A uniformly convergent scheme on a nonuniform mesh for convection–diffusion parabolic problems

Clavero, C.; Jorge, Juan Carlos; Lisbona, F. J.

doi:10.1016/s0377-0427(02)00861-0

Cited by 109 publications

(45 citation statements)

References 10 publications

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“…Assuming certain smoothness and compatibility requirements on s and f , in [18] the authors give the details to obtain these bounds for 0 ≤ j ≤ 3 when the Euler implicit rule is considered. Under the smoothness and compatibility requirements imposed in this paper, the same techniques can be applied to obtain (3.1), (3.2).…”

Section: Asymptotic Behaviour Of the Solution Of Semidiscrete Prmentioning

confidence: 99%

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

Clavero

Gracia

Jorge

2004

Numerical Methods Partial

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In this work we construct and analyse some finite difference schemes used to solve a class of timedependent one-dimensional convection-diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank-Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples according to the theoretical results, in the case of using the Euler implicit method, and a better numerical behaviour than the predicted theoretically, showing order two in time and order N −2 log 2 N in space, if the Crank-Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A-stable SDIRK with two stages and a third order HODIE difference scheme, showing its uniformly convergent behaviour, reaching order three, up to a logarithmic factor. c (Year) John Wiley & Sons, Inc.

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Section: Asymptotic Behaviour Of the Solution Of Semidiscrete Prmentioning

confidence: 99%

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

Clavero

Gracia

Jorge

2004

Numerical Methods Partial

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“…It is necessary to calculate the transport of fluid properties or trace constituent concentrations within a fluid for applications such as water quality modeling, air pollution, meteorology, oceanography and other physical sciences. When velocity field is complex, changing in time and transport process cannot be analytically calculated, and then numerical [1] give a uniform convergent numerical method with respect to the diffusion parameter to solve the one-dimensional time-dependent convection-diffusion problem. The used the implicit Euler method for the time discretization and the simple upwind finite difference scheme on a Shishkin mesh for the spatial discretization in [1].…”

Section: Introductionmentioning

confidence: 99%

“…When velocity field is complex, changing in time and transport process cannot be analytically calculated, and then numerical [1] give a uniform convergent numerical method with respect to the diffusion parameter to solve the one-dimensional time-dependent convection-diffusion problem. The used the implicit Euler method for the time discretization and the simple upwind finite difference scheme on a Shishkin mesh for the spatial discretization in [1]. Ramos presented an exponentially fitted method for singularly perturbed parameter [2].…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving time-dependent convection-diffusion problems

hajaji¹

2017

B Soc Paran Mat

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In this paper, we develop a new numerical method for solving a timedependent convection-diffusion equation with Dirichlet's type boundary conditions. We first propose the θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the back-ward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize the temporal variable, resulting in a linear partial differential equation (PDE). To numerically solve this linear PDE, we develop and we analyze a new cubic spline collocation method for the spatial discretization. To solve the discretized linear system, we design a collocation method and we prove that the method is second order convergent. The computed results are compared wherever possible with those already available in the literature.

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“…Hence, to avoid uniformly fine meshes in many approaches, see e.g. [4,16], the use of a priori refined meshes is suggested to capture the solution behavior in the boundary layers numerically. For steady state problems, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters

2015

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A uniformly convergent scheme on a nonuniform mesh for convection–diffusion parabolic problems

Cited by 109 publications

References 10 publications

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

A numerical method for solving time-dependent convection-diffusion problems

Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters

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