Koichi Anada, Tetsuya Ishiwata

Abstract: We consider initial-boundary value problems for a nonlinear parabolic equation which has Type 2 blow-up solutions. We provide precise blow-up rates for solutions.

Handlovičová Angela

Abstract:

DDFV numerical scheme   to the regularized level set equation is derived.  Numerical analysis, namely  stability and convergence to the weak solution  is presented together with some numerical experiments.  DDFV scheme can be applied for  Heston model too. Some  numerical experiments and  basic properties of the proposed scheme are presented.

Michal Beneš, Alexandr Žák

Abstract:

In the contribution, we discuss the model of solidification of melt occupying pores of the porous medium with grains intact but participating in the heat transfer. The research motivation is given by development of advanced materials as well as by climate changes. The model is based on coupled heat conduction equation and the phase-field. We present the model and several computational studies.

Peter Frolkovič

Abstract: A new class of semi-implicit methods for numerical solutions of advection dominated problems will be presented. An origin approach to derive such schemes will be given. The schemes offer several improvements with respect to fully explicit and/or fully implicit schemes. They are simple to treat in the case of variable velocity field and they have better stability properties. Several benchmarks to document such behaviour will be shown.

Elliott Ginder

Abstract:

Tracking the evolution of multiphase geometries represents a fundamental problem that arises in a variety of scientific simulations. The phenomena under consideration can exhibit additional challenges to its numerical simulation, including the occurrence of topological changes, volume constraints, and inertial effects. We will introduce a method for treating these issues and investigate its application in the simulation of 2D and 3D multiphase parabolic and hyperbolic curvature flows. Volume constrained motions will also be investigated through the use of minimizing movements, and we will remark about aspects related to our method’s numerical realization.

Vít Hanousek, Tomáš Oberhuber

Abstract:

Use of the Intel Xeon Phi coprocessor is becoming more and more popular in high performance computing. In this talk, we present explicit and semi-implicit numerical solvers for parabolic PDEs on the Xeon Phi in offload mode. The solvers are implemented in C++ as part of Template Numerical Library. We also compare efficiency of Xeon Phi, GPU and CPU on results obtained by this library.

Sampei Hirose, Junichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

Abstract:

The local induction equation is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation. In this talk, we present a discrete model of dynamics of vortex filaments. Namely, we formulate a discrete deformation of discrete space curves whose complex curvature satisfies the discrete nonlinear Schrödinger equation proposed by Hirota-Ohta-Tsujimoto.

Tomoyuki Idogawa, Reo Kobayashi

Abstract:

Sudoku is a kind of number-placement puzzles.  The objective of this puzzle is to fill a 9x9 board with digits 1-9 (some numbers were given initially) satisfying each column, raw or 3x3 size block has no duplicate digit.  We are interested in counting all possible grids (arrangements of digits) and will show an algebraic method using Groebner basis.

Kota Ikeda, Takeshi Miki

Abstract:

Many ecological systems are influenced by positive feedbacks between organisms and abiotic environments, which generates multiple stable equilibria of a mathematical model with a hysteresis structure. In addition, discontinuous shifts of system at equilibrium is predicted, which is often called regime shift in ecosystem sciences. This hysteresis structure is unfavorable from environmental management point of view, because the reconstruction of original equilibrium state requests much lower levels of external forcing. Mathematical models proposed in previous works are simple and mathematically tractable. However, it is difficult to extrapolate from such simple models the occurrence likelihood of regime shift in natural environments since temporally dynamic features in ecology and physico-chemical environments, and spatial dimension are less explored in those models.

In this study, we construct a realistic but mathematically tractable model of interaction between phytoplankton and phosphorus, which incorporates (1) 1-dimensional vertical structure of lake ecosystem and (2) seasonal periodic cycle of water mixing. We aim to understand the impact of changes in seasonality in various types of lakes on the occurrence of multiple attractors (periodic solution) and hysteresis structure.

Tetsuya Ishiwata, Katsuhiro Miki

Abstract: We consider a blow-up problem to nonlinear ordinary equations with a delay . For the delayed problem, the standard numerical method for the blow-up problem is not so easy to apply. In this presentation, we propose a simple numerical method for this problem and show the convergence of the numerical blow-up time to the blow-up time.

Viera Kleinová, Peter Frolkovič

Abstract: Optical flow is very important topic in medicine, computer vision and image processing. We present preliminary results of a new numerical method for determining an optical flow based onlevel-set motion between two images. Our examples include synthetic and real data such that medical images of lungs or satellite images of clouds. Some representative results will be presented.

Vladimír Klement, Tomáš Oberhuber

Abstract:

In this contribution we present implementation of 3D Navier-Stokes flow simulation, which is parallelized on the GPU. In order to save memory resources, which are often limiting factor in 3D computations, we describe a matrix-less version of the program, where all matrices are computed on the fly instead of stored in computer memory. This approach would be too computationally demanding for standard processors, but as can be seen from conducted measurements it is very suitable for modern GPUs.

Miroslav Kolář, Michal Beneš, Jan Kratochvíl, Daniel Ševčovič

Abstract:

In this contribution we analyze the problem of dislocation cross-slip considered as a deterministic, stress-driven elementary dislocation process. Our approach to discrete dislocation dynamics (DDD) modeling is based on mathematical theory of smooth curves evolving either in plane or on a two dimensional surface. The motion of dislocation curves is driven by the mean curvature motion law in the form

B v = T κ + F.

Here v denotes the normal velocity, F is the normal component of all external forces acting on the dislocation, and parameters B and T denote the drag coefficient and the line tension, respectively. In the case of planar curves, the κ is the mean curvature, and in the case the dislocation evolving on a surface, κ stands for geodesic curvature.

The cross-slip is considered as a deterministic phenomenon controlled by the repulsive exerted by a spherical obstacle. The sharp edges between the primary planes and the cross-slip plane are regularized to ensure the C2 smoothness of the whole glide surface. For numerical simulations, we employ the parametric description of the evolving dislocation curves and semi-implicit flowing finite volume method. To ensure the numerical stability, the employed semi-implicit scheme is enhanced with the tangential redistribution of the discretization points. Overcoming of a spherical obstacle by double cross-slip is presented as an illustrative example. The results of computational experiments are compared with the results obtained by the estabilished projection method.

Jiří Mikyška

Abstract:

We formulate the general form of the p-phase flash equilibrium problem and discus ways to solve the resulting systems of linear algebraic equations. We show that the commonly used elimination of constraints may not be efficient and propose an alternative technique using the structure of the general system.

Takaaki Minomo, Elliott Ginder

Abstract:

We study a partial difference equation for understanding the self-motion of camphor disks atop a water surface. The model equation

is presented in the form of a reaction–diffusion system and we will show how to simplify the problem by formulating the source term using delta functions. The resulting system is a point mass model for diffusing particles and is a simple reaction-diffusion equation, where the role of the delta functions is to express locations of camphor sources. In our model, source terms interact with each other and move by the gradient of the concentration field. We approximate the model equation’s solution numerically using the finite element method’s approach to the Galerkin method and will show the results of our simulations. We will also discuss properties of the model equation—in particular, we will study properties of traveling pulse solutions, whose existence are reduced to the solution of an ordinary differential equation. The existence of solutions will be shown and we analyze their stability. The method used in constructing such solutions will also be presented.

Hideki Murakawa

Abstract:

Cell adhesion is the binding of a cell to another cell or to an extracellular matrix component. This process is essential in organ formation during embryonic development and in maintaining multicellular structure. Armstrong, Painter and Sherratt [{\em J. Theor. Biol.} {\bf 243} (2006), pp.~98--113] proposed a nonlocal advection-diffusion system as a possible continuous mathematical model for cell-cell adhesion. Although the system is attractive and challenging, it gives biologically unrealistic numerical solutions. We identify the problems and change underlying idea of cell movement from ``cells move randomly" to ``cells move from high to low pressure regions". Then we provide a modified continuous model for cell-cell adhesion. Numerical experiments illustrate that the modified model is able to replicate not only the Steinberg's cell sorting experiments but also some phenomena which can not be captured at all by Armstrong-Painter-Sherratt model. Furthermore, we give theoretical results about the modified model. 

Masaharu Nagayama, Yasuaki Kobayashi, Mitsuhiro Denda, Hiroyuki Kitahata

Abstract:

Mammalian skin not only functions as a boundary separating the body from its environment, but also a barrier function which keeps away foreign substances and retains internal water. Such barrier functions are realized by forming the stratum corneum (SC), the outermost structure of the epidermis, which consists of cornified cells surrounded by inter-cellular lipids. Hence the spatial and temporal stability of the SC is essential for the barrier function.
  Epidermal cells are continually reproduced in the basal layer, moving upward, differentiating into stratum spinosum, then stratum granulosum, and finally undergoing cornification to become a part of the SC. It is remarkable that ordered SC layers are formed through such complicated processes, and that they reform immediately after being subject to damage. How such homeostasis is maintained, however, is still unclear.
  The epidermis is maintained by continual supply and removal of cells, and is a typical example of a dissipative structure. Mathematical descriptions are often quite useful in uncovering mechanisms emergent properties in non-equilibrium systems. In this study, we developed a mathematical model of the epidermis. It consists of cell dynamics, including cell division, differentiation and kinetics, and calcium ion dynamics, which are known to affect the homeostasis of the barrier function. As the result of our study, we found that the localization of calcium ions and cell division plays a particularly important role in the barrier function.

Tomáš Oberhuber

Abstract:

We present Template Numerical Library with native support of CUDA for computations on GPUs. The library is written in C++ and it uses C++ templates extensively. The templated design of TNL allows to develop solvers of PDEs with GPU support relatively easily and almost without any knowledge of GPUs. The aim of TNL is to provide an easy to use tool for numerical mathematicians so that they may concentrate only to numerical methods but they can still profit from modern accelerators and parallel architectures. We will also discuss disadvantages of C++ templates and metaprogramming.

Yuki Ohta, Katsuhisa Ozaki

Abstract:

Convex hull is the smallest convex which encloses all input points.

To obtain the convex hull is one of important tasks in computational geometry.

Assume that given data is represented by not a point but an interval.

We propose iterative algorithms which produce the outer convex hull for all intervals based on floating-point filters.

Takeshi Ohtsuka, Tetsuya Ishiwata

Abstract:

In this talk we introduce a mathematical for evolving polygonal spiral curves by crystalline eikonal-curvature flow with pinned center. It is well-known that the motion of polygonal curve by crystalline curvature flow is expressed by a system of ordinary differential equations for the length of facets of the evolving curve by J. Taylor. We combine this idea and a rule of generation of new facet at the center of the spiral, and then we describe the evolution of a polygonal spiral by crystalline eikonal-curvature flow. We present the existence, uniqueness, self-intersection free result on this formulation, and and some numerical simulations with our formulation.

Mamoru Okamoto, Masaharu Nagayama, Masakazu Akiyama

Abstract:

Many researchers have studied the self-motion of camphor and it is now said that the motion of camphor (atop water) is caused by differences in surface tension. The gradient is induced by a camphor layer development atop the surface. Mathematical models for the camphor motion have been constructed used the above mechanisms, and the models reproduce the motion of camphor disks.

 Convection caused by differences in surface tension has also been observed. Although convection induced by differences in surface tension seems to influence the self-motion of the camphor, there are only a few reports discussing mathematical models that include convection explicitly.

 We have constructed a mathematical model for the self-motion of camphor driven by convection, and have observed motions that the previous models cannot reproduce.

Katsuhisa Ozaki

Abstract:

To solve linear system is one of important tasks in scientific computations. Accuracy of approximate solutions is sometimes monitored by residual. However, basics in numerical linear algebra say that the residual is not enough to show the accuracy of numerical solutions. If the exact solution of the linear system is known in advance, then it is very useful to check the behavior of convergence for iterative methods and tightness of the computed interval by verified numerical computations. We propose an algorithm which produces the linear system with exact solution from a given matrix and the exact solution.

Petr Pauš, Tomáš Oberhuber, Jiří Kafka

Abstract:

In standard MRI, it is possible to detect only the normal direction of the myocardial motion. In tagged MRI, an artificial grid is added to the tissue before scanning. The grid then helps to detect ventricular motion also in tangential direction. However, the grid disappears in short time and the detection becomes difficult after several images. Our method is based on image segmentation by the means of parametric curves driven by mean curvature and the external force dependent on the image intensity. Parametric curves create a grid which is manually fitted to the first segmented image and then the mean curvature flow evolves the grid according to the intensity of the following images.

Ondřej Polívka, Jiří Mikyška

Abstract:

We deal with the numerical modeling of compressible multicomponent two-phase flow in porous media with species transfer between the phases.

The mathematical model is formulated by means of the extended Darcy's laws for all phases, components continuity equations, constitutive relations, and appropriate initial and boundary conditions. The splitting of components among the phases is described using a formulation of the local thermodynamic equilibrium which uses volume, temperature, and moles as specification variables.

The problem is solved numerically using a combination of the mixed-hybrid finite element method for the total flux discretization and the finite volume method for the discretization of continuity equations. These methods ensure the local mass balance. The resulting system of nonlinear algebraic equations is solved by the Newton-Raphson iterative method. The numerical flux is discretized in a way that no phase identification nor determination of correspondence between the phases on adjacent elements is required in contrast to the traditional approaches. This is very important for the simulations of CO2 sequestration because, typically, the CO2 is injected into a reservoir in the supercritical state at which the phase distinction is ambiguous. Moreover, our model performs well in situations where a phase appears or disappears and no switching of variables is needed.

We briefly describe the numerical method and provide several 2D simulations, e.g. CO2 injection into water saturated reservoir.

Ondřej Pártl, Michal Beneš, Peter Frolkovič, Tissa Illangasekare, Kathleen Smits

Abstract:

In this contribution, we shall present a mathematical and numerical model for the description of non-isothermal compressible flow of gas mixtures which is affected by gravity in heterogeneous porous medium and in the free space above its surface. These models are based on the two domain approach. We shall also present latest results of our numerical tests.

Koya Sakakibara

Abstract:

The aim of this talk is to develop mathematical theory of the invariant scheme for the method of fundamental solutions (MFS) used to solve potential problems in doubly-connected regions. Particularly, we prove that an approximate solution actually exists uniquely under some conditions, and that the error decays exponentially when the boundary data are analytic, and algebraically when they are not analytic but belong to some Sobolev spaces. Moreover, we present results of several numerical experiments in order to show the sharpness of our error estimate. 

Takiko Sasaki

Abstract:

We consider a blow-up curve for the one dimensional wave equation ∂t2u − ∂x2u = |∂tu|p with p > 1. The purpose of this  is to show that the blow-up curve is a C1 curve if the initial values are large and smooth enough. To prove the result, we convert the equation into a first order system, and then apply a modification of the method of Caffarelli and Friedman (). Moreover, we present some numerical investigations of the blow-up curves. From the numerical results, we were able to confirm that the blow-up curves are smooth if the initial values are large and smooth enough. Moreover, we can predict that the blow-up curves have singular points if the initial values are not large enough even they are smooth enough. 

Jakub Solovský

Abstract:

This work deals with two phase compositional flow. We present equations describing two phase flow, component transport and interphase mass transfer.

For this type of problems, we propose a numerical method based on the mixed hybrid finite element method. We implement several variations of this method using different approaches to solving resulting system of linear algebraic equations. We use direct and iterative solvers and parallel implementation using MPI.

The method is verified on problems for which exact solutions are known or solutions can be found in literature. Numerical experiments show that the errors are similar for all variations of the method. The method is convergent and the order of convergence is slightly less than one. There are significant differences in the computational time. Iterative solvers are faster and the parallelism is advantageous while using fine meshes.

In the next part, we focus on the compositional flow. Data from two experiments are used and numerical results are compared with measured values.

The first experiment was focused on evaporation of dissolved TCE. For low air flow rates above the water table, there is a good match with experimental data. For higher flow rates, the results differs.

The second, more complex experiment in larger scale examined the influence of water table fluctuation and rainfall events on evaporation and transport of the dissolved TCE. For water table fluctuation, there is a good match with experimental data but for rainfall events there are significant differences.

During the rainfall events there are uncertainties concerning the experimental data.

Differences between the measured values and the numerical simulations indicate certain limits of the mathematical model used or the influence of other processes that are neglected in the current model.

Finally we focus on hypothetical scenarios of vapor intrusion.

In the field scale, we examine effect of water table drop or rainfall events, that were in smaller scale studied experimentally.

Numerical results are similar to the second experiment.

Pavel Strachota, Dejan Kirda, Michal Beneš

Abstract:

We first introduce an outline of the mathematical model of combustion in circulating fluidized bed boilers. The model is concerned with multiphase flow of flue gas, bed material, and fuel particles. It further considers phase interaction resulting in particle attrition, devolatilization and burnout of fuel particles, and energy balance between heat production and consumption (radiative and convective transfer to walls). The numerical solution is carried out by finite volume methods with cell-centered or staggered grid schemes together with a Runge-Kutta class time integration method. Some simulation results of several (partial) implementations of the model in quasi-1D, 2D, and 3D are demonstrated.

Robert Straka

Abstract: Lattice Boltzmann Method (LBM) will be derived and applied for the Navier-Stokes equation in 2D and 3D. Several flavours of LBM will be presented (SRT,MRT,CLBM,KBC and Cumulant approach) together with some numerical results and comparison of the selected flow problems.

Daisuke Tagami

Abstract:

Error estimates of a generalized particle method is considered for convection-diffusion equations. We have introduced a generalized particle method as a class of particle methods, which can describes widely used particle methods like Smoothed Particle Hydrodynamics (SPH), Moving Particle Semi-implicit (MPS), and others, and have established error estimates of the generalized particle method for the Poisson equation and the heat equation have been established.

This presentation is a next step of series of mathematical analysis of generalized particle methods. At this step, the particle motions are considered, which play a key role in practical computational fluid dynamics with particle methods. In general, the

particle motions cause particle distributions unevenness and numerical schemes instability. To overcome these difficulties, the characteristic methods (see, for example, Pironneau (Numer. Math., vol.38 (1982), pp.309-332), and Notsu-Tabata (ESAIM: M2AN, vol.50 (2016), pp.361--380), are introduced into approximatons of material derivative. Owing to the characteristic methods, particle redistributions are not required in our numerical scheme. Some mathematical analysis are shown to confirm the effectiveness of our strategy.

Akitoshi Takayasu, Makoto Mizuguchi, Takayuki Kubo, Shin'ichi Oishi

Abstract:

This talk presents computable estimates of the evolution operator. The evolution operator is the solution operator of homogeneous parabolic initial value problems and was proposed by Tanabe-Sobolevskii in 1960s. By using Tanabe-Sobolevskii's  formulation of  the evolution operator and a fixed-point formulation based on an analytic semigroup, we give two kinds of computable estimates of the evolution operator. Furthermore, we introduce an application of the estimate to verification methods for nonlinear parabolic partial differential equations.

Takeshi Terao, Katsuhisa Ozaki

Abstract:

We focus on verification of positive definiteness of a given symmetric matrix and error bounds for an approximate solution of a linear system.Using grouped block matrix computations, we showed that upper bounds of the residual of Cholesky decomposition can be reduced.Based on the new bounds, the positive definiteness can be verified for a wide range of problem.In addition, we applied the new bounds to the verification of the numerical solution of the linear systems. 

Mária Tješšová, Milan Sokol

Abstract:

The work deals with the influence of the vertebra geometry on the stress analysis of human spine.

The main part of the work focuses on the change of total curvature. The spine is analyzed using FEM.

The geometry can is taken from X-ray pictures.

The change the overall spine geometry, patient attitude and local curvatures of vertebra can be investigated as well as.

Kyoko Tomoeda, Yoshiaki Teramoto

Abstract:

We consider the two dimensional motion of a viscous incompressible fluid flowing down an inclined plane with an angle of inclination $\alpha$, under the effect of gravity. The fluid motion is governed by the Navier-Stokes equations with the free boundary conditions. This problem contains two dimensionless quantities: Reynolds number and Weber number. When the Reynolds number and the angle α is sufficiently small, Nishida-Teramoto-Win (1993) proved the global existence of periodic solutions with an exponential decay rate for sufficiently small initial data. To obtain a specific range of this ``sufficiently small Reynolds number', we examine the spectra of the compact operator arising the linearized problem. In this talk we discuss about a range of the Reynolds number and Weber number, when the linear operator has a non-zero spectral value.

Shigetoshi Yazaki, Tetsuya Ishiwata

Abstract:

An area-preserving crystalline curvature flow is regarded as a simple model of the deformation process of a vapor figure or negative crystal. In this talk, behavior of polygonal curves by area-preserving crystalline curvature flow is discussed. We show ``convexity phenomena'', that is, the solution polygon from a nonconvex initial polygon becomes convex in a finite time. Due to show this assertion, we classify edge-disappearing patterns completely and prove that all zero-curvature edges disappear in a finite time, and we also show that evolution process of the flow can be continued beyond such edge-disappearing singularities.

Jan Šembera, Ivan Bruský

Abstract:

The contribution deals with a result from C1 Task of the DECOVALEX 2015 project. The task considered the detailed experimental work of Yasuhara et al. (2006), wherein a single artificial fracture in novaculite (micro- or crypto-crystalline quartz) is subject to variable fluid flows, mechanical confining pressure and different applied temperatures. This talk presents simulation of the considered experiment performed by a commercial geochemical software and a self made software solving the coupling equations. The experiment, mathematical model, its solution, and interpretation of its important parameters will be presented.

Daniel Ševčovič, S. Pavlíková

Abstract:

In this talk we investigate spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze

the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate  its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this talk.

Alexandr Žák

Abstract:

This contribution deals with a 2D micro-scale model of thermomechanical processes during solidification of a medium within porous material. The solidification problem is described in the Lagrangian framework by means of the heat, Navier, and phase-field equations. Suitable couplings of multi-phase and multi-physics are introduced. Several computational results are presented.