ševčovič

### Comenius University, Bratislava, Slovakia

Abstract: We study a flow of closed curves on a given surface driven by the geodesic curvature and external force. We use a direct method for solving the evolution of surface curves based on vertical projection to the plane. It is shown that this geometric problem can be reduced to a solution of a fully nonlinear system of parabolic differential equations. We prove short time existence of classical solutions. Various Lyapunov like functionals for the flow of surface curves are derived. A special attention is put on the analysis of closed stationary surface curves. We give sufficient conditions for their dynamic stability. We also discuss an important link between the geodesic flow and the edge detection problem in the theory of image segmentation. An efficient numerical scheme for solving the governing system of equations is presented. Several computational examples of evolution of surface curves driven by the geodesic curvature and external force on various surfaces are presented.

# The Mass-Conservative Upwind Finite Element Approximation and its Application to the Density-Dependent Navier-Stokes Equations

## M. Tabata

### Kyushu University, Fukuoka, Japan

Abstract: At first we review the mass-conservative upwind finite element approximation developed by the author et al. in 1981, which is a source of the finite volume method. This approximation has such a nice property that it not only preserves the mass-conservation but also satisfies the discrete maximum principle when the velocity is incompressible. Applying the approximation to the density-dependent Navier-Stokes equations, whose unknowns are the density, the velocity, and the pressure, we show how the approximation works well in proving the convergence of the finite element solutions to the exact one.

# Numerical Computations of Glass Melting Process

## D. Tagami

### Kyushu University, Fukuoka, Japan

Abstract: A class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients, and its application to glass melting process are considered. Such variable coefficients make the diffusion and the buoyancy terms nonlinear, whose estimates are key points in our analysis. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Numerical results are shown that the numerical convergence order agree well with theoretical one. Moreover, some numerical results on glass melting process also are shown.

# Support Splitting, Connecting, and Re-splitting Phenomena in the Flow through an Absorbing Medium

## K. Tomoeda

### Osaka Institute of Technology, Osaka, Japan

Abstract: An interaction between diffusion and absorption shows several interesting phenomena in the dynamical behavior of the flow through an absorbing porous medium. The model equation is written in the form of the nonlinear diffusion equation with strong absorption. In the case where the diffusion is active rather than the absorption, the region which is occupied by the flow never split. In the opposite case, the flow becomes extinct in a finite time and the region, while initially connected, splits into multiple connected components as time increases. Thus support splitting phenomena appear. Moreover, the complicated behavior of the support is given by the numerical computation; that is, after support splitting phenomena appear, the support becomes connected, and thereafter the support splitting phenomena appear again. In this talk the mathematical justification will be stated.

# Pulse Dynamics in Spatially Inhomogeneous Medium

## D. Ueyama

### Department of Mathematical and Life Sciences, Hiroshima University, Japan

Abstract: Recently, both of spacial and temporal inhomogeneity in the nature have been interested in many area. From the view point of an applied mathematics, it is natural to include the inhomogeneous effect to the model equation because of the nature has inhomogeneity all time. We have considered the effect of spatial inhomogeneity on traveling pulses arising in reaction-diffusion (RD) systems. Propagating fronts arising in scalar RD equation with spacial inhomogeneity have been well studied. However, little is known on traveling pulses in RD systems of equations. We treat two component RD systems with mono-stable excitable nonlinearities where the diffusion coefficients periodically change on position. Our interest is motion of pulses in such inhomogeneous medium. Numerical simulations reveal that motion of a pulse has quite different propaties compare with the psedo-traveling frontal wave arising on scalar RD equation with spacial inhomogeneity.

# Multi-Scale Finite Element Analysis of Piezoelectric Ceramics Based on A Crystallographic Homogenization Method

## Y. Uetsuji, S. Ueda and E. Nakamachi

### Department of Mechanical Engineering, Osaka Institute of Technology, Osaka, Japan

Abstract: Polycrystalline piezoelectric materials are an aggregation of crystal grains and domains with uneven forms and orientations. Therefore, their macroscopic ferroelectric characteristics should be evaluated through introducing a microscopic in-homogeneity of crystal morphology. In this study, a multi-scale finite element modeling procedure based on a crystallographic homogenization method has been proposed to describe a macroscopic behaviors of polycrystalline ferroelectrics with consideration of crystal morphology in a microscopic scale, and to evaluate microscopic behaviors in response to arbitrary macroscopic external load. The proposed procedure has been applied to a piezoelectric ceramic, BaTiO3 polycrystal. The crystal orientation dependence of ferroelectric properties has been investigated for BaTiO3 single crystal to obtain the characteristics of crystal grains and domains as a component of polycrystal. Then, the influence of microscopic crystal orientation distribution on macroscopic ferroelectric properties has been revealed for BaTiO3 polycrystal. From the computational results, it was shown that piezoelectric constants of polycrystalline ferroelectrics can be maximized by design of microscopic crystal morphology. Furthermore, microscopic behaviors of BaTiO3 polycrystal under a macroscopic electric field have been characterized by employing the statistical procedure such as the symmetrized dot pattern.

# Fully Two-dimensional HLLEC Riemann Solver and Associated Difference Schemes

## P. Váchal, R. Liska and B. Wendroff

### Czech Technical University in Prague, Czech Republic Los Alamos National Laboratory, Los Alamos, USA

Abstract: Fully two-dimensional sixteen state HLLEC (Harten, Lax, van Leer, Einfeldt, with contact correction) approximate Riemann solver has been developed. The solver is applied to first order Godunov and second order WAF (Weighted Average Flux) finite difference schemes. The results yield improved treatment of contact discontinuities, stationary contacts are resolved exactly.