A generalized 1-dimensional particle transport method for convection diffusion reaction model

James, Makungu; Haario, Heikki; Mahera, William Charles

doi:10.1007/s13370-011-0007-0

Cited by 8 publications

(10 citation statements)

References 12 publications

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“…James and co-workers [12] developed a method based on the unifying Eulerian-Lagrangian particle scheme and operator splitting approach. In [13], a positive preserving explicit scheme is developed for CDR problems with constant advection coefficients and reaction terms that are sum of positive and negative functions of the solution variable; a strategy for the generalization is presented.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Two Finite Dierence Schemes for a Channel Flow Problem

Nwaigwe

Weli

2019

ARJOM

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Aims/ Objectives: To investigate the influence of a model parameter on the convergence of two finite difference schemes designed for a convection-diffusion-reaction equation governing the pressure-driven flow of a Newtonian fluid in a rectangular channel.Methodology: By assuming a uni-directional and incompressible channel flow with an exponentially time-varying suction velocity, we formulate a variable-coefficient convectiondiffusion- reaction problem. In the spirit of the method of manufactured solutions, we first obtain a benchmark analytic solution via perturbation technique. This leads to a modified problem which is exactly satisfied by the benchmark solution. Then, we formulate central and backward difference schemes for the modified problem. Consistency and convergence results are obtained in detail. We show, theoretically, that the central scheme is convergent only for values of a model parameter up to an upper bound, while the backward scheme remains convergent for all values of the parameter. An estimate of this upper bound, as a function of the mesh size, is derived. We then conducted numerical experiments to verify the theoretical results.Results: Numerical results showed that no numerical oscillations were observed for values of the model parameter less than the theoretically derived bound.Conclusion: We therefore conclude that the theoretical bound is a safe value to guarantee non-oscillatory solutions of the central scheme.

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Section: Introductionmentioning

confidence: 99%

Analysis of Two Finite Dierence Schemes for a Channel Flow Problem

Nwaigwe

Weli

2019

ARJOM

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“…In many of these applications, the unknown variables in the governing partial differential equation represent physical quantities that cannot take negative values such as pollutants, population, concentration of chemical compounds [6]. The application of advection-diffusion-reaction (ADR) equation is classified into three processes [9]. The first process is called convection and is due to movement of materials from one region to another.…”

Section: Introductionmentioning

confidence: 99%

“…The last process is called reaction and is due to decay, adsorption and reaction of substances with other components. Qualitatively all the three processes form the ADR model that describe how the disturbed quantity being studied in the medium changes under the influence of these processes [9].…”

Section: Introductionmentioning

confidence: 99%

“…The difficulties in obtaining analytical solution lead many researchers and engineers to resort to numerical methods for approximating the solution to the problem. These numerical methods have been active for about fifty years now and the development of new techniques still attracts more attention [9]. However, there are also some challenges when applying these numerical methods to approximate the solution to problems that are non-linear.…”

Section: Introductionmentioning

confidence: 99%

“…One of the challenges is numerical instabilities [10]. For the ADR model dominated with diffusion, the standard finite difference methods (SFDM) and finite element method (FEM) produce satisfactory results while for problems dominated by convection, numerical instabilities like oscillations and numerical dispersion appear in the solution approximated by these methods [9]. These numerical instabilities can be overcomed for instance by applying the Non-Standard Finite Difference methods (NSFD).…”

Section: Introductionmentioning

confidence: 99%

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Computational Study of Three Numerical Methods for Some Linear and Non-Linear Advection-Diffusion-Reaction Problems

Appadu

Mphephu

2015

PCFD

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In this paper, we use three existing schemes namely, Upwind Forward Euler, Non-Standard Finite Difference (NSFD) and Unconditionally Positive Finite Difference (UPFD) schemes to solve two numerical experiments described by a linear and a non-linear advection-diffusion-reaction equation with constant coefficients. These equations model exponential travelling waves and biofilm growth on a medical implant respectively. We study the exact and numerical dissipative and dispersive properties of the three schemes for both problems. Moreover, L 1 error, dispersion and dissipation errors, at some values of temporal and spatial step sizes have been computed for the three schemes for both problems.

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Canonical Euler splitting method for nonlinear composite stiff evolution equations

2016

Applied Mathematics and Computation

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A generalized 1-dimensional particle transport method for convection diffusion reaction model

Cited by 8 publications

References 12 publications

Analysis of Two Finite Dierence Schemes for a Channel Flow Problem

Analysis of Two Finite Dierence Schemes for a Channel Flow Problem

Computational Study of Three Numerical Methods for Some Linear and Non-Linear Advection-Diffusion-Reaction Problems

Canonical Euler splitting method for nonlinear composite stiff evolution equations

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