Czech-Japanese Seminar in Applied Mathematics 2013
The 6th floor (classroom and presentation space), Nakano Campus
Sep. 5 (Thu)

Sep. 6 (Fri)
9:50-10:15 Tohru Tsujikawa
Global solution branches of some ...
10:15-10:40 Robert Straka
Computational model of ...
10:40-10:50 Break
10:50-11:10 Robert Spir
Tracking of cells in Zebrafish ...
11:10-11:30 Radek Maca
Segmentation of 3D MRI images by ...
11:30-13:00 Lunch

13:00-13:10

Opening

13:10-13:35

Peter Frolkovic

13:00-13:25 Akiko Fukuda
Conserved quantities of the integrable ...
13:25-13:50 Yoshiaki Muroya

Semi-implicit finite volume level ...
13:35-14:00

14:00-14:25

Karel Svadlenka

Global stability of Lotka-Volterra

Numerical computation of ...

systems with infinite delays and ...

with nonlocal constraints

13:50-14:00 Break

Karol Mikula

14:00-14:20 Tereza Jindrova
Computation of two-phase equilibrium ...

Surface evolution with ...
14:25-14:45

Break

14:45-15:10

Kensuke Aihara

15:10-15:35

14:20-14:40 Pedro Polvora
Optimal value of a firm investing in ...

Solving sparse linear systems ...

14:40-14:50 Break

Katsuhisa Ozaki

14:50-15:10 Ondrej Polivka
Compositional modeling of two-phase ...

Accurate algorithms for matrix ...
15:35-15:50

Break

15:50-16:40

Daniel Sevcovic (Plenary talk (1))

17:30-19:30

15:10-15:35 Jiri Mikyska
Modeling diffusion in multicomponent ...

On nonlocal geometric flows and ...

15:35-15:45 Break

Welcome reception

15:45-17:15 Poster session (1)(Restaurance at the 1st floor )18:30-20:30 Conference dinner(The 14th floor, SUNPLAZA )Meeting point: 18:00, Entrance

1Sep. 7 (Sat)Sep. 8 (Sun)9:50-10:15 Masato Kimura

9:50-10:15 Takayasu Matsuo

Continuous and discrete crack ...

A discontinuous Galerkin method ...

10:15-10:40 Takeshi Takaishi

10:15-10:40 Pavol Kutik

Complicated propagation of cracks in ...

A stabilized finite volume numerical ...

10:40-10:50 Break

10:40-10:50 Break

10:50-11:10 Miroslav Kolar

10:50-11:15 Daisuke Furihata

Mathematical model and numerical ...

An iteration method for the numerical ...

11:10-11:30 Vladimir Klement

11:15-11:40 Michal BenesGPU multigrid solver for ...

Numerical simulation of flow in ...

11:30-13:00 Lunch

11:40-11:50 Closing and Poster award ceremony

13:00-14:30 Poster session (2)14:30-14:50 Break
14:50-15:40 Yoshihisa Morita (Plenary talk (2))
Spectral comparison and gradient-like ...
15:40-16:00 Break
Departure to ...

16:00-16:25 Norbert Pozar
A viscosity approach to total ...
16:25-16:50 Petr Knobloch
A nonlinear local projection finite ...
16:50-17:15 Hideki Murakawa
Convergence rates of discrete-time ...
SUNPLAZA (in front of Nakano Station):2

List of posters (in alphabetical order)
Sep. 6 (Fri), 15:45-17:15 Poster session (1)1. Masakazu Akiyama (Hokkaido University, Japan)
A mathematical model of planar cell polarity
2. Koichi Anada (Waseda University Senior high school, Japan)
Some features for blow-up solutions of a nonlinear parabolic equations
3. Michal Benes (Czech Technical University in Prague, Czech Republic)(1) Interaction of dynamically evolving dislocations(2) Numerical simulation of anomalous transport in porous media described by fractional-advection dispersion equation
4. Kota Ikeda (Meiji University, Japan)
Collective motions of particles with diffusive interactions
5. Tomoyuki Idogawa (Shibaura Institute of Technology, Japan)
Research on an expansion to the continuous version of morphological contour decomposition-reconstruction
algorithm
6. Tetsuya Ishiwata (Shibaura Institute of Technology, Japan)(1) Spiral-shaped solutions to crystalline motion with a rotating tip(2) Structure-preserving finite difference scheme for Heisenberg equation and its application to a vortex
filament motion
7. Tereza Jindrova (Czech Technical University in Prague, Czech Republic)
General computation of phase equilibrium in multicomponent mixtures
8. Vladimir Klement (Czech technical University in Prague, Czech Republic)
Algebraic multigrid for Navier-Stokes problems
9. Yasuaki Kobayashi (Hokkaido University, Japan)
Mathematical modeling of epidermal homeostasis
10. Miroslav Kolar (Czech technical University in Prague, Czech Republic)
Computational study of planar curve dynamics
11. Radek Maca (Czech Technical University in Prague, Czech Republic)
CFD simulation of two phase flow in fluidized bed boiler geometry
12. Rhudaina Z. Mohammad (Kanazawa University, Japan)
Multiphase volume-preserving interface motions via localized signed distance vector scheme
13. Harunori Monobe (Meiji Institute for Advanced Study of Mathematical Sciences, Japan)
On a free boundary problem describing cell motility
14. Kei Nishi (Hokkaido University, Japan)
Dynamics of two interfaces in a hybrid system with jump-type heterogeneity315. Ondrej Polivka (Czech Technical University in Prague, Czech Republic)
Numerical simulations of two-phase multicomponent compressible flow in porous media
16. Takiko Sasaki (The University of Tokyo, Japan)
A second-order time-discretization scheme for a system of nonlinear Schrodinger equations
17. Nur Shofianah (Kanazawa University, Japan)
Simulation of triple junction motion Ab0 with arbitrary surface tensions Ab0
18. Robert Spir (Slovak University of Technology in Bratislava, Slovakia)
Tracking of cells in Zebrafish embryogenesis by finding centered paths in 4D spatio-temporal structures
19. Robert Straka (AGH University of Science and Technology, Poland)
Computational model of combustion in rock-melting cupolaSep. 7 (Sat), 13:00-14:30 Poster session (2)20. Takayuki Narumi (Kwansei Gakuin University, Japan)
Three-dimensional pattern formation in a chemotaxis system with logistic source
21. Hirotoshi Sato (Kwansei Gakuin University, Japan)
Kinematic equation for open curves with tangential velocities
22. Kenta Uemichi (Kwansei Gakuin University, Japan)
A mathematical model for honeybee comb construction

4

Abstracts ( P osters and T alks in random order)
Robert Straka (AGH University of Science and Technology, Poland)P

T Computational model of combustion in rock-melting cupola

Rock melting cupolas are special shaft furnaces used in mineral wool production. Their design is similar as ofblast furnaces i.e. tube-like shape with tuyeres and siphon in the bottom, feeding and exhaust pipe in the top.
The furnace is fed by mixture of stone and coke and during combustion of coke the stone is melted in very hightemperature to produce lava which is used to make mineral wool. Due to high temperatures inside the furnace andrapid oxygen consumption, gasification of coke naturally occurs. Gasification is source of unwanted CO emmisionand energy losses. We develop a mathematical model of coke combustion and rock melting in cupola in order toinspect the influence of secondary air tuyeres. The model provide valuable insight in operation of rock-melting
cupolas and is able to predict important parameters as lava temperature, flue gas concentrations and temperatureand heat losses to the water jacket. The validation of the model will be also presented.
Keywords: multiphase reacting flow, coke combustion, rock melting, mineral wool production

Takiko Sasaki (The university of Tokyo, Japan)
P A second-order time-discretization scheme for a system of nonlinear Schrödinger equations
We propose a new time discretization scheme for a system of nonlinear Schrödinger equations which is a modelof the interaction of a non-relativistic particles with different masses. Our scheme is composed of two (complexvalued) linear systems at each time step, and the solution is shown to converge at a second order rate. We reportnumerical example to confirm the theoretical results. Our idea can be applied to a large class of nlinear PDEs.
Keywords: finite difference method, nonlinear Schrödinger equation

Takayasu Matsuo (The University of Tokyo, Japan)
T A discontinuous Galerkin method based on variational structures
We consider partial differential equations with certain variational structures, which have invariants or dissipation
properties in relation to the structure. We propose a unified approach for constructing discontinuous Galerkinschemes based on the variational structure, so that the invariants or dissipation properties are kept in the resulting
schemes.
This is a joint work with Yuto Miyatake and Yoshifumi Aimoto.

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Pavol Kutik (Slovak University of Technology, Slovakia)
T A Stabilized Finite Volume Numerical Scheme for Solving the Partial Di erential Equation in the Heston Model
The article is aimed to provide a stable numerical scheme, based on the finite volume method discretization,
for solving the partial di erential equation arising in the Heston model. In order to build a scheme which does
not violate the discrete minimum-maximum principle a diamondcell approximation of the gradient is used anda splitting into an inflow/implicit and an outflow/explicit part is applied. By the use of appropriate weights, asuciently large set of the outflow fluxes are transfered to the corresponding inflow fluxes of the neighbouring nite
volumes. We illustrate the stability and accuracy of the scheme on two numerical experiments, the former one
with a European binary option and the latter one with a European call option.
Keywords: Finite volume method, Heston model, Stabilized scheme

Petr Knobloch (Charles University, Czech Republic)
T A nonlinear local projection finite element method for convection-diffusion-reaction equations
This contribution is devoted to the numerical solution of convection-diffusion-reaction equations by means of
the finite element method (FEM). It is well known that the standard Galerkin discretization is inappropriate
if convection dominates diffusion since the approximate solution is usually globally polluted by large spurious
oscillations. The usual way of treating dominating convection consists in adding extra terms to the Galerkinformulation, aimed at enhancing the stability of the approximate solution. Among such stabilized FEMs, thelocal projection stabilization (LPS) method has received some attention over the last decade. Originally proposed
for the Stokes problem and extended to the Oseen equations, the LPS method has also been used recently to treat
convection-diffusion equations.
We introduce an extension of the LPS FEM and analyze it both in the steady-state and transient settings. In
addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced, which accounts for the
reduction of spurious oscillations. The time-dependent equation is discretized in time with an implicit one-steptheta-scheme. We prove the existence of a solution and, depending on the choice of the stabilization parameter,
also its uniqueness. To our best knowledge, this is the first nonlinear discretization for convection-diffusion-reaction
equations for which both existence and uniqueness of a solution can be shown. Moreover, we derive error estimates
which are supported by numerical studies. These studies demonstrate also a reduction of the spurious oscillations.
Keywords: finite element method; local projection stabilization; crosswind diffusion; convection-diffusionreaction equation; well posedness; time dependent problem; stability; error estimates

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Katsuhisa Ozaki (Shibaura Institute of Technology, Japan)
T Accurate Algorithms for Matrix Multiplication with Error-Free Splitting
Recently, we proposed accurate algorithms for matrix multiplication. First, the matrix multiplication is transformed into an unevaluated sum of several floating-point matrices. Next, we evaluate the sum of floating-point
matrices by accurate summation algorithms. The algorithm is fast in many cases, compared to component-wise
accurate dot product algorithms. We introduce recent progress of our accurate algorithms for matrix multiplication: improvement of error-free spitting, application to interval arithmetic, reduction of working space and soforth as long as time allows.

Kei Nishi (Hokkaido University, Japan)
P Dynamics of two interfaces in a hybrid system with jump-type heterogeneity
We study the behavior of a localized moving pattern arising in an activator-inhibitor-type reaction-diffusion
system with heterogeneity. It is well known that front solutions typically appear in bistable systems. When the
nonlinearity of the system is slightly perturbed from the odd symmetry, such front solutions are glued together
to form a spatially localized pulse (front-back pulse). The front-back pulse exhibits not just traveling but also
oscillatory motions via the Hopf bifurcations depending the parameters. The global behaviors of such pulse
solutions have already been studied numerically and analytically.
In this study, we consider the traveling pulse dynamics in the jump- type heterogeneous media, in which one
constant parameter value jumps up (or down) to another constant value at one point in space. By numerical simulation, it is found that five different behaviors occur through the collision between the pulse and the heterogeneity.
Especially, we focus on the sliding motion of an oscillating pulse observed in some parameter regime.
To clarify the mechanism for this behavior, the system is first reduced to a singular limit system, which we calla hybrid system. For hybrid system, the equation for the activator in the original PDEs is replaced by two ODEs
for the motion of the interfaces, which are coupled with a PDE for the inhibitor field. It is numerically confirmedthat this reduced hybrid system reproduces the pulse dynamics for the original PDEs qualitatively very well.
By applying the center manifold theory near Hopf bifurcation point, the hybrid system is further reduced tofinite dimensional ODEs to analyze the sliding motion. The hybrid system facilitates the calculation needed forthe reduction. The details will be shown in the poster session.
Keywords: Dynamical System, Reaction-Diffusion Equations, Center Manifold Reduction

Kensuke Aihara (Tokyo University of Science, Japan)
T Solving sparse linear systems with an enhanced implementation of IDRstab
The IDRstab method is often more effective than the conventional Krylov subspace methods for solving large
nonsymmetric linear systems. In this talk, we propose an alternative implementation of IDRstab to improve theaccuracy of the approximate solutions. Our variant also reduces the computational costs by saving the number7

of vector updates. Numerical experiments show that our variant is more effective than the original IDRstab forsparse linear systems.
Keywords: linear systems, Krylov subspace method, Induced Dimension Reduction, IDRstab method

Akiko Fukuda (Tokyo University of Science, Japan)
T Conserved quantities of the integrable discrete hungry Lotka-Volterra system
In this talk, we derive new conserved quantities of the integrable discrete hungry Lotka-Volterra system, which
has an application to computing eigenvalues of banded matrices, from the characteristic polynomial of the matrices.
Conserved quantities of the integrable discrete hungry Toda equation and the box and ball system with M kindsof balls are also presented.
Keywords: Conserved quantities Discrete hungry Lotka-Volterra system Discrete hungry Toda equation Box
and ball system

Takeshi Takaishi (Hiroshima Kokusai Gakuin University, Japan)
T Complicated propagation of cracks in a 3 dimensional phase-field model
Complicated phenomenon of crack propagation in 3 dimensional material is studied. Usefulness of the phasefield model for 3 dimensional crack growth that Kimura and the author introduce from Francfort-Marigo’s energy
is verified.
Keywords: Numerical simulation, phase-field, mathematical modeling

Karol Mikula (Slovak University of Technology, Slovakia)
T Surface evolution with tangential redistribution
Authors: Karol Mikula, Mariana Remesikova, Peter Sarkoci and Daniel Sevcovic
We propose several techniques for tangential redistribution of points on evolving surfaces. This is an important
issue in numerical approximation of any evolution model, since the quality of the mesh has a significant impact
on the result of the computation. We explain the volume-oriented and length-oriented tangential redistribution
methods in a general setting of an m-dimensional manifold evolving in an n-dimensional manifold. After, weapply the proposed techniques to the particular case of mean curvature evolution of surfaces in R3 . We explain
the numerical approximation of the model and present several experiments illustrating the performance of theredistribution techniques.

8Radek Maca (Czech Technical University in Prague, Czech Republic)
P CFD simulation of two phase flow in fluidized bed boiler geometry
This contribution presents the CFD simulation of two phase flow in fluidized bed boiler geometry. The CFD
simulation is performed using a free, open source CFD software package OpenFOAM. OpenFOAM provides awide range of features to solve complex fluid flows. Computations were performed both for 2D and 3D geometry.
Partially, 1D model is discussed.
Acknowledgement: Advanced Control and Optimization of Biofuel Co-Firing in Energy Production, project No.
TA01020871 of the Technological Agency of the Czech Republic
Keywords: fluidized bed, two phase flow, OpenFOAM
T Segmentation of 3D MRI Images by Degeneare Diffusion Method
The contribution presents a 3D (2D+time) segmentation of the real cardiac MRI data using the level set
formulation of the geodesic active contour model and its semi-implicit complementary volume discretization. In
particular, the application is focused on the segmentation of the heart ventricles from the cine MRI data. The
algorithm is compared to the Allen-Cahn approach, software package SEGMENT and to manually segmented
data.
Acknowledgement: Advanced Supercomputing Methods for Implementation of Mathematical Models, projectof the Student Grant Agency of the Czech Technical University in Prague No. SGS11/161/OHK4/3T/14
Keywords: image processing, image segmentation, level set, MRI

Masato Kimura (Kanazawa University, Japan)
T Continuous and discrete crack propagation models with energy gradient property
Keywords: crack propagation, gradient flow

Takayuki Narumi (Kwansei Gakuin University, Japan)
P Three-Dimensional Pattern Formation in a Chemotaxis System with Logistic Source
We study three-dimensional pattern formation in a chemotaxis-diffusion-growth system. In one- and twodimensional domains we have found various nonuniform spatial patterns; however, we still do not know muchabout pattern formations in three-dimensional domains. In our poster presentation, we will review two-dimensional
pattern formations near bifurcation points, and discuss some sufficient conditions for stability of piled-hexagonal
patterns in three-dimensional rectangular parallelepiped.
Keywords: chemotaxis, pattern formation, three-dimensional pattern, bifurcation, numerical calculation

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Kenta Uemichi (Kwansei Gakuin University, Japan)
P A Mathematical Model for Honeybee Comb Construction
Honeycombs consist of several combs, of which structure is made of the hexagonal two sides. The hexagonal
structure has fascinated lots of people, but the mechanism of the emergence has not been unraveled yet. We
have examined the behavior of European honeybees at the first stage of comb construction. From the result, we
propose a mathematical model for honeybee comb construction.
Keywords: pattern formation, insect, honeybee, taxis

Tohru Tsujikawa (University of Miyazaki, Japan)
T Global solution branches of some shadow system
We consider the stationary problem of a reaction-diffusion-advection system. In order to find clues for the
global structure of non-constant solutions of the system, we introduce a shadow system in the limiting casethat the advectoin coefficients tends to infinity. Our methods to solve it are based on the bifurcation, singular
perturbation, a level set analysis.
Keywords: bifurcation, stationary solution, reaction-diffusion system

Hirotoshi Sato (Kwansei Gakuin University, Japan)
P Kinematic Equation for Open Curves with Tangential Velocities
The kinematic equation was proposed by Brazhnik, Davydov, Mikhailov and Zykov for analyzing the spiralwave motion in an excitable medium. In their paper, they pointed out that if the wave sprouts up or retracts
from its free end(s), there is a need to reset the arc-length variable at each time in the equation. In our study, we
will refine this ambiguity in the mathematical model of kinematic equation.
Keywords: kinematic equation, pattern formation, spiral wave, BZ reaction

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Robert Spir (Slovak University of Technology, Slovakia)
P

T Tracking of cells in zebrafish embryogenesis by finding centered paths in 4D spatio-temporal structures

Authors: Karol Mikula and Robert Spir
The paper presents numerical algorithms leading to automatic cell tracking and reconstruction of the cell lineage
tree during the first hours of animal embryogenesis. We present results obtained for large-scale 3D+time confocal
microscopy image sequences of early stages of Zebrafish embryo development. Our approach consists of threebasic steps - the image filtering, the cell centers detection and cell trajectories extraction yielding the lineage tree
reconstruction. In all three steps we use nonlinear partial differential equations. For the filtering the geodesic mean
curvature flow in level set formulation is used, for the cell center detection the motion of level sets by a constant
speed regularized by mean curvature flow is used and the solution of the eikonal equation is essential for the celltrajectories extraction. The core of our new tracking method is an original approach to cell trajectories extraction
based on finding a centered paths inside the spatio-temporal tubular structures representing cell movement anddivisions. The paths are found by using a properly designed distance functions from cell centers detected inevery time step and by a backtracking in steepest descent direction of a potential field built from these distancefunctions. We also present efficient and naturally parallelizable discretizations of the aforementioned nonlinear
PDEs and discuss properties and results of our new tracking method.
Keywords: 4D spatio-temporal tracking, cell tracking

Masakazu Akiyama (Hokkaido University, Japan)
P A Mathematical Model of Planar Cell Polarity
In considering the morphogenesis of tissue of multicellular organisms, the orientation of the cell placement isvery important. For example, in the inner ear of a person, a function

hear the soundis produced by cells

called hair cell arranged in a particular direction. The phenomenon caused by individual cells, such as having
an asymmetric spatial pattern are called

PCP(Planar cell polarity). Not only in hair cells but also in thevarious organisms and tissue, PCP is seen as a universal phenomenon. Hence many researchers have been study
about PCP phenomena. Since easy to control experimental environment and abundance of generic information,
There are a lot of study of PCP in Drosophila (Fly). For instance, there is small hair on the wing of Drosophila,
the hair has same orientations. This is also function of PCP. Protein that includes the seven-pass transmembrane
receptor Frizzled (Fz), the four-pass transmembrane protein Strabismus (Stbm), the golgi-kinase Four-jointed (Fj)and atypical cadherin Dachsous (Ds) in the cell is critical for PCP to work. In Drosophila, PCP molecules such as
Ds and Fj may function as global directional cues orienting cellular asymmetry, which is manifested as polarized
localization of the PCP core proteins such as Fz. However, the relationship between Ds/Fj gradient slopes and Fz
asymmetry in the eye is opposite to that in the wing, thereby causing controversy about how these two systems areconnected. In this study, we show our mathematical model which is constructed from only 3 term. Mathematicalmodel is very simple, various aspect of PCP are shown using numerical simulations. To construct model, we

11

proposed the key concept of GLOBAL and LOCAL rules. In the poster, we also show about the relationship theserules and proteins in the cell.
Keywords: Cell Polarity, Mathematical Model, Numerical Simulation

Daisuke Furihata (Osaka University, Japan)
T An iteration method for the numerical solution of the discrete variational derivative schemes
The discrete variational derivative method (DVDM) is a structure-preserving numerical method for partial
differential equations.
The obtained schemes inherit the conservation property or dissipation one.
If the original equation is nonlinear, the normal DVDM schemes are nonlinear.
This nonlinearity means that we should pay much computational cost to obtain numerical solutions.
There exists a linear-decomposition technique to consist linear DVDM schemes, but this technique is applicable
to only low order polynomial equations.
We consider a new generalized decomposition technique and to use the obtained extended scheme as a predictorof the iteration solver for the original DVDM scheme.
Keywords: discrete variational derivative method, structure preserving, iteration method, predictor

Yasuaki Kobayashi (Hokkaido University, Japan)
P Mathematical modeling of epidermal homeostasis
We present a mathematical model of the epidermis, consisting of keratinocyte dynamics including cell divisionand hard-core repulsive interactions, and the calcium dynamics based on the one-pool model. We show thatexperimentally observed calcium localization phenomena under the stratum corneum, which is considered to beimportant for the epidermal homeostasis, can be reproduced by our model. We further introduce an evaluation
function that quantifies the stability of the epidermis and investigate what is responsible for a stable epidermal
structure.
Keywords: partial differential equation, dynamical system, reaction-diffusion systems

12

Yoshiaki Muroya (Waseda University, Japan)
T Global stability of Lotka-Volterra systems with infinite delays and feedback controls
In this talk, to multiple species Lotka-Volterra systems with infinite delays and feedback controls, applying
Lyapunov functional techniques, we establish sufficient conditions of the global stability for the unique saturated
equilibrium. The feedback controls can be eliminated from the Lyapunov functional, whereas feedback controls
have influence on boundedness conditions for the positive density of saturated equilibrium and change the positionnof a unique saturated equilibrium, and for the saturated equilibrium on the boundary of R+, we obtain new sharper

criteria on stability conditions. Moreover, we extend this to a model with patch structure and obtain the similar
result to multi-group SIR epidemic model with patch structure.
Keywords: Lotka-Volterra system, feedback control, saturated equilibrium, global stability, infinite delay
MSC: 34K20, 34K25, 92D30

Tomoyuki Idogawa (Shibaura Institute of Technology, Japan)
P Research on an Expansion to the Continuous Version of Morphological Contour Decomposition-Reconstruction
Algorithm
We had proposed an algorithm to decompose and reconstruct binary images by using morphological contours.
In this algorithm, images were treated as binary discrete ones, namely, were regarded as subsets of Z 2 . In thisresearch, we will try to expand the algorithm to binary but continuous settings (subsets of R2 ) and consider themeanings of our “2 contours” in the Euclidean space R2 .
Keywords: mathematical morphology, image decomposition-reconstruction, continuous setting

Hideki Murakawa (Kyushu University, Japan)
T Convergence rates of discrete-time schemes to approximate nonlinear cross-diffusion systems
This talk is concerned with nonlinear and linear discrete-time algorithms for cross-diffusion systems. The
nonlinear scheme corresponds to backward differences in time. The linear algorithm is a very easy to implement
scheme we proposed. The main purpose of this talk is to derive convergence rates of the discrete-time schemes.
We obtain the same orders for both the nonlinear and the linear schemes. Moreover, these orders are optimal.
We also establish uniqueness and regularity results of weak solutions of the cross-diffusion systems.
Keywords: cross-diffusion systems, discrete-time schemes, optimal error estimates, uniqueness13

Tetsuya Ishiwata (Shibaura Institute of Technology, Japan)
P Spiral-shaped solutions to crystalline motion with a rotating tip
P Structure-preserving finite difference scheme for Heisenberg equation and its application to a vortex filament motion

Harunori Monobe (Meiji University, Japan)
P On a free boundary problem describing cell motility

Norbert Pozar (Kanazawa University, Japan)
T A viscosity approach to total variation flows of non-divergence type

Kota Ikeda (Meiji University, Japan)
P Collective motions of particles with diffusive interactionsPeter Frolkovic (Slovak University of Technology, Slovakia)
T Semi-implicit finite volume level set methods
In this talk we introduce new semi-implicit finite volume discretization method for the numerical solutions oflevel set advection equation to capture moving interfaces. The method is based on the implicit in time treatmentof fluxes at inflow boundaries and the explicit in time treatment of outflow boundaries. The resulting algebraic
systems of equations can be solved efficiently using Gauss-Seidel iterative method in four (2d) or nine (3d)directions. Several numerical examples will illustrate the advantages of this method. Some applications likewildland fire spreading or mobing groundwater table will be briefly discussed. This talk is based on the joint workwith Karol Mikula and Jozef Urban.
Keywords: advection equation, level set method, finite volume method, semi-implicit method

Koichi Anada (Waseda University Senior High School, Japan)
P Some Features for Blow-up Solutions of a Nonlinear Parabolic Equations
In previous studies we have shown some conjectures for behavior of blow-up solutions to a nonlinear parabolic
equations. They are very important features to investigate behavior of solutions near their blow-up time.The
purpose of our talk/poster is to prove one of them that we call “weak eventual monotonicity”.
Keywords: parabolic equations, blow-up solutions, eventual monotonicity

14

Daniel Sevcovic (Comenius University, Slovakia)
Plenary T

On nonlocal geometric flows and minimization of isoperimetric ratio in relative geometries

We analyze a gradient flow of closed planar curves minimizing the soperimetric ratio in the relative Finsler
geometry. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends
on the total interfacial energy and enclosed area of the curve. The governing system of equation consists ofnonlinear parabolic equations with nonlocal terms. In contrast to the gradient flow for the isoperimetric ratio, weshow there exist initial curves for which the enclosed area is decreasing with respect to time. We also derive amixed anisoperimetric inequality for the product of total interfacial energies corresponding to different anisotropy
functions. Finally, we present several computational examples illustrating theoretical results. This is a joint workwith Shigetoshi Yazaki.
References[1] D. Sevcovic and S.Yazaki: On a gradient flow of plane curves minimizing the anisoperimetric ratio, IAENG
International Journal of Applied Mathematics 43(3) (2013), 160-171.
[2] D. Sevcovic and S.Yazaki: Computational and qualitative aspects of motion of plane curves with a curvatureadjusted tangential velocity, Mathematical Methods in the Applied Sciences, 35(15) (2012), 1784-1798.
[3] D. Sevcovic and S.Yazaki: Evolution of plane curves with a curvature adjusted tangential velocity, Japan J.
Indust. Appl. Math., 28(3) (2011), 413-442.

15

Plenary talkSpectral comparison and gradient-like
property for some reaction-diﬀusion
systems
Yoshihisa Morita
Department of Applied Mathematics and Informatics
Ryukoku University
Seta Otsu 520-0803, Japan
morita@rins.ryukoku.ac.jp
ABSTRACT
We are dealing with the FitzHugh-Nagumo equations ut = d∆u + f (u) −v, vt = ∆v − γv + u, and the equations ut = d∆u − g(u + v) + v, vt =∆v+g(u+v)−v which have a conservation property, in a bounded domain
with the Neumann boundary condition. Those systems with appropriate
nonlinearities share the common feature such that the Turing instability
takes places, namely a stable constant solution loses its stability in thepresence of the diﬀusion terms. Moreover, the stationary equations ofeach system can be reduced to a scalar elliptic equation with a nonlocal
term, whose solution is given by a critical point of an energy functional.
Our aim is to show the gradient-like dynamics of the semiﬂow generated by the solutions of each system and the coincidence of the dimensionof the unstable manifold of any equilibrium solution and the Morse index
of the corresponding critical point of the energy functional. The former
result is proved by the presence of the Lyapunov function while the latteris done by the spectral comparison method. These results are based on
the recent joint works with Chao-Nien Chen and Shuichi Jimbo.
Key words: Reaction-diﬀusion system, spectral comparison, Lyapunov
function, FitzHugh-Nagumo equations, conservation property.

16

Talk

Numerical Simulation of Flow in Fluidized Beds
Michal Beneš
Czech Technical University in Prague, Czech Republicmichal.benes@fjﬁ.cvut.cz

Pavel Strachota
Czech Technical University in Prague, Czech Republic
pavel.strachota@fjﬁ.cvut.cz

Tomáš Oberhuber
Czech Technical University in Prague, Czech Republic
tomas.oberhuber@fjﬁ.cvut.cz

Radek Máca
Czech Technical University in Prague, Czech Republicradek.maca@fjﬁ.cvut.cz

Radek Fučı́kCzech Technical University in Prague, Czech Republicradek.fucik@fjﬁ.cvut.cz

Keywords: Two-phase flow; conservation laws; finite-volume method; fluidized bed.
Oral presentation. In the contribution, we present the mathematical model of two-phase flow in
fluidized-bed combustors. The model consists of conservation laws for mass, momentum and energy ofboth phases. The fluidized bed is formed due to the interaction with gravity. The model reflects the bed
formation and behaviour. The one-dimensional model describes the integral behaviour of corresponding
quantities in the horizontal cross-section of the combustor volume whereas the multidimensional model
can provide more detailed information. Numerical solution is obtained by means of the finite-volume
method in space and by a higher-order time solver. The model is discussed with respect of the bed
formation, reaction to the control parameters and the energy transfer. A series of computational results
is presented to demonstrate the model capabilities.
Acknowledgement. The research is supported by the Technological Agency of the Czech Republicthrough the project No. TA102871.
References[1] Beneš M., Oberhuber T., Strachota P., Straka R. and Havlena V.: Mathematical modelling of combustion
and biofuel co-ﬁring in industrial steam generators, RIMS Kokyuroku Bessatsu B35 (2012), 141–157.
[2] Hoang Dieu H., Mach J.: Fluidized Bed Design and Applications. Functional Design Speciﬁcation MMG
1-12, Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, 2012[3] Máca R. and Strachota P.: Solid Gas Flow One-dimensional Numerical Simulation. Functional Design Speciﬁcation MMG 2-12, Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,
Czech Technical University in Prague, 2012

117

PosterInteraction of Dynamically Evolving Dislocations
Michal Beneš
Czech Technical University in Prague, Czech Republicmichal.benes@fjﬁ.cvut.cz

Petr Pauš
Czech Technical University in Prague, Czech Republicpetr.paus@fjﬁ.cvut.cz

Jan Kratochvı́lCzech Technical University in Prague, Czech Republickratochvil@fsv.cvut.cz

Keywords: Mean curvature ﬂow; degenerate diﬀusion; ﬂowing ﬁnite-volume method; defects in materials.
Poster. The contribution describes the mathematical model of the glide dislocation motion, interaction
with other objects such as dipolar dislocation loops or precipitates, and the topological changes - generation of
glide dislocations from the Frank-Read source or annihilation. The model represents the glide dislocations asparametrically described curves moving in the glide planes. The motion law considers the normal velocity givenby the line tension and by the forces of external origin as well as the mutual interaction with other objects. The
law is reformulated into the parametric form. The redistribution along the tangential direction is incorporated
which allows to stabilize the dislocation discretization in the long term simulations using the numerical algorithm
based on the ﬂowing ﬁnite volume method. The model is generalized to handle the topological changes of thedislocation curves across the glide planes such as merging or splitting which appears during the generation of
dislocations or during the annihilation by cross slip. As the computational complexity increases with the numberof interacting objects considered, the simulation model is designed by advanced computational techniques and theparallel implementation. The ability of the model is demonstrated in a series of computational studies of various
dislocation and object conﬁgurations including the interaction of glide dislocations with obstacles, encountersin the band channels, sweeping of dipolar loops by glide dislocations, loop clustering and dislocation cross slip
annihilation.
Acknowledgement. The research is supported by the project No. P108/12/1463 of the Grant Agency of
the Czech Republic.
References[1] Beneš M., Kratochvı́l J., Křišt’an J., Minárik V. and Pauš P.: A parametric simulation method for discrete
dislocation dynamics, European Phys. J. ST 177 (2009) 177–192.
[2] Minárik V., Beneš M., Kratochvı́l J.: Simulation of dynamical interaction between dislocations and dipolar
loops, J. Appl. Phys. 107, 1 (2010).
[3] Pauš P., Beneš M., Kratochvı́l J.: Simulation of dislocation annihilation by cross-slip, Acta Physica Polonica
A, 122, No. 3 (2012) 509–511.

218

Poster

Numerical simulation of anomalous transport in porous media described byfractional-advection dispersion equation
Michal Beneš
Czech Technical University in Prague, Czech Republicmichal.benes@fjﬁ.cvut.cz

Tissa H. Illangasekare
Colorado School of Mines, Golden, U.S.A.
tissa@mines.edu

Dave Dean
Colorado School of Mines, Golden, U.S.A.
deandw@mail.com

Keywords: FADE; anomalous transport; Levy ﬂights; porous media.
Poster. In the contribution, we present computational studies of the fractional-advection dispersion equation containing the fractional-derivative multidirectional diﬀusion term which can be responsible for anomaloustransport eﬀects. Known justiﬁcation of this model relies on the relation with the Levy stochastic processes.
The solution of the model exhibits anisotropic features and variety of interesting phenomena not observed in the
Brownian diﬀusion. Numerical solution of the transport equation leads to the linear systems of equations with fullmatrices which slows down the solution process. We couple the transport equation to the single-phase saturated
ﬂow in a heterogeneous medium and study the anomalous contaminant transport in it.
Acknowledgement. The research was supported by the project of Czech Ministry of Education, Youth andSports Kontakt ME10009, and by the project No. SGS11/161/OHK4/3T/14 of the Student Grant Agency of the
Czech Technical University in Prague.
References[1] D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert, Application of a Fractional Advection-Dispersion
Equation, Water Resources Reasearch, 36, no. 6 (2000), 1403–1412.
[2] D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert, The Fractional-Order Governing Equation of Lévy
Motion, Water Resources Reasearch, 36, no. 6 (2000), 1413–1423.
[3] R.F. Holub, M. Beneš, and Honeyman, B.D., Interfacial Phenomena in Fluid Dynamics: Linking Atomistic
and Macroscopic Properties: Can They Explain the Transport Anomalies? in Beneš, M., Mikyška, J. andOberhuber, T., editors, Proceedings of the Czech Japanese Seminar in Applied Mathematics. Prague:
Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, 2005. ISBN
80-01-03181-0, pages 57–62.

3

19

Poster

General computation of phase equilibrium in multicomponent mixtures
Tereza Jindrová
Czech Technical University in Prague, Czech Republic
jindrter@fjfi.cvut.cz

Jiřı́ Mikyška
Czech Technical University in Prague, Czech Republicjiri.mikyska@fjfi.cvut.cz

Mathematical models of phase stability and phase equilibrium computation of multicomponent fluids
play an important role in large-scale compositional hydrocarbon reservoir simulations, in interpretationof laboratory experiments which are performed to understand the behaviour of chemical species in hydrocarbon reservoirs, or in multiphase-flow simulations in porous media. Dealing with phase stability
and phase equilibrium at constant volume, temperature and moles (the so-called V T -stability and V T flash),we have recently developed a new algorithm for the constant-volume two-phase split calculation(see [1]) based on the direct minimization of the total Helmholtz free energy of the mixture with respect
to the mole- and volume-balance constraints. In the contribution, we focus on a general strategy for N phase equilibrium computation at constant volume, temperature and moles, where N ∈ N is the numberof phases (see [2]). The problem is formulated using the Helmholtz free energy and two equations ofstate: the Peng-Robinson equation of state and the Cubic-Plus-Association equation of state. Finally, we
present several numerical examples of two-phase and three-phase equilibrium calculations showing theperformance of the general algorithm on multicomponent mixtures of different complexity.

Key words
phase equilibrium, constant-volume flash, V T -flash, constant-volume stability, V T -stability, Helmholtz freeenergy minimization.

References[1] T. Jindrová and J. Mikyška: Fast and Robust Algorithm for Calculation of Two-Phase Equilibria at Given
Volume, Temperature, and Moles, Fluid Phase Equilibria (2013) 353, 101–114.
[2] T. Jindrová: Computational Methods in Thermodynamics of Multicomponent Mixtures, master degree
thesis Czech Technical University in Prague (2013).

1

20

Talk

Computation of two-phase equilibrium in multicomponent mixtures
Tereza Jindrová
Czech Technical University in Prague, Czech Republic
jindrter@fjfi.cvut.cz

Jiřı́ Mikyška
Czech Technical University in Prague, Czech Republicjiri.mikyska@fjfi.cvut.cz

The contribution is focused on the investigation of two-phase equilibrium at constant volume, temperature and moles (the so-called V T -flash) in a multicomponent mixture. The problem is formulatedusing the Helmholtz free energy and two equations of state: the Peng-Robinson equation of state andthe Cubic-Plus-Association equation of state. Recently, we have developed a new algorithm for theconstant-volume phase-split calculation (see [1]) based on the direct minimization of the total Helmholtz
free energy of the mixture with respect to the mole- and volume-balance constraints. The algorithm uses
the Newton-Raphson minimization method with line-search and modified Cholesky decomposition of the
Hessian matrix to produce a sequence of states with decreasing values of the total Helmholtz free energy.
To initialize the algorithm, an initial guess is constructed using the results of stability testing at constantvolume, temperature and moles (see [2]). The performance and efficiency of the algorithm are shownon several examples of two-phase equilibrium calculations. In the end, we discuss reasons supporting
the fact that the V T -approach seems more natural than the classical formulation at constant pressure,temperature and moles (P T -flash), especially when using pressure-explit equations of state.

Key words
two-phase equilibrium, constant-volume flash, V T -flash, Helmholtz free energy minimization, Newton-Raphson
method, modified Cholesky factorization.

References[1] T. Jindrová and J. Mikyška: Fast and Robust Algorithm for Calculation of Two-Phase Equilibria at Given
Volume, Temperature, and Moles, Fluid Phase Equilibria (2013) 353, 101–114.
[2] A. Firoozabadi and J. Mikyška: Investigation of Mixture Stability at Given Volume, Temperature, andNumber of Moles, Fluid Phase Equilibria (2012) 321, 1–9.

121

Poster

Algebraic multigrid for Navier-Stokes Problems
Petr Bauer
Czech Technical University in Prague, Czech Republicbauerpet@kmlinux.fjfi.cvut.cz

Vladimı́r Klement
Czech Technical University in Prague, Czech Republic
wlada@post.cz

Tomáš Oberhuber
Czech Technical University in Prague, Czech Republic
tomas.oberhuber@fjfi.cvut.cz

Vı́tězslav Žabka
Czech Technical University in Prague, Czech Republiczabkav@gmail.com

Algebraic multigrid is an advanced method for solving systems of equations arising from various
numerical problems. On this poster we will describe this method and its implementation (based on [1])for the simulation of heat equation. We will also point out main differences compared to geometricmultigrid, compare the speed of both methods and present modifications needed to implement algebraic
multigrid on GPU.
Keywords: Algebraic multigrid, GPU.

References[1] D. Göddeke: Dissertation thesis: Fast and Accurate Finite-Element Multigrid Solvers for PDE Simulations
on GPU Clusters, Technischen Universität Dortmund, 2010[2] Nils Klimanis: Generic Programming and Algebraic Multigrid, VDM Verlag Dr. Mueller e.K., 2008

1

22

Talk

GPU multigrid solver for Navier-Stokes problems
Petr Bauer
Czech Technical University in Prague, Czech Republicbauerpet@kmlinux.fjfi.cvut.cz

Vladimı́r Klement
Czech Technical University in Prague, Czech Republic
wlada@post.cz

Tomáš Oberhuber
Czech Technical University in Prague, Czech Republic
tomas.oberhuber@fjfi.cvut.cz

Vı́tězslav Žabka
Czech Technical University in Prague, Czech Republiczabkav@gmail.com

In this contribution we present implementation of geometric multigrid solver on GPU. This method
will be used to solve problem of air flow simulation in atmospheric boundary layer [4], which is governedby the Navier-Stokes equations for incompressible flow. The problem is discretized by finite elements
method [3] with the Crouzeix-Raviart elements [2]. We will describe basic aspects of GPU programming.
Based on them we will present modified multigrid algorithm from [1] and obtained speed-up.
Keywords: Multigrid, GPU, Navier-Stokes flow.

References[1] V. V. Shaidurov: Multigrid Methods for Finite Elements. Kluwer Academic Publishers, Dordrecht, 1995.
[2] M. Crouzeix, P. A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Rev. Franc. Automat. Inform. Rech. Operat, 1973
[3] F. Brezzi and M. Fortin: Mixed and hybrid finite-element methods, Springer Verlag, NewYork, 1991
[4] P. Bauer: Dissertation thesis: Mathematical modelling of pollution transport in urban canopy, CTUFNSPE, 2011[5] V. Klement: Master’s thesis: Implementation of the sparse matrix solvers on the GPU, CTU-FNSPE, 2008

123

Poster

Computational Study of Planar Curve Dynamics
Miroslav Kolář, Michal Beneš
Czech Technical University in Prague, Czech Republickolarmir@fjfi.cvut.cz, michal.benes@fjfi.cvut.cz

Keywords: Planar curve, mean curvature flow, parametric approach, level set method.
The contribution deals with the numerical solution of curve evolution in plane. The smooth curves
are described by means of the parametric approach [3] and the level set formulation [1]. Their evolution
is given by the equation for the mean curvature flow, which reads as
v = −k + F,
where v is the normal velocity, k is the mean curvature, and F is the spatially dependent force termbringing nonlinearities to the motion. Parametrized equations for the mean curvature flow are numerically solved by means of semi-implicit finite differences method and flowing finite volumes method [2].
We improve the behaviour and the stability of the numerical algorithm by adding the tangential redistribution of curve discretization nodes. The level set equation is solved by means of the method of lines,
and arisen system of ODEs is integrated via the Runge-Kutta-Merson method with adaptive time step.
Numerical results contain the computational study of both open and closed curves, parametric study of
the redistribution algorithm, and numerical tests of area and length conservation. The comparison ofparametric and level set approach is presented.
Acknowledgement. The authors were partly supported by the project ”Advanced Supercomputing
Methods for Implementation of Mathematical Models”, project of the Student Grant Agency of the
Czech Technical University in Prague No. SGS11/161/OHK4/3T/14 of the Grant Agency of the Czech
Republic.

References[1] K. Deckelnick and G. Dziuk: Numerical Approximation of mean Curvature Flow of Graphs and Level Sets,
Mathematical Aspects of Evolving Interfaces (2003), 53–87[2] M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki: Application of a Curvature
Adjusted Method in Image Segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New
Series) (2003) No. 4, 509–523. 437–453.
[3] D. Ševčovič and S. Yazaki: Evolution of plane curves with a curvature adjusted tangential velocity. Japan
J. Industrial and Applied Mathematics (2011) 28, 413–442.

1

24

Talk

Mathematical Model and Numerical Simulation of Dislocation Dynamics
Miroslav Kolář, Michal Beneš
Czech Technical University in Prague, Czech Republickolarmir@fjfi.cvut.cz, michal.benes@fjfi.cvut.cz

Keywords: Dislocation, mean curvature flow, parametrization, redistribution, PSB channel.
This contribution deals with the numerical simulation of dislocation dynamics. In the field of solidstate physics, the term dislocation describes some irregularity in crystal structure of material (see [1,2]).
The presence of dislocations considerably affects many of material properties. The dislocation itself
can be modelled as a line defect in crystalline lattice causing the disturbance of the regularity of thecrystallographic arrangement of atoms.
From the mathematical point of view, the dislocations are defined as smooth closed or open planar
curves which evolve in time. The motion itself is only two-dimensional and is driven by the equation
for the mean curvature flow, which reads as Bv = −T k + F , where v is the normal velocity, k is themean curvature, and F is the spatially and time dependent external force term acting on the dislocation
curve in normal direction. The force F is caused by various stress fields. The coefficient B denotes dragcoefficient, and T is the line tension, which is the force causing straightening of dislocation curve.
In this contribution, we describe the evolving curves by parametric (also called direct or Lagrangian)approach, where the family of planar curves is described by a two-dimensional smooth spatially andtime-dependent vector function X = X(u, t) called parametrization. This model is solved by means ofsemi-implicit finite differences method and flowing finite volumes method. However, this aproach exhibits
unintended behaviour, since during the time evolution, the grid points can accumulate somewhere, andthen can be sparse somewhere else. One way to overcome this problem is to add the tangential velocity
to the dislocation motion law, which moves the grid points along the curve (see [3]). Since the tangential
terms do not affect the shape of the curve, we can achieve better numerical stability, especially in longterm computation.
The presented numerical results contains the motion of dislocation curves in the so called PSB (persistent slip band) channel, which is a pattern consisting of areas with very high and very low dislocation
density.
Acknowledgement. The authors were partly supported by the project ”Two scales discrete-continuum
approach to dislocation dynamics”, project No. P108/12/1463 of the Grant Agency of the Czech Republic.

References[1] P. Pauš, M. Beneš, J. Kratochvı́l: Simulation of dislocation annihilation by cross-slip, Acta Physica Polonica
A, 122, No. 3 (2012) 509–5011[2] V. Minárik, M. Beneš, J. Kratochvı́l: Simulation of Dynamical Interaction between Dislocations and Dipolar
Loops, Journal of Applied Physics 107, 061802 (2010)125

[3] M. Beneš, J. kratochvı́l, J. Křišt’an, V. Minárik and P. Pauš: A parametric simulation method for discrete
dislocation dynamics, European Physical Journal ST, Special Topics 177 (2009) 177–192

2

26

Talk

Modeling diffusion in multicomponent mixtures in porous media: overview
of available theories, their limitations, and introduction of a generalized
Maxwell-Stefan-Darcy approach
Jiřı́ Mikyška
Czech Technical University in Prague, Czech Republicjiri.mikyska@fjfi.cvut.cz

Abbas Firoozabadi
Reservoir Engineering Research Institute, Palo Alto, CA

Diffusion of components in a multicomponent mixture flowing through a porous medium is an important transport process that needs to be taken into account when simulating problems of gas injection into
reservoirs [1]. These models find applications in enhanced oil recovery or CO2 geological sequestration. In
the contribution we discuss available diffusion theories including Fick’s law and its generalization, and theMaxwell-Stefan approach to mass transfer [2]. We will show several limitations of these models related
to modeling of the molecular diffusion in porous media. Although some of these limitations have been
described in the literature, it seems that many authors in the porous medium community are not aware ofthem. Identifying these limitations leads us to the proposition of a new generalized Maxwell-Stefan-Darcy
model describing transport of several chemical species in the porous medium. We will discuss the basic
assumptions and formulation of this model. At the end, we will present some desirable features of the
new model and indicate some future extensions of this approach.
Keywords
diffusion, multicomponent mixture, Fick’s law, Maxwell-Stefan model, Darcy’s law, compositional model

References[1] H. Hoteit, and A. Firoozabadi: Numerical modeling of diffusion in fractured media for gas-injection andrecycling schemes, SPE Journal (June, 2009) 323–337.
[2] R. Krishna, and J. A. Wesselingh: The Maxwell-Stefan Approach to Mass Transfer, Chem. Eng. Sci (1997)
52 861–911.

1

27

Talk

Compositional Modeling of Two-Phase Multicomponent
Compressible Flow in Porous Media
Ondřej Polı́vka
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Czech Republic
ondrej.polivka@fjfi.cvut.cz

Jiřı́ Mikyška
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Czech Republicjiri.mikyska@fjfi.cvut.cz

Keywords
Two-Phase Compressible Multicomponent Flow, Mixed-Hybrid Finite Element Method, Finite Volume
Method, Newton-Raphson Method, Phase-by-Phase Upwinding, Constant-Volume Phase Splitting.

Abstract
In the contribution, we present the numerical model of two-phase compressible multicomponent flowin porous media. The mathematical model is formulated by means of partial differential equations
representing the conservation laws, extended Darcy’s laws, and by means of the conditions of local
thermodynamic equilibrium in a form of algebraic relations describing the pressure and the distributionof components between the phases. Appropriate initial and boundary conditions are prescribed.
The formulated problem is solved numerically using the Mixed-Hybrid Finite Element Method [1, 2]
for Darcy’s law discretization, and the Finite Volume Method [3] for the component transport equations
discretization. To discretize the component fluxes across the boundaries of the elements a special upwind
technique is used. The time discretization is carried out by the backward Euler method [3]. The resulting
system of nonlinear algebraic equations is solved by the Newton-Raphson iterative method. The derived
numerical scheme ensures the local mass balance and correct treatment of the phase fluxes between
elements in a way that no phase identification and decisions on the corresponding phases is necessary.

References[1] F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York Inc. (1991).
[2] G. Chavent, J. E. Roberts. A unified physical presentation of mixed, mixed-hybrid finite elements and standard
finite difference approximations for the determination of velocities in waterflow problems. Advances in Water
Resources, 14(6) (1991), pp. 329–348.
[3] R. J. Leveque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge(2002).

128

Poster

Numerical Simulations of Two-Phase Multicomponent
Compressible Flow in Porous Media
Ondřej Polı́vka
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Czech Republic
ondrej.polivka@fjfi.cvut.cz

Jiřı́ Mikyška
Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Czech Republicjiri.mikyska@fjfi.cvut.cz

Keywords
Two-Phase Compressible Multicomponent Flow, Enhanced Oil Recovery, CO2 Sequestration, MixedHybrid Finite Element Method, Finite Volume Method, Newton-Raphson Method.

Abstract
On the poster, we show a few numerical simulations of two-phase compressible multicomponent flow inporous media which occurs in many engineering problems like enhanced oil recovery or CO2 sequestration.
We simulate injection of different gases into horizontal and vertical reservoirs filled with different species
and two-phase flow of the mixture.
Equations describing this kind of flow consist of the conservation laws, extended Darcy’s laws, andmeans of the conditions of local thermodynamic equilibrium. Our numerical solution of the problem isbased on a combination of the Mixed-Hybrid Finite Element Method [1, 2] for Darcy’s law discretization,and the Finite Volume Method [3] for the component transport equations discretization. The system of
equations is linearized by the Newton-Raphson iterative method.

References[1] F. Brezzi, M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York Inc. (1991).
[2] G. Chavent, J. E. Roberts. A unified physical presentation of mixed, mixed-hybrid finite elements and standard
finite difference approximations for the determination of velocities in waterflow problems. Advances in Water
Resources, 14(6) (1991), pp. 329–348.
[3] R. J. Leveque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge(2002).

1

29

Talk

Optimal Value of a Firm Investing in Exogenous Technology
Pedro Pólvora
September 2013
Tokyo,Japan

Abstract
We study the optimal value for a Firm whose value is function of an exogenous technology
level. At any point in time the Firm can invest in a new technology, incurring in an immediate
cost and in return it will become able to use that technology yielding profit through a given
profit flow function. The technology is modelled by a discrete stochastic process with atime-dependent arrival rate. We study the optimal stopping time that will correspond to
the point in time when the firm will invest. We use a dynamic programming approach,finding the Hamilton-Jacobi-Bellman equation whose solution gives us the optimal value ofthe firm. We particularise for two cases, one with a constant arrival rate and the other witha time-dependent and non-monotonic arrival.

130

Poster

Multiphase Volume-preserving Interface Motions via Localized Signed
Distance Vector Scheme
Rhudaina Z. Mohammad1 and Karel Svadlenka2
1

Graduate School of Natural Science and Technology, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
rhudaina@polaris.s.kanazawa-u.ac.jp
2

Institute of Science and Engineering, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
kareru@staff.kanazawa-u.ac.jp

Abstract

We develop a signed distance vector approach for approximating volume-preserving mean
curvature motions of interfaces separating multiple phases – a variant of the MBO (MerrimanBence-Osher) thresholding dynamics. We use vector-type discrete Morse flow and adopt avariational method, which allows us to easily treat volume constraint via penalization without
having to change the threshold value. Moreover, the method is designed to allow subgrid accuracy on uniform grids without adaptive refinement; thereby, alleviating the well-known MBO
time and grid restrictions.
Keywords: multiphase flow, signed distance vector, mean curvature motion, volume preservation, constrained variational method, MBO thresholding

1

31

Poster

Simulation of Triple Junction Motion
with Arbitrary Surface Tensions
1

1

Nur Shofianah, 2Rhudaina Z. Mohammad and 3Karel Svadlenka

Graduate School of Natural Science and Technology, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
viena_shofianah@yahoo.com

2

Graduate School of Natural Science and Technology, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
rhudaina@polaris.s.kanazawa-u.ac.jp
3Institute of Science and Engineering, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
kareru@staff.kanazawa-u.ac.jp
Abstract

We aim at simulating triple junction motion that is given by the gradient flow of surface
energy with arbitrary surface tensions of the individual interfaces. The fundamental
method is the diffusion-based BMO algorithm in vector-valued formulation. We modify
the original BMO method, so that it can accommodate motions for any triple of given
surface tensions. The main idea of the modification is to change the coefficients in theunderlying diffusion system and the reference vectors, which correspond to the positionof wells in the phase-field method. Analysis and numerical results of this generalized
BMO algorithm are presented.

Keywords: triple junction motion, gradient flow of surface energy, surface tension,
vector-valued thresholding.

32

Talk

Numerical computation of motion of interface
networks with nonlocal constraints
Karel Svadlenka1, Rhudaina Z. Mohammad2, Nur Shofianah2
1Institute of Science and Engineering, Kanazawa University
Kakuma-machi, Kanazawa, Japan 920-1192
kareru@staff.kanazawa-u.ac.jp

Graduate School of Natural Science and Technology, Kanazawa University

2

Kakuma-machi, Kanazawa, Japan 920-1192

Abstract
Numerical approximation of gradient flow of surface energy for networks ofinterfaces including junctions is considered. Moreover, the volume of each region
surrounded by the interfaces is to be preserved. The main tool is the thresholding
algorithm due to Merriman, Bence and Osher, which is reformulated in a vectorvalued fashion and suitably modified. Analysis of interface velocity and junction
stability under this new algorithm will be presented. The treatment of arbitrary
surface tensions will also be mentioned.
Keywords: curvature-dependent interface motion, nonlocal constraints, triple
junction stability, thresholding method

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SurnameNameJapanese expression

Institution

Talk

PosterAbe
AiharaAkiyama
Anada
Benes
Chalupecky
Frolkovic
FukudaFurihata
Idogawa
Ikeda
Ishiwata
Ishiwata
Ito
Jindrova
KashimaKimuraKlement
Knobloch
Kobayashi
Kolar
Kutik
Maca
MatsuoMikyska
MikulaMohammadMonobeMoritaMurakawa
MuroyaNagayama
Nakamura
Narumi
Nishi
Nishiura
Ogawa
Omata
Ozaki
Polivka
PolvoraPozar
Sakamoto
Sasaki
Sato
Sevcovic
Shofianah
SpirStrakaSvadlenka
TakaishiTsujikawa
UemichiYazakiKuniyoshi
Kensuke
Masakazu
KoichiMichalVladimir
Peter
Akiko
DaisukeTomoyuki
Kota
Emiko
Tetsuya
Takahiro
TerezaYohei
MasatoVladimir
Petr
Yasuaki
Miroslav
Pavol
Radek
Takayasu
Jiri
Karol
Rhudaina Z.
Harunori
Yoshihisa
Hideki
Yoshiaki
Masaharu
MasaakiTakayuki
Kei
Yasumasa
Toshiyuki
Seiro
Katsuhisa
OndrejPedro
NorbertTakashiTakikoHirotoshi
Daniel
Nur
RobertRobertKarel
Takeshi
Tohru
Kenta
Shigetoshi

阿部邦美
相原研輔
秋山正和
穴田浩一
ミハル・ベネシュ
ヴラディミール・ハルペツキー
ペテル・フロルコヴィッチ
福田亜希子
降旗大介
井戸川知之
池田幸太
石渡恵美子
石渡哲哉
伊藤貴大
テレザ・インドロバー
鹿島洋平
木村正人
ヴラディミール・クレメント
ペトル・クノブロフ
小林康明
ミロスラフ・コラージュ
パボル・クチーク
ラデック・マーツァ
松尾宇泰
イジー・ミキシュカ
カロル・ミクラルダイナ・ムハンマド
物部治徳
森田善久
村川秀樹
室谷義昭
長山雅晴
中村正彰
鳴海孝之
西慧
西浦廉政
小川知之
小俣正朗
尾崎克久
オンドレィ・ポリーフカ
ペドロ・パルヴォラ
ノルベルト・ポジャール
坂元孝志
佐々木多希子佐藤宏俊
ダニエル・シェフチョヴィッチ
ショフィアナ・ヌル
ロベルト・シュピール
ロベルト・ストラッカ
カレル・シュヴァドレンカ
高石武史
辻川亨
上道賢太
矢崎成俊

Gifu Shotoku University
Tokyo University of Science
Hokkaido University
Waseda University Senior High school
Czech Technical University in Prague
Fujitsu
Slovak University of Technology
Tokyo University of Science
Osaka University
Shibaura Institute of Technology
Meiji University
Tokyo University of Science
Sibaura Institute of Technology
Kanazawa University
Czech Technical University in Prague
University of Tokyo
Kanazawa University
Czech Technical University in Prague
Charles University
Hokkaido University
Czech Technical University in Prague
Slovak University of Technology
Czech Technical University in Prague
The University of Tokyo
Czech Technical University in Prague
Slovak University of Technology
Kanazawa University
Meiji Institute for Advanced Study of Mathematical Sciences
Ryukoku University
Kyushu University
Waseda University
Hokkaido University
Nihon University
Kwansei Gakuin University
Hokkaido University
Tohoku University
Meiji University
Kanazawa University
Shibaura Institute of Technology
Czech Technical University in Prague
Technical University of Lisbon
Kanazawa University
Meiji University
The University of Tokyo
Kwansei Gakuin University
Comenius University
Kanazawa University
Slovak University of Technology in Bratislava
AGH University of Science and Technology
Kanazawa University
Hiroshima Kokusai Gakuin University
University of Miyazaki
Kwansei Gakuin University
Meiji University

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