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Discrete & Continuous Dynamical Systems - S

2015 , Volume 8 , Issue 5

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Preface: Special Issue on recent topics in industrial and applied mathematics
Michal Beneš, Tetsuya Ishiwata, Takashi Sakamoto and Shigetoshi Yazaki
2015, 8(5): i-i doi: 10.3934/dcdss.2015.8.5i +[Abstract](203) +[PDF](86.6KB)
This issue consists of 16 carefully refereed papers dealing with mathematical modeling, numerical analysis, numerical simulation and mathematical analysis for ordinary/partial differential equations in several fields: fluid dynamics, chemistry, crystal physics, material science, medical science and biology. The authors in the present issue are mainly from Czech Republic, Slovakia and Japan. Scientific and mathematical exchanges began in 2004 among the authors and their friends. Since then, fruitful relationships and effective exchanges have been promoted and developed. Professor Tatsuyuki Nakaki strongly supported these exchanges and established the Czech-Slovak-Japanese joint seminars on industrial and applied mathematics. Unfortunately, he passed away in 2011. In 2013 in Tokyo, we had an opportunity to have a meeting in honor of Professor Nakaki. Recent topics in industrial and applied mathematics were widely discussed and we could exchange new ideas and various kinds of new problems. The present issue includes a part of those results.
    Finally we would like to thank the referees for their valuable comments in reviewing the papers.
A local discontinuous Galerkin method based on variational structure
Yoshifumi Aimoto, Takayasu Matsuo and Yuto Miyatake
2015, 8(5): 817-832 doi: 10.3934/dcdss.2015.8.817 +[Abstract](251) +[PDF](696.9KB)
We present a special variant of the local discontinuous Galerkin (LDG) method for time-dependent partial differential equations with certain variational structures and associated conservation or dissipation properties. The method provides a way to construct fully-discrete LDG schemes that retain discrete counterparts of the conservation or dissipation properties. Numerical results confirm the accuracy and effectiveness of the method.
Numerical simulation of flow in fluidized beds
Petr Bauer, Michal Beneš, Radek Fučík, Hung Hoang Dieu, Vladimír Klement, Radek Máca, Jan Mach, Tomáš Oberhuber, Pavel Strachota, Vítězslav Žabka and Vladimír Havlena
2015, 8(5): 833-846 doi: 10.3934/dcdss.2015.8.833 +[Abstract](268) +[PDF](632.6KB)
The article provides a brief overview of a one-dimensional model of two-phase flow in the geometry of a circulating fluidized bed combustor exhibiting vertical variability of cross-section. The model is based on numerical solution of conservation laws for mass, momentum and energy of gas and solid components of the fluidized-bed system by means of the finite-volume method in space and of a multistep higher-order solver in time. The presented computational results reproduce characteristic behavior of fluidized beds in the given geometry.
Reduced model from a reaction-diffusion system of collective motion of camphor boats
Shin-Ichiro Ei, Kota Ikeda, Masaharu Nagayama and Akiyasu Tomoeda
2015, 8(5): 847-856 doi: 10.3934/dcdss.2015.8.847 +[Abstract](228) +[PDF](320.1KB)
Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.
Annihilation of two interfaces in a hybrid system
Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura and Takashi Teramoto
2015, 8(5): 857-869 doi: 10.3934/dcdss.2015.8.857 +[Abstract](221) +[PDF](954.7KB)
We consider the mixed ODE-PDE system called a hybrid system, in which the two interfaces interact with each other through a continuous medium and their equations of motion are derived in a weak interaction framework. We study the bifurcation property of the resulting hybrid system and construct an unstable standing pulse solution, which plays the role of a separator for dynamic transition from standing breather to annihilation behavior between two interfaces.
Distance function and extension in normal direction for implicitly defined interfaces
Peter Frolkovič, Karol Mikula and Jozef Urbán
2015, 8(5): 871-880 doi: 10.3934/dcdss.2015.8.871 +[Abstract](182) +[PDF](4603.3KB)
In this paper we present a novel application of extrapolation procedure for three popular numerical algorithms to compute the distance function for an interface that is given only implicitly. The methods include the fast marching method [8], the fast sweeping method [10] and the linearization method [10]. The extrapolation procedure removes the necessity of a special initialization procedure for the grid nodes next to the interface that is used so far with the methods, thus it represents a natural extension of these methods. The extrapolation procedure can be used also for an extension of a function that is defined only locally on the interface in the direction given by the gradient of distance function [2].
On spiral solutions to generalized crystalline motion with a rotating tip motion
Tetsuya Ishiwata
2015, 8(5): 881-888 doi: 10.3934/dcdss.2015.8.881 +[Abstract](182) +[PDF](350.4KB)
In our previous paper we proposed a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation and discussed global existence of spiral solutions to the proposed model. In this paper we extend the previous results for generalized crystalline curvature flow with a suitable tip motion. We show that solution curves never intersect a trajectory of a tip and has no self-intersections. We also show that any facet never disappear during time evolution. Finally we show a time-global existence of the spiral-shaped solutions.
Conserved quantities of the integrable discrete hungry systems
Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa Nakamura
2015, 8(5): 889-899 doi: 10.3934/dcdss.2015.8.889 +[Abstract](193) +[PDF](318.2KB)
In this paper, conserved quantities of the discrete hungry Lotka-Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.
Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations
Petr Knobloch
2015, 8(5): 901-911 doi: 10.3934/dcdss.2015.8.901 +[Abstract](183) +[PDF](391.4KB)
A recently proposed local projection stabilization (LPS) finite element method containing a nonlinear crosswind diffusion term is analyzed for a transient convection-diffusion-reaction equation using a one-step $\theta$-scheme as temporal discretization. Both the fully nonlinear method and its semi-implicit variant are considered. Solvability of the discrete problem is established and a priori error estimates in the LPS norm are proved. Uniqueness of the discrete solution is proved for the semi-implicit approach or for sufficiently small time steps.
Diamond--cell finite volume scheme for the Heston model
Pavol Kútik and Karol Mikula
2015, 8(5): 913-931 doi: 10.3934/dcdss.2015.8.913 +[Abstract](197) +[PDF](457.6KB)
The objective of this article is to propose a novel numerical scheme for solving the partial differential equation arising in the Heston stochastic volatility model. We discretize the governing advection-diffusion-reaction equation using the finite volume technique. The diffusion tensor is treated by means of the diamond--cell approximation. A theoretical result concerning the existence and uniqueness of the solution to the corresponding system of linear equations is proved. Numerical experiments regarding accuracy and order of convergence are shown.
Truss structure design using a length-oriented surface remeshing technique
Karol Mikula, Mariana Remešíková and Peter Novysedlák
2015, 8(5): 933-951 doi: 10.3934/dcdss.2015.8.933 +[Abstract](221) +[PDF](1917.0KB)
We present a method that can be used for designing truss structures representing either minimal surface shapes or general free-form shapes. The structures are designed so that they meet some specific criteria concerning their aesthetic properties and especially the lengths of the truss elements. We explain a technique for tangential redistribution of points on evolving surfaces that allows to obtain equally sized truss elements in selected subsets of the structure. This technique is applied to surfaces evolving by their mean curvature yielding constructions that approximate minimal surface shapes. Afterwards, we show how to remesh static free-form surfaces.
Numerical algorithm for tracking cell dynamics in 4D biomedical images
Karol Mikula, Róbert Špir and Nadine Peyriéras
2015, 8(5): 953-967 doi: 10.3934/dcdss.2015.8.953 +[Abstract](155) +[PDF](3019.6KB)
The paper presents new numerical algorithm for an automated cell tracking from large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. The cell trajectories are extracted as centered paths inside segmented spatio-temporal tree structures representing cell movements and divisions. Such paths are found by using a suitably designed and computed constrained distance functions and by a backtracking in steepest descent direction of a potential field based on these distance functions combination. The naturally parallelizable discretization of the eikonal equation which is used for computing distance functions is given and results of the tracking method for real 4D image data are presented and discussed.
Multiphase volume-preserving interface motions via localized signed distance vector scheme
Rhudaina Z. Mohammad and Karel Švadlenka
2015, 8(5): 969-988 doi: 10.3934/dcdss.2015.8.969 +[Abstract](310) +[PDF](722.9KB)
We develop a signed distance vector approach for approximating volume-preserving mean curvature motions of interfaces separating multiple phases -- a variant of the BMO (Bence-Merriman-Osher) thresholding dynamics. We adopt a variational method employing the idea of a vector-type discrete Morse flow, which allows us to easily treat volume constraint via penalization without having to change the threshold value. Moreover, employing a vector-valued analogue of the signed distance function, the scheme is designed to allow subgrid accuracy on uniform grids without adaptive refinement; thereby, alleviating the well-known BMO time and grid restrictions. Finally, we present numerical tests and examples.
Behavior of radially symmetric solutions for a free boundary problem related to cell motility
Harunori Monobe
2015, 8(5): 989-997 doi: 10.3934/dcdss.2015.8.989 +[Abstract](180) +[PDF](320.2KB)
We consider a free boundary problem related to cell motility. In the previous work, the author [5] replaced the boundary condition, in the original problem, with a simple boundary condition and studied the behavior of radially symmetric solutions for the modified problem. In this paper, we consider the original mathematical model and show that the behavior of solutions for the model is similar to the one of solutions for the modified model under the certain condition.
A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model)
Yoshiaki Muroya
2015, 8(5): 999-1008 doi: 10.3934/dcdss.2015.8.999 +[Abstract](182) +[PDF](346.1KB)
In this paper, for a Lotka-Volterra system with infinite delays and patch structure related to a multi-group SI epidemic model, applying Lyapunov functional techniques without using the form of diagonal dominance of the instantaneous negative terms over the infinite delay terms, we establish the complete global dynamics by a threshold parameter $s(M(0))$, that is, the trivial equilibrium is globally asymptotically stable if $s(M(0)) \leq 0$ and the positive equilibrium is globally asymptotically stable if $s(M(0))>0$, respectively. This offer new type condition of global stability for Lotka-Volterra systems with patch structure.
Towards modelling spiral motion of open plane curves
Koichi Osaki, Hirotoshi Satoh and Shigetoshi Yazaki
2015, 8(5): 1009-1022 doi: 10.3934/dcdss.2015.8.1009 +[Abstract](159) +[PDF](770.8KB)
We propose a simple evolution law for the motion of open curves with the boundary conditions towards realizing spiral growth, and derive the so-called kinematic equation. The role of the tangential velocities is studied and proved that they can be chosen arbitrarily for given boundary values. From this fact, a curvature adjusted tangential velocity for open curves is introduced. We present a numerical example which provides spiral motion starting from a line segment. This is a contrast to the case where an expanding line segment is the exact solution without the boundary conditions.
Stationary solutions for some shadow system of the Keller-Segel model with logistic growth
Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto and Hirofumi Izuhara
2015, 8(5): 1023-1034 doi: 10.3934/dcdss.2015.8.1023 +[Abstract](221) +[PDF](627.7KB)
From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.

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