A Green's function formulation for finite-differences schemes

Alvarez-Ramı́rez, José; Valdés‐Parada, Francisco J.; Álvarez, Jesús; Ochoa‐Tapia, J. Alberto

doi:10.1016/j.ces.2007.03.013

Cited by 15 publications

(5 citation statements)

References 7 publications

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“…The main task is to show that a Green's function formulation offers a natural framework to derive non-standard FD schemes without resorting into heuristic rules. The approach followed is a generalization of the methodology used for Cartesian coordinates in [1]. In that work, we showed that traditional FD schemes for Cartesian coordinates can be derived from a Green's function formulation of the reaction-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Alvarez-Ramı́rez

Valdés‐Parada

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tReaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. In this work we design, in a systematic way, non-standard finite-differences (FD) schemes for a class of reaction-diffusion equations of the form 1, where σ is the shape power that accounts for the complexity of the domain geometry. The proposed FD scheme, that is derived from a Green's function formulation, replicates the underlying geometry and reduces to traditional FD schemes for sufficiently small values of the grid spacing. Numerical results show that the non-standard FD scheme offers smaller approximation errors with respect to traditional schemes, specially for coarse grids.

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Section: Introductionmentioning

confidence: 99%

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Alvarez-Ramı́rez

Valdés‐Parada

2009

Journal of Computational and Applied Mathematics

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“…In order to circumvent such difficulties, some attempts have been made to obtain numerical Green's functions [17][18][19][20][21], but high cost for computing is still inevitable. In contrast, if we focus on 1D boundary value problems, then the Green's function method can be applied to a much broader class of problems because the closed forms of 1D fundamental solution and the corresponding Green's function are obtainable with more ease [22,23].…”

Section: Introductionmentioning

confidence: 99%

Axial Green's function method for multi‐dimensional elliptic boundary value problems

Kim

Park

Jun

2008

Numerical Meth Engineering

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Axial Green's function method is proposed to solve multi-dimensional second-order elliptic partial differential equations with the geometrical complexity of solution domains. Instead of directly dealing with 2D or 3D Green's functions, the proposed method systematically decomposes a multi-dimensional elliptic differential operator into 1D ones. Consequently, the method only requires 1D Green's function associated with each coordinate axis. Theoretical formulation and simple axial discretization result in the system of equations of which the solutions represent the numerical solution of the original multi-dimensional elliptic problem, which is otherwise intractable to obtain by the traditional multi-dimensional Green's function technique. The method employs analytical form of 1D Green's function and integral schemes so that it can potentially be more robust than both the finite difference method and the boundary element method. In addition, the simple discretization procedure, which uses only axially orthogonal straight lines and boundary data, demonstrates its convenience over the finite element method of comparable accuracy. Numerical examples verify these advantages by demonstrating the second-order convergence in various discretized models of complex geometries. AXIAL GREEN'S FUNCTION METHOD where the bracket implies [•] t= = lim t→ + •−lim t→ − •. (ii) Continuity: G(t, ) is continuous in I with respect to the variable t.

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“…Numerical analysis of several finite differences (FD) schemes for reaction-diffusion systems is given in the research monograph [1]. Apart from the classical approach, Ramirez et al [2] derived nonstandard FD schemes for steady state reaction-diffusion equations based on Green's integral formulation of the exact solution. In this work, we will derive so-called exponentially fitted finite difference schemes that can be obtained by only changing denominators in discrete derivative formulas.…”

Section: Introductionmentioning

confidence: 99%

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

Erdoğan

Akarbulut

Tan

2016

MCA

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Abstract:The purpose of this work is to introduce a new kind of finite difference formulation inspired from Fourier analysis, for reaction-diffusion equations. Compared to classical schemes, the proposed scheme is much more accurate and has interesting stability properties. Convergence properties and stability of the scheme are discussed. Numerical examples are provided to show better performance of the method, compared with other existing methods in the literature.

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A Green's function formulation for finite-differences schemes

Cited by 15 publications

References 7 publications

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Axial Green's function method for multi‐dimensional elliptic boundary value problems

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

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