A fourth order difference method for the one‐dimensional general quasilinear parabolic partial differential equation

Jain, M. K.; Jain, Rakesh K.; Mohanty, R. K.

doi:10.1002/num.1690060403

Cited by 43 publications

(37 citation statements)

References 17 publications

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“…The ENSD due to Chawla, 14,15 Jain et al, 16 and Mohanty et al 17 can be, in turn, represented by the formula:…”

Section: The Csd and The Ensdmentioning

confidence: 99%

“…The ENSD suggested by Chawla 14,15 for the solution of boundary value problems, and later used by Jain et al 16 and Mohanty et al 17 for the solution of parabolic initial boundary value problems, appears convenient and computationally economical. In addition, this ENSD is argued 16 to produce oscillation-free solutions for all values of the cell Reynolds number, in the case of constant coefficient convection-diffusion equations, which promises a safety of use even under the extreme conditions of convection dominated transport. The examination of the utility of this ENSD for the solution of example models of electrochemical kinetics is the goal of the present study.…”

Section: Introductionmentioning

confidence: 98%

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Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

Bieniasz

2004

J Comput Chem

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The fourth-order accurate, three-point finite-difference Numerov spatial discretization provides accurate and efficient solutions to the time-dependent governing differential equations of electrochemical kinetics in one-dimensional space geometry, when the equations contain first time derivatives of the solution, second spatial derivatives, and homogeneous reaction terms only. However, the original Numerov discretization is not applicable when the governing equations involve first spatial derivative terms. To overcome this limitation, an appropriately extended Numerov discretization is required. We examine the utility of one of such extensions, first described by Chawla. Relevant discrete formulae are outlined for systems of linear governing equations involving first derivative terms, and applied to five representative example models of electrochemical transient experiments. The extended Numerov discretization proves to have an accuracy and efficiency comparable to the original Numerov scheme, and its accuracy is typically up to four orders of magnitude higher, compared to the conventional, second-order accurate spatial discretization, commonly used in electrochemistry. This results in a considerable improvement of efficiency. Therefore, the application of the extended Numerov discretization to the electrochemical kinetic simulations can be fully recommended.

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“…The ENSD due to Chawla, 14,15 Jain et al, 16 and Mohanty et al 17 can be, in turn, represented by the formula:…”

Section: The Csd and The Ensdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

Bieniasz

2004

J Comput Chem

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“…(1.1) has been given recently by Jain et al [1] and this scheme can be written in the form It is thus seen that the above scheme is equivalent to a central difference scheme applied to the equation…”

Section: Introductionmentioning

confidence: 96%

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

Bhuruth¹

1998

Numer. Methods Partial Differential Eq.

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The solution of the linear system Ax = b by iterative methods requires a splitting of the coefficient matrix in the form A = M − N where M is usually chosen to be a diagonal or a triangular matrix. In this article we study relaxation methods induced by the Hermitian and skew-Hermitian splittings for the solution of the linear system arising from a compact fourth order approximation to the one dimensional convectiondiffusion equation and compare the convergence rates of these relaxation methods to that of the widely used successive overrelaxation (SOR) method. Optimal convergence parameters are derived for each method and numerical experiments are given to supplement the theoretical estimates. For certain values of the diffusion parameter, a relaxation method based on the Hermitian splitting converges faster than SOR. For two-dimensional problems a block form of the iterative algorithm is presented.

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“…The fundamental properties, stability, and consistency were precisely analyzed in these articles so as to obtain accurate numerical results. The methods described in [1][2][3][4] are of high orders; however, those methods are not directly applicable to solve singular parabolic equations. A special technique is required to handle the difference scheme without losing the accuracy of the scheme.…”

Section: Introductionmentioning

confidence: 98%

“…Our previous work [1][2][3][4] was concerned with two-level implicit finite difference schemes of order 2 in time and 4 in space for the nonlinear parabolic partial differential equation:…”

Section: Introductionmentioning

confidence: 99%

An O(k² + kh² + h⁴) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

Mohanty

Karaa

Arora

2006

Numerical Methods Partial

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This article develops a new two-level three-point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation u xx ϭ f (x, t, u, u x , u t ), 0 Ͻ x Ͻ 1, t Ͼ 0 subject to appropriate initial and Dirichlet boundary conditions, where Ͼ 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (Ѩu/Ѩx). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas.

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A fourth order difference method for the one‐dimensional general quasilinear parabolic partial differential equation

Cited by 43 publications

References 17 publications

Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

An O(k² + kh² + h⁴) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

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A fourth order difference method for the one‐dimensional general quasilinear parabolic partial differential equation

Cited by 43 publications

References 17 publications

Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

Improving the accuracy of the spatial discretization in finite‐difference electrochemical kinetic simulations, by means of the extended Numerov method

A note on Hermitian splitting induced relaxation methods for convection-diffusion equations

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

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An O(k² + kh² + h⁴) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations