An unconditionally positivity preserving scheme for advection–diffusion reaction equations

Chen-Charpentier, Benito M.; Kojouharov, Hristo V.

doi:10.1016/j.mcm.2011.05.005

Cited by 78 publications

(54 citation statements)

References 10 publications

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“…(17) and (30) with h = 0.1, the relative phase error is given by (5) and (6). We note that the relative phase error for the scheme is not one when ω = 0.…”

Section: Spectral Analysismentioning

confidence: 99%

“…First, we consider the advection-diffusion-reaction equation for modelling the exponential travelling waves [6]. This equation is given by…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We note that in [6], they have used k = 0.26 when h is chosen as 0.01. However, from our stability analysis, we observe that the Upwind Forward Euler and NSFD schemes are not stable for k > 0.25 and k > 0.253 respectively.…”

Section: Stabilitymentioning

confidence: 99%

“…These equations are often used to predict and control the extent of contamination in ground water systems [7]. 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].…”

Section: Introductionmentioning

confidence: 99%

“…The novelty in this work in regard to work in [6] is that the stability of the schemes have been obtained and hence it allows a more effective way of comparing the performance of the three schemes. Also, dissipation and dispersion properties of the schemes are studied.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems

Appadu

Lubuma

Mphephu

2017

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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“…(17) and (30) with h = 0.1, the relative phase error is given by (5) and (6). We note that the relative phase error for the scheme is not one when ω = 0.…”

Section: Spectral Analysismentioning

confidence: 99%

“…First, we consider the advection-diffusion-reaction equation for modelling the exponential travelling waves [6]. This equation is given by…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems

Appadu

Lubuma

Mphephu

2017

PCFD

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A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

Kovács

Nagy

Saleh

2022

Advcd Theory and Sims

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This paper reports on a novel explicit numerical method for the spatially discretized diffusion or heat equation. After discretizing the space variables as in conventional finite difference methods, this method does not use a finite difference approximation for the time derivatives, it instead combines constant-neighbor and linear-neighbor approximations, which decouple the ordinary differential equations, thus they can be solved analytically. In the obtained three-stage method, the time step size appears in exponential form with negative coefficients in the final expression. This property guarantees unconditional stability, as it is shown using von Neumann stability analysis. It is also proved that the convergence of the method is third order in the time step size. After verification, by solving Fisher's and Huxley's equations, it is demonstrated that it works for nonlinear equations as well. The new algorithm is tested against widely used numerical solvers for cases where the media is strongly inhomogeneous. According to the results, the new method is significantly more effective than the traditional explicit or implicit methods, especially for extremely large stiff systems. It is believed that this new method is unique in the sense that it is the first unconditionally stable explicit method with third-order convergence.

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A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

Qin

Ding

2014

Math Methods in App Sciences

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In this note, a non-standard finite difference (NSFD) scheme is proposed for an advection-diffusion-reaction equation with nonlinear reaction term. We first study the diffusion-free case of this equation, that is, an advection-reaction equation. Two exact finite difference schemes are constructed for the advection-reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection-reaction equation is combined with a finite difference space-approximation of the diffusion term to provide a NSFD scheme for the advection-diffusion-reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. f inite difference scheme; non-standard f inite difference scheme; positivity; boundedness As the exact solution of Equation (2) difficult to obtain, numerical scheme that provides a discretization approximation of the exact solution is widely used [1]. For a good numerical scheme, the numerical solution is qualitatively the same as the exact solution, such as the number and stability of the fixed points, the positivity, the boundedness, and the monotonicity of the exact solution. However, traditional numerical schemes are often unable to achieve this, while non-standard finite difference (NSFD) scheme by Mickens et

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An unconditionally positivity preserving scheme for advection–diffusion reaction equations

Cited by 78 publications

References 10 publications

Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems

Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems

A New Stable, Explicit, Third‐Order Method for Diffusion‐Type Problems

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

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