American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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Discrete & Continuous Dynamical Systems - S

March 2021 , Volume 14 , Issue 3

Issue on recent topics in material, computer and life sciences

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2021, 14(3): i-i doi: 10.3934/dcdss.2021009 +[Abstract](687) +[HTML](396) +[PDF](68.56KB)
Abstract:
2021, 14(3): 769-783 doi: 10.3934/dcdss.2020232 +[Abstract](1506) +[HTML](515) +[PDF](776.33KB)
Abstract:

This paper is concerned with the simulation of multiphase flow hydrodynamics in an experimental oxyfuel fluidized bed combustor designed for biomass fuels. The aim is to perform cross-validation between several models and solvers that differ in the description of some phenomena in question. We focus on the influence of turbulence modeling, inter-phase drag force models, the presence of biomass in the mixture. Also the possibility to simplify the full 3D description to a quasi-1D model is tested. However, the results indicate that such simplification is not suitable for chaotic phenomena in considered scenarios. The models were developed using ANSYS Fluent and OpenFOAM CFD software packages as well as our in-house CFD code CFBSim. The quantities relevant for comparison (the densities of the dispersed solid phases and the phase velocities) are presented in the form of cross-section averaged vertical profiles.

2021, 14(3): 785-801 doi: 10.3934/dcdss.2020333 +[Abstract](1173) +[HTML](394) +[PDF](8850.25KB)
Abstract:

The paper presents a numerical study of the efficiency of the newly proposed far-field boundary simulations of wall-bounded, stably stratified flows. The comparison of numerical solutions obtained on large and truncated computational domain demonstrates how the solution is affected by the adopted far-field conditions. The mathematical model is based on Boussinesq approximation for stably stratified viscous variable density incompressible fluid. The three-dimensional numerical simulations of the steady flow over an isolated hill were performed using a high-resolution compact finite difference code, with artificial compressibility method used for pressure computation. The mutual comparison of the full domain reference solution and the truncated domain solution is provided and the influence of the newly proposed far-field boundary condition is discussed.

2021, 14(3): 803-817 doi: 10.3934/dcdss.2020229 +[Abstract](1471) +[HTML](470) +[PDF](1102.98KB)
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A system of ordinary differential equations that describes the motion of a self-propelling rigid disk is studied. In this system, the disk moves along a straight-line and reflects from a boundary. Interestingly, numerical simulation shows that the angle of reflection is greater than that of incidence. The purpose of this study is to present a mathematical proof for this attractive phenomenon. Moreover, the reflection law is numerically investigated. Finally, existence and asymptotic stability of a square-shaped closed orbit for billiards in square table with inelastic reflection law are discussed.

2021, 14(3): 819-833 doi: 10.3934/dcdss.2020349 +[Abstract](1567) +[HTML](332) +[PDF](3773.47KB)
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In this article, we deal with the numerical immersed boundary-lattice Boltzmann method for simulation of the fluid-structure interaction problems in 2D. We consider the interaction of incompressible, Newtonian fluid in an isothermal system with an elastic fiber, which represents an immersed body boundary. First, a short introduction to the lattice Boltzmann and immersed boundary method is presented and the combination of these two methods is briefly discussed. Then, the choice of the smooth approximation of the Dirac delta function and the discretization of the immersed body is discussed. One of the significant drawbacks of immersed boundary method is the penetrative flow through the immersed impermeable boundary. The effect of the immersed body boundary discretization is investigated using two benchmark problems, where an elastic fiber is deformed. The results indicate that the restrictions placed on the discretization in literature are not necessary.

2021, 14(3): 835-850 doi: 10.3934/dcdss.2020387 +[Abstract](1241) +[HTML](266) +[PDF](3768.8KB)
Abstract:

In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient \begin{document}$d$\end{document} of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as \begin{document}$d$\end{document} increases the speed of propagation slows down and the parameter region of propagation narrows down.

2021, 14(3): 851-863 doi: 10.3934/dcdss.2020347 +[Abstract](1080) +[HTML](302) +[PDF](2227.86KB)
Abstract:

We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.

2021, 14(3): 865-879 doi: 10.3934/dcdss.2020350 +[Abstract](1407) +[HTML](316) +[PDF](3949.11KB)
Abstract:

A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation is proposed without numerically computing the gradient itself at the faces of computational cells in order to find a normal diffusive flux. To apply the method for parallel computations with a 1-ring neighborhood, we use an iterative method to solve the obtained system of algebraic equations. Several numerical examples illustrate some advantages of the proposed method.

2021, 14(3): 881-891 doi: 10.3934/dcdss.2020233 +[Abstract](1572) +[HTML](523) +[PDF](2306.95KB)
Abstract:

We propose a simple and accurate procedure how to extract the values of model parameters in a flame/smoldering evolution equation from 2D movie images of real experiments. The procedure includes a novel method of image segmentation, which can detect an expanding smoldering front as a plane polygonal curve. The evolution equation is equivalent to the so-called Kuramoto-Sivashinsky (KS) equation in a certain scale. Our results suggest a valid range of parameters in the KS equation as well as the validity of the KS equation itself.

2021, 14(3): 893-907 doi: 10.3934/dcdss.2020390 +[Abstract](1336) +[HTML](268) +[PDF](3679.4KB)
Abstract:

In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized \begin{document}$L^1$\end{document} norm of the difference of step-like functions which are branches of \begin{document}$\arg (x)$\end{document} whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.

2021, 14(3): 909-918 doi: 10.3934/dcdss.2020391 +[Abstract](1196) +[HTML](329) +[PDF](3790.88KB)
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We consider the blow-up problems of the power type of stochastic differential equation, \begin{document}$dX = \alpha X^p(t)dt+X^q(t)dW(t)$\end{document}. It has been known that there exists a critical exponent such that if \begin{document}$p$\end{document} is greater than the critical exponent then the solution \begin{document}$X(t)$\end{document} blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive time step and analyze it mathematically. Finally we show the numerical result by using the proposed scheme.

2021, 14(3): 919-933 doi: 10.3934/dcdss.2020228 +[Abstract](1846) +[HTML](460) +[PDF](457.37KB)
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We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

2021, 14(3): 935-951 doi: 10.3934/dcdss.2020382 +[Abstract](1349) +[HTML](301) +[PDF](1070.41KB)
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This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.

2021, 14(3): 953-970 doi: 10.3934/dcdss.2020384 +[Abstract](874) +[HTML](266) +[PDF](420.11KB)
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The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot class. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy value is. In this article, we discuss two topics on the O'Hara energies. First, we slightly generalize the O'Hara energies and consider a characterization of its finiteness. The finiteness of the O'Hara energies was considered by Blatt in 2012 who used the Sobolev-Slobodeckij space, and naturally we consider a generalization of this space. Another fundamental problem is to understand the minimizers of the O'Hara energies. This problem has been addressed in several papers, some of them based on numerical computations. In this direction, we discuss a discretization of the O'Hara energies and give some examples of numerical computations. Particular one of the O'Hara energies, called the Möbius energy thanks to its Möbius invariance, was considered by Kim-Kusner in 1993, and Scholtes in 2014 established convergence properties. We apply their argument in general since the argument does not rely on Möbius invariance.

2021, 14(3): 971-985 doi: 10.3934/dcdss.2020351 +[Abstract](1437) +[HTML](323) +[PDF](3736.38KB)
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Here, we report a novel method of 3D image segmentation, using surface reconstruction from 3D point cloud data and 3D digital image information. For this task, we apply a mathematical model and numerical method based on the level set algorithm. This method solves surface reconstruction by the application of advection equation with a curvature term, which gives the evolution of an initial condition to the final state. This is done by defining the advective velocity in the level set equation as the weighted sum of distance function and edge detector function gradients. The distance function to the shape, represented by the point cloud, is computed using the fast sweeping method. The edge detector function is applied to the presmoothed 3D image. A crucial point for efficiency is the construction of an initial condition by a simple tagging algorithm, which allows us also to highly speed up the numerical scheme when solving PDEs. For the numerical discretization, we use a semi-implicit co-volume scheme in the curvature part and implicit upwind scheme in the advective part. The method was tested on representative examples and applied to real data representing 3D biological microscopic images of developing mammalian embryo.

2021, 14(3): 987-999 doi: 10.3934/dcdss.2020381 +[Abstract](1227) +[HTML](294) +[PDF](8057.72KB)
Abstract:

The finite volume method (FVM) as a numerical method can be straightforwardly applied for global as well as local gravity field modelling. However, to obtain precise numerical solutions it requires very refined discretization which leads to large-scale parallel computations. To optimize such computations, we present a special class of numerical techniques that are based on a physical decomposition of the computational domain. The domain decomposition (DD) methods like the Additive Schwarz Method are very efficient methods for solving partial differential equations. We briefly present their mathematical formulations, and we test their efficiency in numerical experiments dealing with gravity field modelling. Since there is no need to solve special interface problems between neighbouring subdomains, in our applications we use the overlapping DD methods. Finally, we present the numerical experiment using the FVM approach with 93 312 000 000 unknowns that would not be possible to perform using available computing facilities without aforementioned methods that can efficiently reduce a numerical complexity of the problem.

2021, 14(3): 1001-1015 doi: 10.3934/dcdss.2020380 +[Abstract](975) +[HTML](279) +[PDF](330.35KB)
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We consider a boundary value problem for the stationary Stokes problem and the corresponding pressure-Poisson equation. We propose a new formulation for the pressure-Poisson problem with an appropriate additional boundary condition. We establish error estimates between solutions to the Stokes problem and the pressure-Poisson problem in terms of the additional boundary condition. As boundary conditions for the Stokes problem, we use a traction boundary condition and a pressure boundary condition introduced in C. Conca et al (1994).

2021, 14(3): 1017-1032 doi: 10.3934/dcdss.2020348 +[Abstract](1387) +[HTML](326) +[PDF](3360.0KB)
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In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains the term expanding the curve in its outer normal direction up to the region of habitat boundary edges, the term attracting the curve accurately to the edges and the smoothing term given by the influence of local curvature. For the numerical solution, we use the flowing finite volume method discretizing the arising advection-diffusion intrinsic partial differential equation including the asymptotically uniform tangential redistribution of curve grid points. We present segmentation results for satellite data from a selected area of Western Slovakia (Záhorie) where the so-called riparian forests represent the important European Natura 2000 habitat. The automatic segmentation results are compared with the semi-automatic segmentation performed by the botany expert and with the GPS tracks obtained in the field. The comparisons show the ability of our numerical model to segment the habitat areas with the accuracy comparable to the pixel resolution of the Sentinel-2 optical data.

2021, 14(3): 1033-1046 doi: 10.3934/dcdss.2020231 +[Abstract](2149) +[HTML](527) +[PDF](4490.7KB)
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In this paper we introduce mathematical model and real-time numerical method for segmentation of Natura 2000 habitats in satellite images by evolving open planar curves. These curves in the Lagrangian formulation are driven by a suitable velocity vector field, projected to the curve normal. Besides the vector field, the evolving curve is influenced also by the local curvature representing a smoothing term. The model is numerically solved using the flowing finite volume method discretizing the arising intrinsic partial differential equation with Dirichlet boundary conditions. The time discretization is chosen as an explicit due to the ability of real-time edge tracking. We present the results of semi-automatic segmentation of various areas across Slovakia, from the riparian forests to mountainous areas with scrub pine. The numerical results were compared to habitat boundaries tracked by GPS device in the field by using the mean and maximal Hausdorff distances as criterion.

2021, 14(3): 1047-1062 doi: 10.3934/dcdss.2020405 +[Abstract](1204) +[HTML](251) +[PDF](901.07KB)
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Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.

2021, 14(3): 1063-1078 doi: 10.3934/dcdss.2020230 +[Abstract](2185) +[HTML](528) +[PDF](5737.97KB)
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Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmission boundary condition, an inequality for the energy estimate is proved under some assumptions to be satisfied in practical computation. In the numerical part, based on the theoretical results, the energy estimate of the SWEs with a transmission boundary condition is confirmed numerically by a finite difference method (FDM). The choice of a positive constant \begin{document}$c_0$\end{document} used in the transmission boundary condition is investigated additionally. Furthermore, we present numerical results by a Lagrange-Galerkin scheme, which are similar to those by the FDM. The theoretical results along with the numerical results strongly recommend that the transmission boundary condition is suitable for the boundaries in the open sea.

2021, 14(3): 1079-1092 doi: 10.3934/dcdss.2020352 +[Abstract](1592) +[HTML](337) +[PDF](2379.25KB)
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We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series obtained from the model approximates the actual one and that some characteristics of the chaotic invariant set mimic the actual ones. We investigate the appropriate choice of the delay-coordinate, especially the delay-time and the dimension, which enables us to construct a model having a relatively high-dimensional attractor with low computational costs.

2021, 14(3): 1093-1102 doi: 10.3934/dcdss.2020385 +[Abstract](967) +[HTML](256) +[PDF](298.86KB)
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In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.

2021, 14(3): 1103-1110 doi: 10.3934/dcdss.2020392 +[Abstract](1146) +[HTML](273) +[PDF](500.51KB)
Abstract:

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation

where \begin{document}$f(x) = r\sin x$\end{document} with \begin{document}$r>0$\end{document}. It is shown that the well studied second order ordinary differential equation, known as the nonlinear pendulum equation, derives a symmetric periodic solution of period \begin{document}$2$\end{document}, expressed in terms of the Jacobi elliptic functions, for the delay differential equation. We here apply the approach in Kaplan and Yorke (1974) for a differential equation with discrete delay to the distributed delay differential equation.

2021, 14(3): 1111-1122 doi: 10.3934/dcdss.2020383 +[Abstract](1133) +[HTML](277) +[PDF](348.47KB)
Abstract:

Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.

2021, 14(3): 1123-1132 doi: 10.3934/dcdss.2020389 +[Abstract](1210) +[HTML](269) +[PDF](9659.55KB)
Abstract:

Automatic detection of objects in photos and images is beneficial in various scientific and industrial fields. This contribution suggests an algorithm for segmentation of color images by the means of the parametric mean curvature flow equation and CIE94 color distance function. The parametric approach is enriched by the enhanced algorithm for topological changes where the intersection of curves is computed instead of unreliable curve distance. The result is a set of parametric curves enclosing the object. The algorithm is presented on a test image and also on real photos.

2021, 14(3): 1133-1143 doi: 10.3934/dcdss.2020388 +[Abstract](1190) +[HTML](321) +[PDF](347.14KB)
Abstract:

We consider a blow-up phenomenon for \begin{document}${ \partial_t^2 u_ \varepsilon}$\end{document} \begin{document}${- \varepsilon^2 \partial_x^2u_ \varepsilon }$\end{document} \begin{document}${ = F(\partial_t u_ \varepsilon)}.$\end{document} The derivative of the solution \begin{document}$\partial_t u_ \varepsilon$\end{document} blows-up on a curve \begin{document}$t = T_ \varepsilon(x)$\end{document} if we impose some conditions on the initial values and the nonlinear term \begin{document}$F$\end{document}. We call \begin{document}$T_ \varepsilon$\end{document} blow-up curve for \begin{document}${ \partial_t^2 u_ \varepsilon}$\end{document} \begin{document}${- \varepsilon^2 \partial_x^2u_ \varepsilon }$\end{document} \begin{document}${ = F(\partial_t u_ \varepsilon)}.$\end{document} In the same way, we consider the blow-up curve \begin{document}$t = \tilde{T}(x)$\end{document} for \begin{document}${\partial_t^2 u}$\end{document} \begin{document}$=$\end{document} \begin{document}${F(\partial_t u)}.$\end{document} The purpose of this paper is to show that, for each \begin{document}$x$\end{document}, \begin{document}$T_ \varepsilon(x)$\end{document} converges to \begin{document}$\tilde{T}(x)$\end{document} as \begin{document}$\varepsilon\rightarrow 0.$\end{document}

2021, 14(3): 1145-1160 doi: 10.3934/dcdss.2020386 +[Abstract](1225) +[HTML](275) +[PDF](964.63KB)
Abstract:

In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of object of interest and its representation by the signed-distance function (OF\begin{document}$^{dist}$\end{document} method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of \begin{document}$T_1$\end{document} relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF\begin{document}$^{dist}$\end{document} method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM\begin{document}$^{dist}$\end{document}). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF\begin{document}$^{dist}$\end{document} and MIM\begin{document}$^{dist}$\end{document} provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM\begin{document}$^{dist}$\end{document} provided a larger registration error, but more homogeneously distributed, compared to OF\begin{document}$^{dist}$\end{document}. For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF\begin{document}$^{dist}$\end{document} registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.

2021, 14(3): 1161-1180 doi: 10.3934/dcdss.2020227 +[Abstract](1350) +[HTML](510) +[PDF](8694.93KB)
Abstract:

In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles (\begin{document}$VTN$\end{document}-specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.

2021, 14(3): 1181-1195 doi: 10.3934/dcdss.2020226 +[Abstract](1596) +[HTML](478) +[PDF](432.5KB)
Abstract:

The aim of the paper is to study problem of financial derivatives pricing based on the idea of the Heston model introduced in [9]. Following the approach stated in [6] and in [7] we construct the regularised version of the Heston model and the discrete duality finite volume (DDFV) scheme for this model. The numerical analysis is performed for this scheme and stability estimates on the discrete solution and the discrete gradient are obtained. In addition the convergence of the DDFV scheme to the weak solution of the regularised Heston model is proven. The numerical experiments are provided in the end of the paper to test the regularisation parameter impact.

2021, 14(3): 1197-1212 doi: 10.3934/dcdss.2020234 +[Abstract](1830) +[HTML](484) +[PDF](1865.43KB)
Abstract:

The purposes of this work are to study the \begin{document}$L^{2}$\end{document}-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically. The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2020 CiteScore: 3.1