A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems

Mohanty, R. K.; Singh, Swarn

doi:10.1016/j.amc.2005.08.023

Cited by 45 publications

(20 citation statements)

References 13 publications

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“…For the derivation of the new method, we simply follow the techniques given by Mohanty and Singh [11] and Chawla and Shivakumar [1].…”

Section: Derivation Of the Methodsmentioning

confidence: 99%

“…We do not need more than 9-spatial grid points to discretize the differential equation (1). Recently, Mohanty and Singh [10,11] have proposed new high accuracy arithmetic average discretizations for singularly perturbed 1-D parabolic and 2-D elliptic non-linear partial differential equations. In next two sections, we give mathematical details of the methods.…”

Section: Introductionmentioning

confidence: 99%

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A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
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We propose a new two-level implicit difference method of O(k 2 + kh 2 + h 4 ) for the solution of singularly perturbed non-linear parabolic differential equation (u xx +u yy )=f (x, y, t, u, u x , u y , u t ), 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are grid sizes in time and space directions, respectively, and > 0 is a small parameter. We also develop new methods of O(kh 2 + h 4 ) for the estimates of (ju/jx) and (ju/jy). In all cases, we use 9-spatial grid points and a single computational cell. The proposed methods are directly applicable to singular problems. We do not require any special scheme to solve singular problems. We also discuss alternating direction implicit (ADI) method for solving diffusion equation in polar cylindrical coordinates. This method permits multiple use of the one-dimensional tri-diagonal algorithm with a considerable saving in computing time, and produces a very efficient solver. It is shown that the ADI method is unconditionally stable. Numerical experiments are conducted to test the high accuracy of the proposed methods and compared with the exact solutions.

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“…For the derivation of the new method, we simply follow the techniques given by Mohanty and Singh [11] and Chawla and Shivakumar [1].…”

Section: Derivation Of the Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
Self Cite
11
0
0
0
View full text Add to dashboard Cite

show abstract

“…SCMBT can be easily used to the Sinccollocation patching domain decomposition methods. (26) The numerical solution of two-dimensional singularly perturbed non-linear elliptic partial differential equation is taken from Mohanty and Singh [72] is given by…”

Section: ð20:2þmentioning

confidence: 99%

A collection of computational techniques for solving singular boundary-value problems

Kumar

Singh

2009

Advances in Engineering Software

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“…The proposed formulas are directly applicable to both singular and nonsingular parabolic problems, which is the main attraction of this piece of work, and we do not require any fictitious points to avoid difficulties near the boundaries. Recently, Mohanty and Singh [5] have developed a 9-point fourth-order arithmetic average discretization for the solution of nonlinear elliptic partial differential equations on a uniform mesh. Bieniasz [6] has discussed high-accuracy finite difference electrochemical kinetic simulations by means of the extended Numerov method.…”

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 new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems

Cited by 45 publications

References 13 publications

A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
Self Cite
11
0
0
0
View full text Add to dashboard Cite

A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
Self Cite
11
0
0
0
View full text Add to dashboard Cite

A collection of computational techniques for solving singular boundary-value problems

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

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A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems

Cited by 45 publications

References 13 publications

A new two-level implicit discretization of O(k2+kh2+hMohanty1, Singh2 2007Journal of Computational and Applied MathematicsSelf Cite11000View full textAdd to dashboardCite

A new two-level implicit discretization of O(k2+kh2+hMohanty1, Singh2 2007Journal of Computational and Applied MathematicsSelf Cite11000View full textAdd to dashboardCite

A collection of computational techniques for solving singular boundary-value problems

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

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Resources

About

A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
Self Cite
11
0
0
0
View full text Add to dashboard Cite

A new two-level implicit discretization of O(k2+kh2+h
Mohanty
¹
,
Singh
²

2007
Journal of Computational and Applied Mathematics
Self Cite
11
0
0
0
View full text Add to dashboard Cite

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