Numerical algorithms for a variational problem of the spatial segregation of reaction–diffusion systems

Arakelyan, Avetik; Bozorgnia, Farid

doi:10.1016/j.amc.2013.03.074

Cited by 9 publications

(17 citation statements)

References 13 publications

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“…The second approach is a finite difference method, but lack of its analysis in [6]. This finite difference method has been generalized in [9] for the case of nonnegative f i . In [9] the authors present a numerical consistent variational system with strong interaction, and provide disjointness condition of populations during the iteration of the scheme.…”

Section: Known Resultsmentioning

confidence: 99%

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

Arakelyan

Barkhudaryan

2016

Computers & Mathematics with Applications

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In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction-diffusion systems with m ≥ 2 population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case m = 2, we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.2000 Mathematics Subject Classification. 35R35, 65N06, 65N22, 92D25.

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Section: Known Resultsmentioning

confidence: 99%

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

Arakelyan

Barkhudaryan

2016

Computers & Mathematics with Applications

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Section: Which Satisfymentioning

confidence: 99%

“…For the numerical approximation of the system (1.2) we refer to [3,4]. In [3] the authors propose a numerical scheme for a class of reaction-diffusion system with m densities having disjoint supports and are governed by a minimization problem. The proposed numerical scheme is applied for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions.…”

Section: Which Satisfymentioning

confidence: 99%

On a Class of Singularly Perturbed Elliptic Systems with Asymptotic Phase Segregation

Bozorgnia¹,

Burger²

2019

Preprint

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This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, we present some numerical simulations of the asymptotic problem.

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“…Numerical studies of this type of problem have so far been limited to the planar case. We mention in particular the study of Chang et al (2004) and some special algorithms in the case of small m given by Bozorgnia and Arakelyan (2013) and Bozorgnia (2009). Also Bourdin, Bucur and Oudet (2010) considered the problem for large values of m using a fictitious domain approach.…”

Section: )mentioning

confidence: 99%

A computational approach to an optimal partition problem on surfaces

Elliott¹,

Ranner²

2015

Interfaces Free Bound.

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We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace-Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces.

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Numerical algorithms for a variational problem of the spatial segregation of reaction–diffusion systems

Cited by 9 publications

References 13 publications

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

A numerical approach for a general class of the spatial segregation of reaction–diffusion systems arising in population dynamics

On a Class of Singularly Perturbed Elliptic Systems with Asymptotic Phase Segregation

A computational approach to an optimal partition problem on surfaces

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