Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations

Kaya, Adem

doi:10.1016/j.jcp.2015.01.024

Cited by 32 publications

(20 citation statements)

References 20 publications

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“…We also note that, in 3-D domains, 124 sub-grid nodes are added into the initial stencil to derive the augmented discretization and the coordinates of the sub-grid nodes are determined by applying the same procedure as we did in the 2-D case (see [26] for details).…”

Section: Remarkmentioning

confidence: 99%

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Kaya

Sendur

2015

Journal of Computational Physics

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A numerical scheme for the convection-diffusion-reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind PetrovGalerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.

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Section: Remarkmentioning

confidence: 99%

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Kaya

Sendur

2015

Journal of Computational Physics

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“…These two generalized HOC ADI schemes are second order accurate in time, fourth order accurate in space and unconditionally stable. More recently, finite difference method is also employed by some authors to solve multidimensional unsteady convection diffusion reaction equations [35,36].…”

Section: Introductionmentioning

confidence: 99%

A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

Ge¹

2018

ACM

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In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa's HOC ADI method [26], Cao and Ge's HOC ADI method [27], and our previous exponential HOC ADI method [28].

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“…Oliveira presents a particle‐based Lagrangian‐Eulerian FEM algorithm for the unsteady ADR problems considering the phase change . Kaya proposes a finite difference scheme for multidimensional steady and unsteady convection‐diffusion‐reaction problems . Gavete considers a finite‐volume FDM for ADR problems .…”

Section: Introductionmentioning

confidence: 99%

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

Reutskiy

Lin

2017

Numerical Meth Engineering

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The aim of this paper is to present a new semi-analytic numerical method for strongly nonlinear steady-state advection-diffusion-reaction equation (ADRE) in arbitrary 2-D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.

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Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations

Cited by 32 publications

References 20 publications

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

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