Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation

Zhang, Rongpei; Yu, Xijun; Zhu, Jiang; Loula, Abimael F. D.

doi:10.1016/j.apm.2013.09.008

Cited by 39 publications

(21 citation statements)

References 26 publications

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“…Turing in his seminal paper [1] presented a consistent account of the details and mathematical formalism showing that reaction-diffusion systems can be responsible for the emergence of pattern formation in nature. It has become an attractive area of research for scholars in applied mathematics [2,3,4,5,6,7,8] to investigate and quantify the behaviour of a set of reaction-diffusion equations as an evolving dynamical system. Systems of reaction-diffusion equations that model the evolution of pattern formation in nature are often a set of non-linear parabolic equations [5,7,9,10,17,18,21,23], whose solution is seldom analytically retrievable.…”

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confidence: 99%

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

Sarfaraz

Madzvamuse

2018

Int. J. Bifurcation Chaos

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This work explores the influence of domain size of a non-compact two dimensional annular domain on the evolution of pattern formation that is modelled by an activator-depleted reaction-diffusion system. A closed form expression is derived for the spectrum of Laplace operator on the domain satisfying a set of homogeneous conditions of Neumann type both at inner and outer boundaries. The closed form solution is numerically verified using the spectral method on polar coordinates. The bifurcation analysis of activator-depleted reaction-diffusion system is conducted on the admissible parameter space under the influence of two bounds on the parameter denoting the thickness of the annular region. The admissibility of Hopf and transcritical bifurcations is proven conditional on the domain size satisfying a lower bound in terms of reaction-diffusion parameters. The admissible parameter space is partitioned under the proposed conditions corresponding to each case, and in turn such conditions are numerically verified by applying a method of polynomials on a quadrilateral mesh. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived conditions on the domain size for all types of admissible bifurcations.

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confidence: 99%

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

Sarfaraz

Madzvamuse

2018

Int. J. Bifurcation Chaos

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“…Consider the Schnakenberg equations on the unit square domain = [0, 1] × [0, 1] with homogenous Neumann boundary conditions applied to both u and v as before [67]. Let the parameter values be determined by a = 0.1305, b = 0.7695, d 1 = 0.05, d 2 = 1 and γ = 100.…”

Section: Examplementioning

confidence: 99%

“…In [69], discontinuous Galerkin (DG) finite element methods, coupled with Strang type symmetrical operator splitting methods, have been used for solving reaction-diffusion systems in domains with complex geometry. The authors of [67] have presented numerical solutions of Schnakenberg model and chlorite-iodidemalonic acid (CIMA) reactive model, by combining direct discontinuous Galerkin (DDG) finite element method with implicit integration factor (IIF) time integration method. In [10], variational multiscale element-free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods have been applied for solving two-dimensional Brusselator reactiondiffusion system with and without cross-diffusion.…”

Section: Introductionmentioning

confidence: 99%

Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models

Ilati

Dehghan

2016

Engineering with Computers

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“…The similar method was extended in [7] to solve the compressible miscible displacement problem. The DGFE techniques have been applied by the authors of this paper [13,14,23,24], to nonlinear partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A combined discontinuous Galerkin finite element method for miscible displacement problem

Zhang

Zhu

Zhang

et al. 2017

Journal of Computational and Applied Mathematics

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A combined discontinuous Galerkin finite element method for miscible displacement problem, Journal of Computational and Applied Mathematics (2016), http://dx. Abstract.A new combined method is constructed for solving incompressible miscible displacement in porous media. In this procedure, a hybrid mixed element method is constructed for the pressure and velocity equation, while a symmetric discontinuous Galerkin finite element method is proposed for the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also conserves the local mass balance. The stability and consistency of the method are analyzed and the optimal error estimate in L ∞ (L 2 ) for velocity and concentration and the super convergence in L ∞ (H 1 ) for pressure are derived. Finally, some numerical results are presented.

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Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation

Cited by 39 publications

References 26 publications

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

Domain-Dependent Stability Analysis of a Reaction–Diffusion Model on Compact Circular Geometries

Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models

A combined discontinuous Galerkin finite element method for miscible displacement problem

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