Accurate discretization of advection-diffusion equations

Grima, Ramon; Newman, T. J.

doi:10.1103/physreve.70.036703

Cited by 26 publications

(19 citation statements)

References 23 publications

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“…Diffusion in a potential field obeys the Nernst-Einstein equation [14], and the resulting advection-diffusion equations for the ad-particle concentration show, in general, both diffusion and drift [10]. We show that previous results in the 2D nucleation and growth literature [6,7] correspond to this type of equation and solutions.…”

Section: Diffusion In 2d Potential Fields: Analytic Formulationmentioning

confidence: 66%

“…for Ge/Si(001). An intriguing point is that Ovesson [6] and Grima and Newman [10] have studied the same problem from opposite ends. Grima and Newman started from the general continuum equation (11), and arrived at their MED as a very efficient numerical method of solving this whole class of problems.…”

Section: Capture Numbers In Radial and Rectangular Geometrymentioning

confidence: 99%

“…The velocity field can be specialized to be proportional to a potential gradient as v(r) = −α∇V(r) [15]. With this constraint, the last term of equation (11) The main algorithm proposed in [10] …”

Section: Discretization Of Advection-diffusion Equationsmentioning

confidence: 99%

“…The second aim is to illustrate numerical time-dependent solutions for the diffusion field in both radial and rectangular geometry, the latter case with anisotropic diffusion coefficients [9]. We adapt a new algorithm [10] to this rectangular case, and show how the processes taking place at the cluster edges influence the simulated growth of 2D quasi-rectangular clusters arranged on a lattice.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion in Potential Fields: Time-Dependent Capture on Radial and Rectangular Substrates

Venables

Yang

2004

MRS Proc.

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In rate equation models of nucleation and growth on surfaces, and it has often been assumed that the energy surface of the substrate is flat, that diffusion is isotropic, and that capture numbers can be calculated in the diffusion-controlled limit. We lift these restrictions analytically, and illustrate the results using a hybrid discrete FFT method of solving for the 2D timedependent diffusion field of ad-particles, which has been implemented in Matlab ® 6.5. A general substrate energy surface is included by transformation of the field. The method can work with any boundary conditions, but is particularly clear for periodic boundary conditions, such as might be appropriate following nucleation on a regular (rectangular) array of defects. The method is instructive for visualizing potential and diffusion fields, and for demonstrating the time-dependence of capture numbers in the initial stages of deposition and annealing.

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Section: Diffusion In 2d Potential Fields: Analytic Formulationmentioning

confidence: 66%

Section: Capture Numbers In Radial and Rectangular Geometrymentioning

confidence: 99%

Section: Discretization Of Advection-diffusion Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Diffusion in Potential Fields: Time-Dependent Capture on Radial and Rectangular Substrates

Venables

Yang

2004

MRS Proc.

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“…To integrate numerically the advection-diffusion equation (Eq. [3]) a modified version proposed by Grima and Newman (2004) was used. The first derivative (both temporal and spatial) was approximated as a simple forward finite difference, whereas a central differences scheme was used for the higher order derivatives.…”

Section: Deterministic Climate Modelmentioning

confidence: 99%

Analysis of the simulated global temperature using a simple energy balance stochastic model

Moreles

Martínez‐López

2016

ATM

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Este trabajo presenta un estudio de la respuesta de la variabilidad simulada de la temperatura global a parametrizaciones estocásticas aditivas y multiplicativas de flujos de calor, junto con una descripción de la variabilidad de largo plazo en términos de procesos autorregresivos simples. Para simular la temperatura global de la Tierra se utilizó un modelo climático de balance de energía promediado globalmente, acoplado a un modelo oceánico termodinámico. Se encontró que procesos autorregresivos simples explican la variabilidad de la temperatura en el caso de parametrizaciones aditivas; sin embargo, en el caso de parametrizaciones multiplicativas, la descripción de la variabilidad de la temperatura involucraría procesos autorregresivos de orden superior, lo cual sugiere la presencia de mecanismos complejos de retroefecto originados por el forzamiento multiplicativo. Asimismo, se encontró que las parametrizaciones multiplicativas produjeron una estructura compleja que emula de manera cercana procesos climáticos observados. Finalmente, se propone un nuevo enfoque para describir la estabilidad de un sistema estocástico general unidimensional en estado estacionario, a través de su función potencial. A partir de una expresión analítica de la función potencial se profundizó en la descripción de un sistema estocástico. ABSTRACTThis work presents a study of the response of the simulated global temperature variability to additive and multiplicative stochastic parameterizations of heat fluxes, along with a description of the long-term variability in terms of simple autoregressive processes. The Earth's global temperature was simulated using a globally averaged energy balance climate model coupled to a thermodynamic ocean model. It was found that simple autoregressive processes explain the temperature variability in the case of additive parameterizations; whereas in the case of multiplicative parameterizations, the description of the temperature variability would involve higher order autoregressive processes, suggesting the presence of complex feedback mechanisms originated by the multiplicative forcing. Also, it was found that multiplicative parameterizations produced a rich structure that emulates closely observed climate processes. Finally, a new approach to describe the stability in the steady state of a general one-dimensional stochastic system, through its potential function, was proposed. From an analytical expression of the potential function, further insight into the description of a stochastic system was provided.

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A stochastic model of corneal epithelium maintenance and recovery following perturbation

2018

Self Cite

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Various biological studies suggest that the corneal epithelium is maintained by active stem cells located in the limbus, the so-called Limbal Epithelial Stem Cell (LESC) hypothesis. While numerous mathematical models have been developed to describe corneal epithelium wound healing, only a few have explored the process of corneal epithelium homeostasis. In this paper we present a purposefully simple stochastic mathematical model based on a chemical master equation approach, with the aim of clarifying the main factors involved in the maintenance process. Model analysis provides a set of constraints on the numbers of stem cells, division rates, and the number of division cycles required to maintain a healthy corneal epithelium. In addition, our stochastic analysis reveals noise reduction as the epithelium approaches its homeostatic state, indicating robustness to noise. Finally, recovery is analysed in the context of perturbation scenarios.

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Accurate discretization of advection-diffusion equations

Cited by 26 publications

References 23 publications

Diffusion in Potential Fields: Time-Dependent Capture on Radial and Rectangular Substrates

Diffusion in Potential Fields: Time-Dependent Capture on Radial and Rectangular Substrates

Analysis of the simulated global temperature using a simple energy balance stochastic model

A stochastic model of corneal epithelium maintenance and recovery following perturbation

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