A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

Jiwari, Ram; Tomasiello, Stefania; Tornabene, Francesco

doi:10.1108/ec-02-2017-0067

Cited by 15 publications

(5 citation statements)

References 48 publications

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“…Furthermore, it is demonstrated that the nonuniform grids improve the accuracy of the solution considerably without any extra computational cost (see Table , Figure a,b). The present scheme offered better results than for the linear model, and similar patterns to for the Gray–Scott and Brusselator models.…”

Section: Resultssupporting

confidence: 71%

“…Some well‐known forms of this model, such as the Gray–Scott model, Brusselator model, Schnakenberg model, and prey–predator model have gained much attention from the research community due to their important applications to biology and chemistry. In this order, the Gray–Scott model has been analyzed and solved by several methods , as is the case with the Brusselator model , Schnakenberg model and prey–predator model .…”

Section: Introductionmentioning

confidence: 99%

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A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Yadav

Jiwari

2018

Numerical Methods Partial

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In this article, we establish the existence and uniqueness of solutions to the coupled reaction-diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank-Nicolson and the predictor-corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second-order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme. KEYWORDS a priori error estimate, Brusselator, coupled reaction-diffusion models, Crank-Nicolson method, existence and uniqueness, Galerkin method, Gray-Scott, predictor-corrector method INTRODUCTIONThe coupled reaction-diffusion models frequently arise in the field of chemistry [1, 2], biology [3,4], sociology [5], physics, geology, ecology, and so forth. These models are naturally applied in chemistry, for example, the Brusselator model. However, such models can also describe dynamical processes of nonchemical nature, for instance, the predator-prey model. In 1952, in his pioneering work on morphogenesis, Turing [6] first proposed that the reaction-diffusion systems may be used to study the replicating patterns such as stripes, spots, and dappling, seen on the skin of many animals such as zebras, lions, and cats. In [6], Turing explained that the involvement of diffusion, Numer Methods Partial Differential Eq. 2019;35:830-850. wileyonlinelibrary.com/journal/num

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Section: Resultssupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Yadav

Jiwari

2018

Numerical Methods Partial

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“…All the computations are performed with N = 100 and the DG scheme with third-order accuracy. In Table 1, L ∞ -norms are estimated for the u variable, and the computed results are compared against the existing numerical results [61][62][63]. In addition, a profile comparison between the exact and numerical solutions for u and v at t = 1 is illustrated in Figure 2.…”

Section: Accuracy Test For Solvermentioning

confidence: 99%

On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems

Singh,

Msmali

2023

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Nonlinear coupled reaction–diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray–Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge–Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications.

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“…All the computations are done with N = 100 and the DG scheme with third-order of accuracy. In Table 1, L ∞ -norms are estimated for the u variable, and the computed results are compared against the existing numerical results [60,61,62]. Also, a profile comparison between the true and numerical solutions for u and v at t = 1 is illustrated in Fig.…”

Section: One-dimensional Linear Rd Systemmentioning

confidence: 99%

Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

Singh¹,

Battiato²,

Kumar³

2022

Preprint

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The nonlinear coupled reaction-diffusion (NCRD) systems are important in the formation of spatiotemporal patterns in many scientific and engineering fields, including physical and chemical processes, biology, electrochemical processes, fractals, viscoelastic materials, porous media, and many others. In this study, a mixed-type modal discontinuous Galerkin approach is developed for one-and two-dimensional NCRD systems, including linear, Gray-Scott, Brusselator, isothermal chemical, and Schnakenberg models to yield the spatiotemporal patterns. These models essentially represent a variety of complicated natural spatiotemporal patterns such as spots, spot replication, stripes, hexagons, and so on. In this approach, a mixed-type formulation is presented to address the second-order derivatives emerging in the diffusion terms. For spatial discretization, hierarchical modal basis functions premised on the orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge-Kutta algorithm. The spatiotemporal patterns generated with the present approach are very comparable to those found in the literature. This approach can readily be expanded to

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A numerical algorithm for computational modelling of coupled advection-diffusion-reaction systems

Cited by 15 publications

References 48 publications

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems

Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

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