Combined finite volume element method for singularly perturbed reaction–diffusion problems

Phongthanapanich, Sutthisak; Dechaumphai, Pramote

doi:10.1016/j.amc.2008.10.047

Cited by 14 publications

(6 citation statements)

References 20 publications

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“…Analogously, one easily shows that b2∇ 4 y u q , w p 2 = − b2∇ 4 y w p , u q 2 . This proves equation (35) in Lemma 3.1. Lemma 3.2.…”

supporting

confidence: 64%

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Unconditional Stability and Fourth-Order Convergence Of A Two-Step Time Split Explicit/Implicit Scheme For Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation With Variable Coefficients

Ngondiep

Tedjani

2022

Preprint

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This paper deals with an efficient two-step time split explicit/implicit scheme applied to a two-dimensional nonlinear unsteady convection-diffusion-reaction equation with variable coefficients. The computational cost of the new algorithm at each time level is equivalent to solving a pentadiagonal matrix equation with strictly dominant diagonal elements. Such a bandwidth matrix can be easily inverted using the Gaussian Decomposition and the corresponding linear system should be solved by the back substitution method. The proposed approach is unconditionally stable, temporal second-order accuracy and fourth-order convergence in space. These results suggest that the developed technique is faster and more efficient than a large class of numerical methods studied in the literature for the considered initial-boundary value problem. Numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the performance of the constructed numerical scheme.

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“…Analogously, one easily shows that b2∇ 4 y u q , w p 2 = − b2∇ 4 y w p , u q 2 . This proves equation (35) in Lemma 3.1. Lemma 3.2.…”

supporting

confidence: 64%

“…To overcome these drawbacks, a broad range of stabilized numerical techniques have been developed in an approximate solution of such complex equations. For more details, the readers can consult the works discussed in [36,23,31,41,13,16,35,27,29,32,9,11,12,19,1] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Unconditional Stability and Fourth-Order Convergence Of A Two-Step Time Split Explicit/Implicit Scheme For Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation With Variable Coefficients

Ngondiep

Tedjani

2022

Preprint

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“…The authors provide evidence that the mathematical model and the computational approach used in the paper are capable of representing and reproducing ecologically significant phenomena. This strengthens the understanding and applicability of the model in studying the dynamics of model populations for tackling the reaction-diffusion problem [5], [13], [14]. The finite volume approach has been presented and researched during the last decades by Garvie et al (2007) [13], and Phongthanapanich et.…”

Section: B Backgroundmentioning

confidence: 73%

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Saqib,

Akhtar,

Hasnain

et al. 2024

IEEE Access

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In this research, we present two Numerical algorithms for studying the dynamics of spatially extended coupled nonlinear reaction-diffusion model. The difference in this work is to model a system of reacting and diffusing chemicals, and how to predict the dynamics of ecologically relevant behavior, including chaos. The goal can be achieved by applying a proposed staggered grid finite volume method for solving the Reaction model, rigorously validating the numerical method and grid generation techniques against established results. Additionally, we integrate an Alternating Direction Implicit formulation to compare the efficiency and accuracy of the model's results. Dynamical system analysis, including linear stability analysis, is applied to comprehend the fundamental qualitative features of the inhibitor-activator system inherent in the coupled reaction-diffusion equations. Results, presented in tables and graphical representations, show that the schemes' high accuracy at fourth-order precision in space and second-order in time, with conditional stability. Numerical results showed that the proposed finite volume approach was exceptionally proficient and accurate for tackling the two-dimensional nonlinear coupled reactiondiffusion model. Overall, the dialog highlights the significance of the proposed staggered grid finite approach and calculations for fathoming coupled reaction-diffusion equations are widely studied in biological and chemical systems of nonlinear differential equations.

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“…In this paper, the gradient term that used to calculate the curvature, ∇ n ij , is approximated by the weighted residuals method, which is commonly employed in the finite element technique [20][21][22]. First, the ∇ n l is assumed to distribute linearly over an arbitrary l in the form (17) where N k ( x) denotes the linear interpolation functions for convex polygon.…”

Section: Characteristic Level Set Equationmentioning

confidence: 99%

Explicit characteristic‐based finite volume element method for level set equation

Phongthanapanich

Dechaumphai

2010

Numerical Methods in Fluids

Self Cite

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SUMMARYAccurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two-phase flow problems. This paper presents an explicit characteristic-based finite volume element method for solving the two-dimensional level set equation. The method is applicable for the case of non-divergence-free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.

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Combined finite volume element method for singularly perturbed reaction–diffusion problems

Cited by 14 publications

References 20 publications

Unconditional Stability and Fourth-Order Convergence Of A Two-Step Time Split Explicit/Implicit Scheme For Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation With Variable Coefficients

Unconditional Stability and Fourth-Order Convergence Of A Two-Step Time Split Explicit/Implicit Scheme For Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation With Variable Coefficients

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Explicit characteristic‐based finite volume element method for level set equation

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