Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

Kumar, Devendra; Kumari, Parvin

doi:10.1002/num.22553

Cited by 13 publications

(5 citation statements)

References 24 publications

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“…The solution u(x, t) of Eq. (1.1) is divided as in [15] into layers and regular components as u(x, t) = v(x, t) + w l (x, t) + w r (x, t). The regular component v(x, t) satisfies the following equation:…”

Section: A Priori Derivatives Bounds Of the Solutionmentioning

confidence: 99%

“…M. Chandru et al in [4] considered the problem (1.1) and proved that the upwind scheme on space on Shishkin mesh and backward Euler scheme on time is almost first-order accurate. D. Kumar et al in [15] gave a numerical method with parameter uniform convergence of order two in time and almost order one in space for the problem of the same type. They used the Crank-Nicolson method in time on a uniform mesh and the upwind method in space on a Shishkin mesh.…”

Section: Introductionmentioning

confidence: 99%

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Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term

Roy¹,

Jha²

2022

Preprint

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In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters and µ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss the problem for ratio µ 2. We used an upwind finite difference approach on a Shishkin-Bakhvalov mesh in the space and the Crank-Nicolson method in time on uniform mesh. At the point of discontinuity, a three-point formula was used. This method is uniformly convergent with second order in time and first order in space. Shishkin-Bakhvalov mesh provides first-order convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.

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Section: A Priori Derivatives Bounds Of the Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term

Roy¹,

Jha²

2022

Preprint

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“…where the denominator function is given in (14). Discretizing the boundary conditions by using forward and backward finite difference approximations using the theory of non-standard difference method, we have the discrete initial and boundary conditions as…”

Section: Fully Discrete Problemmentioning

confidence: 99%

Accelerated Non-standard Numerical Methodfor Two-Parameter Singularly Perturbed Parabolic Problems with Robin Boundary Condition

Gelu

Duressa

2022

Preprint

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This study devised a robust second-order numerical scheme tosolve two-parameter singularly perturbed parabolic problems with Robin boundary condition. On uniform meshes, the problem is discretized using the implicit Euler method in the time directionand the non-standard finite difference method in the space direction. Also, Robin boundary condition is discretized using the concept of the non-standard finite difference method. The first-order ε-uniform convergence is investigated in both time and space. Applying the Richardson extrapolation technique, the accuracy and convergence rate of the present method have improved from first-orderto second-order. Both theoretical findings and numerical results areagreed upon based on computations of two numerical examples.

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“…Quadratic B-spline collocation method on exponentially graded mesh for two-parameter singularly perturbed problem is presented by [17]. An implicit computational method on a predefined Shishkin mesh is presented for solving two-parameter parabolic singularly perturbed boundary value problems with nonsmooth data by [18]. [19] suggested parameter uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay

Daba,

Melesse,

Kebede

2023

Computational and Mathematical Methods

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This paper is aimed at constructing and analyzing a fitted approach for singularly perturbed time delay parabolic problems with two small parameters. The proposed computational scheme comprises the implicit Euler and especially finite difference method for the time and space variable discretization, respectively, on uniform step size. The stability and convergence analysis of the method is provided and is first-order parameter uniform convergent. Further, the numerical results depict that the present method is more convergent than some methods available in the literature.

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Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

Cited by 13 publications

References 24 publications

Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term

Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term

Accelerated Non-standard Numerical Methodfor Two-Parameter Singularly Perturbed Parabolic Problems with Robin Boundary Condition

A Fitted Numerical Approach for Singularly Perturbed Two-Parameter Parabolic Problem with Time Delay

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