Upcoming volume is Volume 45, Number 4.
It will include the following papers:
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Finite Volume Schemes for the Generalized Subjective Surface Equation in Image Segmentation | |
| Karol Mikula and Mariana Remešíková | |
| Abstract: In this paper, we describe an efficient method for 3D image segmentation. The method uses a PDE model -- the so called generalized subjective surface equation which is an equation of advection-diffusion type. The main goal is to develop an efficient and stable numerical method for solving this problem. The numerical solution is based on semi-implicit time discretization and flux-based level set finite volume space discretization. The space discretization is discussed in details and we introduce three possible alternatives of the so called diamond cell finite volume scheme for this type of 3D nonlinear diffusion equation. We test the performance of the method and all its variants introduced in the paper by determining the experimental order of convergence. Finally we show a couple of practical applications of the method. | |
| Pages: xx-xx | |
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On the Singular Limit of Solutions to the Cox-Ingersoll-Ross Interest Rate Model with Stochastic Volatility | |
| Beáta Stehlíková and Daniel Ševčovič | |
| Abstract: In this paper we are interested in term structure models for pricing zero coupon bonds under rapidly oscillating stochastic volatility. We analyze solutions to the generalized Cox--Ingersoll-Ross two factors model describing clustering of interest rate volatilities. The main goal is to derive an asymptotic expansion of the bond price with respect to a singular parameter representing the fast scale for the stochastic volatility process. We derive the second order asymptotic expansion of a solution to the two factors generalized CIR model and we show that the first two terms in the expansion are independent of the variable representing stochastic volatility. | |
| Pages: xx-xx | |
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Direct Approach to Mean-Curvature Flow with Topological Changes | |
| Petr Pauš and Michal Beneš | |
| Abstract: This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves Γ(t) : S ---> R2, t >= 0. The curves are driven by the normal velocity v which is the function of curvature κ and the position. The evolution law reads as: v = -κ + F. The motion law is treated using direct approach numerically solved by two schemes, i.e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved by tangential redistribution of curve points which allows long time computations and better accuracy. The results of dislocation dynamics simulation are presented (e.g., dislocations in channel or Frank-Read source). We also introduce an algorithm for treatment of topological changes in the evolving curve. | |
| Pages: xx-xx | |
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High Order Finite Volume Schemes for Numerical Solution of 2d and 3d Transonic Flows | |
| Petr Furmánek, Jiří Fürst and Karel Kozel | |
| Abstract: The aim of this article is a qualitative analysis of two modern finite volume (FVM) schemes. First one is the so called Modified Causon's scheme, which is based on the classical MacCormack FVM scheme in total variation diminishing (TVD) form, but is simplified in such a way that the demands on computational power are much smaller without loss of accuracy. Second one is implicit WLSQR (Weighted Least Square Reconstruction) scheme combined with various types of numerical fluxes (AUSMPW+ and HLLC). Two different test cases were chosen for the comparison - 1) two-dimensional transonic inviscid nonstationary flow over an oscillating NACA 0012 profile and 2) three-dimensional transonic inviscid stationary flow around the Onera M6 wing. Nonstationary effects were simulated with the use of Arbitrary Lagrangian-Eulerian Method (ALE). Experimental results for these regimes of flow are easily available and so the numerical results are compared both in-between and with experimental data. The obtained numerical results in all considered cases (2D and 3D) are in a good agreement with experimental data. | |
| Pages: xx-xx | |
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Phase Field Model for Mode III Crack Growth in Two Dimensional Elasticity | |
| TAKESHI TAKAISHI and MASATO KIMURA | |
| Abstract: A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. We introduce a phase field to represent the shape of the crack with a regularization parameter ε > 0 and we approximate the Francfort-Marigo type energy using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient flow of this regularized energy. We show several numerical examples of the crack growth computed with an adaptive mesh finite element method. | |
| Pages: xx-xx | |
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The Dynamics of Weakly Interacting Fronts in an Adsorbate-induced Phase Transition Model | |
| Shin-Ichiro Ei and Tohru Tsujikawa | |
| Abstract: Hildebrand et al. proposed an adsorbate-induced phase transition model. For this model, Takei et al. found several stationary and evolutionary patterns by numerical simulations. Due to bistability of the system, there appears a phase separation phenomenon and an interface separating these phases. In this paper, we introduce the equation describing the motion of two interfaces in R2 and discuss an application. Moreover, we prove the existence of the traveling front solution which approximates the shape of the solution in the neighborhood of the interface. | |
| Pages: xx-xx | |
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On a Variant of the Local Projection Method Stable in the Supg Norm | |
| Petr Knobloch | |
| Abstract: We consider the local projection finite element method for the discretization of a scalar convection--diffusion equation with a divergence--free convection field. We introduce a new fluctuation operator which is defined using an orthogonal L2 projection with respect to a weighted L2 inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf--sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution. | |
| Pages: xx-xx | |
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A Note on the Optimal Portfolio Problem in Discrete Processes | |
| Naoyuki ISHIMURA and Yuji MITA | |
| Abstract: We deal with the optimal portfolio problem in discrete-time setting. Employing the discrete Ito formula, which is developed by Fujita, we establish the discrete Hamilton-Jacobi-Bellman (d-HJB) equation for the value function. Simple examples of the d-HJB equation are also discussed. | |
| Pages: xx-xx | |
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Implementation of the MR Tractography Visualization Kit Based on the Anisotropic Allen-Cahn Equation | |
| Pavel Strachota | |
| Abstract: Magnetic Resonance Diffusion Tensor Imaging (MR-DTI) is a noninvasive in vivo method capable of examining the structure of human brain, providing information about the position and orientation of the neural tracts. After a short introduction to the principles of MR-DTI, this paper describes the steps of the proposed neural tract visualization technique based on the DTI data. The cornerstone of the algorithm is a texture diffusion procedure modeled mathematically by the problem for the Allen-Cahn equation with diffusion anisotropy controlled by a tensor field. Focus is put on the issues of the numerical solution of the given problem, using the finite volume method for spatial domain discretization. Several numerical schemes are compared with the aim of reducing the artificial (numerical) isotropic diffusion. The remaining steps of the algorithm are commented on as well, including the acquisition of the tensor field before the actual computation begins and the postprocessing used to obtain the final images. Finally, the visualization results are presented. | |
| Pages: xx-xx | |
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How to Unify the Total/Local-Length-Constraints of the Gradient Flow for the Bending Energy of Plane Curves | |
| Yuki Miyamoto, Takeyuki Nagasawa and Fumito Suto | |
| Abstract: The gradient of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic behavior has not been studied. For the later one, the global existence was guaranteed for only curves with the rotation number one, and the behavior was well studied. It is desirable to compensate the results with each other. In this note, it is proposed how to unify the two flows. | |
| Pages: xx-xx | |
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A Solution of Nonlinear Diffusion Problems by Semilinear Reaction-Diffusion Systems | |
| Hideki Murakawa | |
| Abstract: This paper deals with nonlinear diffusion problems involving degenerate parabolic problems, such as the Stefan problem and the porous medium equation, and cross-diffusion systems in population ecology. The degeneracy of the diffusion and the effect of cross-diffusion, that is, nonlinearities of the diffusion, complicate its analysis. In order to avoid the nonlinearities, we propose a reaction-diffusion system with solutions that approximate those of the nonlinear diffusion problems. The reaction-diffusion system includes only a simple reaction and linear diffusion. Resolving semilinear problems is typically easier than dealing with nonlinear diffusion problems. Therefore, our ideas are expected to reveal new and more effective approaches to the study of nonlinear problems. | |
| Pages: xx-xx | |