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Journal articles on the topic 'Numerical Diffusion'

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Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

Jaichuang, Atit, and Wirawan Chinviriyasit. "Numerical Modelling of Influenza Model with Diffusion." International Journal of Applied Physics and Mathematics 4, no. 1 (2014): 15–21. http://dx.doi.org/10.7763/ijapm.2014.v4.247.

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2

LACHTIOUI, Y., M. MAZROUI, and Y. BOUGHALEB. "COMMENSURABILITY EFFECTS ON DIFFUSION PROCESS IN STEPPED STRUCTURES." Modern Physics Letters B 25, no. 21 (August 20, 2011): 1749–60. http://dx.doi.org/10.1142/s0217984911027005.

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This study deals with the investigation of diffusion process of one-dimensional system with steps for adsorbates interacting via the nearest-neighbor harmonic forces. The results are obtained from numerical studies, utilizing the method of stochastic Langevin dynamics. To study commensurability effects and the role of steps in the behavior of the diffusing particles, we have computed the diffusion coefficient for large concentrations and several interaction strengths. Our numerical results show that the diffusive behavior is reduced for commensurate structure case when the ground state has only one particle per one period of the substrate potential and enhanced for incommensurate density. Furthermore, the dynamic is qualitatively similar to that obtained in the case of no steps but with a clear reduction of the diffusion rate. Implications of these findings are discussed.

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3

Raymond, William H. "Diffusion and Numerical Filters." Monthly Weather Review 122, no. 4 (April 1994): 757–61. http://dx.doi.org/10.1175/1520-0493(1994)122<0757:danf>2.0.co;2.

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4

D'Isidoro, M., A. Maurizi, and F. Tampieri. "Effects of resolution on the relative importance of numerical and physical horizontal diffusion in atmospheric composition modelling." Atmospheric Chemistry and Physics 10, no. 6 (March 24, 2010): 2737–43. http://dx.doi.org/10.5194/acp-10-2737-2010.

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Abstract. Numerical diffusion induced by advection has a large influence on concentration of substances in atmospheric composition models. At coarse resolution numerical effects dominate, whereas at increasing model resolution a description of physical diffusion is needed. A method to investigate the effects of changing resolution and Courant number is defined here and is applied to the WAF advection scheme (used in BOLCHEM), evidencing a sub-diffusive process. The spread rate from an instantaneous source caused by numerical diffusion is compared to that produced by the physical diffusion necessary to simulate unresolved turbulent motions. The time at which the physical diffusion process overpowers the numerical spread is estimated, and it is shown to reduce as the resolution increases, and to increase with wind velocity.

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5

Bengtsson, Lisa, Sander Tijm, Filip Váňa, and Gunilla Svensson. "Impact of Flow-Dependent Horizontal Diffusion on Resolved Convection in AROME." Journal of Applied Meteorology and Climatology 51, no. 1 (January 2012): 54–67. http://dx.doi.org/10.1175/jamc-d-11-032.1.

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AbstractHorizontal diffusion in numerical weather prediction models is, in general, applied to reduce numerical noise at the smallest atmospheric scales. In convection-permitting models, with horizontal grid spacing on the order of 1–3 km, horizontal diffusion can improve the model skill of physical parameters such as convective precipitation. For instance, studies using the convection-permitting Applications of Research to Operations at Mesoscale model (AROME) have shown an improvement in forecasts of large precipitation amounts when horizontal diffusion is applied to falling hydrometeors. The nonphysical nature of such a procedure is undesirable, however. Within the current AROME, horizontal diffusion is imposed using linear spectral horizontal diffusion on dynamical model fields. This spectral diffusion is complemented by nonlinear, flow-dependent, horizontal diffusion applied on turbulent kinetic energy, cloud water, cloud ice, rain, snow, and graupel. In this study, nonlinear flow-dependent diffusion is applied to the dynamical model fields rather than diffusing the already predicted falling hydrometeors. In particular, the characteristics of deep convection are investigated. Results indicate that, for the same amount of diffusive damping, the maximum convective updrafts remain strong for both the current and proposed methods of horizontal diffusion. Diffusing the falling hydrometeors is necessary to see a reduction in rain intensity, but a more physically justified solution can be obtained by increasing the amount of damping on the smallest atmospheric scales using the nonlinear, flow-dependent, diffusion scheme. In doing so, a reduction in vertical velocity was found, resulting in a reduction in maximum rain intensity.

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6

GEORGE, E., J. GLIMM, X. L. LI, A. MARCHESE, Z. L. XU, J. W. GROVE, and DAVID H. SHARP. "Numerical methods for the determination of mixing." Laser and Particle Beams 21, no. 3 (July 2003): 437–42. http://dx.doi.org/10.1017/s0263034603213239.

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We present a Rayleigh–Taylor mixing rate simulation with an acceleration rate falling within the range of experiments. The simulation uses front tracking to prevent interfacial mass diffusion. We present evidence to support the assertion that the lower acceleration rate found in untracked simulations is caused, at least to a large extent, by a reduced buoyancy force due to numerical interfacial mass diffusion. Quantitative evidence includes results from a time-dependent Atwood number analysis of the diffusive simulation, which yields a renormalized mixing rate coefficient for the diffusive simulation in agreement with experiment. We also present the study of Richtmyer–Meshkov mixing in cylindrical geometry using the front tracking method and compare it with the experimental results.

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7

Abrashina-Zhadaeva, Natali. "A SPLITTING TYPE ALGORITHM FOR NUMERICAL SOLUTION OF PDES OF FRACTIONAL ORDER." Mathematical Modelling and Analysis 12, no. 4 (December 31, 2007): 399–408. http://dx.doi.org/10.3846/1392-6292.2007.12.399-408.

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Fractional order diffusion equations are generalizations of classical diffusion equations, treating super‐diffusive flow processes. In this paper, we examine a splitting type numerical methods to solve a class of two‐dimensional initial‐boundary value fractional diffusive equations. Stability, consistency and convergence of the methods are investigated. It is shown that both schemes are unconditionally stable. A numerical example is presented.

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8

D'Isidoro, M., A. Maurizi, and F. Tampieri. "Effects of resolution on the relative importance of numerical and physical diffusion in atmospheric composition modelling." Atmospheric Chemistry and Physics Discussions 9, no. 5 (October 28, 2009): 22865–81. http://dx.doi.org/10.5194/acpd-9-22865-2009.

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Abstract. Numerical diffusion induced by advection has a large influence on concentration of substances in atmospheric composition models. At coarse resolutions numerical effects dominate, whereas at increasing model resolutions a description of physical diffusion is needed. The effects of changing resolution and Courant number are investigated for the WAF advection scheme (used in BOLCHEM), evidencing a sub-diffusive process. The spreading rate from an instantaneous source is compared with the physical diffusion necessary to simulate unresolved turbulent motions. The time at which the physical diffusion process overpowers the numerical spreading is estimated, and is shown to reduce as the resolution increases, and to increase with wind velocity.

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9

Hwang, Sooncheol, and Sangyoung Son. "Development of an Advection-diffusion Model Using Depth-integrated Equations Based on GPU Acceleration." Journal of the Korean Society of Hazard Mitigation 21, no. 1 (February 28, 2021): 281–89. http://dx.doi.org/10.9798/kosham.2021.21.1.281.

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A scalar transport model is developed by adding a depth-averaged advection-diffusion equation to Celeris Advent, which is a Boussinesq-type numerical model that utilizes GPU acceleration. The hybrid finite volume-finite difference method is used to guarantee numerical stability along with the high accuracy of the Boussinesq equation. The advective and diffusive terms are numerically discretized using the finite volume and finite difference methods, respectively. Results of a one-dimensional scalar advection benchmark test showed that the scalar advection by the proposed model was very close to the analytical solution without any remarkable numerical diffusion. In addition, two benchmark tests using experimental data from different hydraulic experiments were numerically reproduced, and the computed results and observed data for scalar transport were found to be in good agreement. The developed model is expected to contribute to real-time disaster prediction for contaminant spills and can assist in preparing countermeasures for these types of disasters.

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10

Somjaivang, Dussadee, and Settapat Chinviriyasit. "Numerical Modeling of an Influenza Epidemic Model with Vaccination and Diffusion." International Journal of Applied Physics and Mathematics 4, no. 1 (2014): 68–74. http://dx.doi.org/10.7763/ijapm.2014.v4.257.

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11

Bonito, Andrea, Juan Pablo Borthagaray, Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado. "Numerical methods for fractional diffusion." Computing and Visualization in Science 19, no. 5-6 (March 7, 2018): 19–46. http://dx.doi.org/10.1007/s00791-018-0289-y.

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12

Khujadze, George, and Martin Oberlack. "Turbulent diffusion: Direct numerical simulation." PAMM 9, no. 1 (December 2009): 451–52. http://dx.doi.org/10.1002/pamm.200910198.

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13

Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (March 2, 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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14

Cai, Yongli, Shuling Yan, Hailing Wang, Xinze Lian, and Weiming Wang. "Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission." International Journal of Bifurcation and Chaos 25, no. 08 (July 2015): 1550099. http://dx.doi.org/10.1142/s0218127415500996.

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In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

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15

Le Roux, Marie-Noëlle. "Numerical solution of fast diffusion or slow diffusion equations." Journal of Computational and Applied Mathematics 97, no. 1-2 (September 1998): 121–36. http://dx.doi.org/10.1016/s0377-0427(98)00106-x.

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16

TSENG, HSEN-CHE, and HUNG-JUNG CHEN. "THE EFFECTS OF QUENCHED DISORDER ON THE CHAOTIC DIFFUSION FOR SIMPLE MAPS." International Journal of Modern Physics B 13, no. 01 (January 10, 1999): 83–95. http://dx.doi.org/10.1142/s0217979299000072.

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That both normal and anomalous chaotic diffusions are suppressed by the presence of quenched disorder for a large class of maps was established by G. Radons.1 In this paper, we consider simple maps (which exhibit normal diffusion) modified by discrete disorder. By decomposing the mean square displacement (MSD) σ2(t) of the system into three terms, namely, [Formula: see text], we find that the MSD of the random walk which corresponds to disorder, [Formula: see text], enhances that of the original unmodified map, [Formula: see text] and that the term 2σ01(t), which describes the correlation between the diffusion fronts of the previous two diffusive processes, just essentially cancels the sum of [Formula: see text] and [Formula: see text]. In consequence, the trajectories of the system are effectively localized. In this formalism, exact numerical calculations without any round-off error can be achieved, the numerical errors coming only from the limited sampling of the initial conditions.

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17

Sengupta, Tapan K., and Ashish Bhole. "Error dynamics of diffusion equation: Effects of numerical diffusion and dispersive diffusion." Journal of Computational Physics 266 (June 2014): 240–51. http://dx.doi.org/10.1016/j.jcp.2014.02.021.

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18

Jaimes, Manuel, and Roel Snieder. "Illustration of diffusion and equipartitioning as local processes: A numerical study using the scalar radiative transfer equation." Journal of the Acoustical Society of America 153, no. 4 (April 2023): 2148–64. http://dx.doi.org/10.1121/10.0017805.

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We study the transition from ballistic to diffusive to equipartitioned waves in scattering media using the acoustic radiative transfer equation. To solve this equation, we first transform it into an integral equation for the specific intensity and then construct a time stepping algorithm with which we evolve the specific intensity numerically in time. We handle the advection of energy analytically at the computational grid points and use numerical interpolation to deal with advection terms that do not lie on the grid points. This approach allows us to reduce the numerical dispersion, compared to standard numerical techniques. With this algorithm, we are able to model various initial conditions for the intensity field, non-isotropic scattering, and uniform scatterer density. We test this algorithm for an isotropic initial condition, isotropic scattering, and uniform scattering density, and find good agreement with analytical solutions. We compare our numerical solutions to known two-dimensional diffusion approximations and find good agreement. We use this algorithm to numerically investigate the transition from ballistic to diffusive to equipartitioned wave propagation over space and time, for two different initial conditions. The first one corresponds to an isotropic Gaussian distribution in space and the second one to a plane wave segment. We find that diffusion and equipartitioning must be treated as local rather than global concepts. This local behavior of equipartitioning has implications for Green's functions reconstruction, which is of interest in acoustics and seismology.

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19

Araújo, Adérito, Amal K. Das, Cidália Neves, and Ercília Sousa. "Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential." Communications in Computational Physics 13, no. 2 (February 2013): 502–25. http://dx.doi.org/10.4208/cicp.280711.010312a.

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AbstractNumerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.

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20

MAEKAWA, Isamu. "Numerical diffusion in multi-dimensional thermal-hydraulic analysis, (I) Numerical diffusion and upwind differencing scheme." Journal of the Atomic Energy Society of Japan / Atomic Energy Society of Japan 28, no. 5 (1986): 444–54. http://dx.doi.org/10.3327/jaesj.28.444.

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21

Hasal, Pavel, and Vladimír Kudrna. "Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process." Collection of Czechoslovak Chemical Communications 61, no. 4 (1996): 512–35. http://dx.doi.org/10.1135/cccc19960512.

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Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.

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22

Zhussanbayeva, A. K., V. Mukamedenkyzy, V. N. Kossov, and A. A. Akzholova. "Numerical research of characteristic mixing times of isothermal three-component steam-gas systems." Bulletin of the Karaganda University. "Physics" Series 106, no. 2 (June 30, 2022): 133–40. http://dx.doi.org/10.31489/2022ph1/133-140.

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Multicomponent diffusion in gases is characterized by a number of effects that are not observed in binary diffusion. Analysis of existing works shows that convective instability may occur in some systems with significantly different diffusion coefficients with certain geometric and thermophysical characteristics. Stability analysis allows determining the spectrum of parameters at which a transition from a diffusive state to a convective is possible. However, this approach does not allow the researchers to investigate the dynamics of the process. Therefore, this work aims to describe emergence and evolution of convective flows in threecomponent systems and assess the influence of the initial composition on the occurrence of concentration gravitational convection. The main part of the work presents a mathematical model describing the occurrence of convective flows based on the splitting scheme according to physical parameters. Numerical data on the concentration fields of the gas with the highest molecular weight at various time points is obtained. It is established that curvature of the isoconcentration lines of the diffusing components can be associated with instability of the mechanical equilibrium of the system. Degree of curvature is determined by the initial concentration of components of the mixture. The obtained data can be used to determine the main characteristics of mass transfer used in calculations related to combined heat and mass transfer in a wide range of thermophysical parameters.

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23

Somathilake, L. W. "Numerical solutions of reaction-diffusion systems with coupled diffusion terms." Ruhuna Journal of Science 4 (September 28, 2009): 1. http://dx.doi.org/10.4038/rjs.v4i0.55.

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24

Zhang, Guoyan, Shengyong Liu, Jie Lu, Jiong Wang, and Yongtao Ma. "Numerical Simulation of Diffusion Absorption Refrigerator." E3S Web of Conferences 233 (2021): 01044. http://dx.doi.org/10.1051/e3sconf/202123301044.

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Based on Fluent software, a mathematical model of thermosyphon pump is established and numerical simulation is carried out to study the influence of riser tube length, tube diameter and immersion ratio on liquid lifting capacity and efficiency. The results showed that: the liquid lifting volume increased with the increase of immersion ratio, whereas the lifting efficiency showed a trend of increasing followed by decreasing. The highest lifting efficiency for a 340mm long, 6mm diameter riser achieved when the immersion ratio is 0.35. With the increasing of the height in riser, the velocity of the gas phase close to the wall in the thermosyphon pump was higher than the velocity along the radial direction. In order to enhance fluid interchange, corners of the refrigeration box were designed to be arc-shaped with a higher corner speed and lower temperature.

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25

Wu, S. H., and L. D. Chen. "Numerical simulation of wick diffusion flames." Journal of Propulsion and Power 8, no. 5 (September 1992): 921–26. http://dx.doi.org/10.2514/3.23573.

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26

Dhawan, S., and S. Kapoor. "Numerical simulation of advection-diffusion equation." International Journal of Mathematical Modelling and Numerical Optimisation 2, no. 1 (2011): 13. http://dx.doi.org/10.1504/ijmmno.2011.037197.

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27

Dutykh, Denys. "Numerical Simulation of Feller’s Diffusion Equation." Mathematics 7, no. 11 (November 6, 2019): 1067. http://dx.doi.org/10.3390/math7111067.

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This article is devoted to Feller’s diffusion equation, which arises naturally in probability and physics (e.g., wave turbulence theory). If discretized naively, this equation may represent serious numerical difficulties since the diffusion coefficient is practically unbounded and most of its solutions are weakly divergent at the origin. In order to overcome these difficulties, we reformulate this equation using some ideas from the Lagrangian fluid mechanics. This allows us to obtain a numerical scheme with a rather generous stability condition. Finally, the algorithm admits an elegant implementation, and the corresponding Matlab code is provided with this article under an open source license.

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28

Tian, X., and A. J. Strojwas. "Numerical integral method for diffusion modeling." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 10, no. 9 (1991): 1110–24. http://dx.doi.org/10.1109/43.85757.

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29

Malik, Nadeem A. "Turbulent particle pair diffusion: Numerical simulations." PLOS ONE 14, no. 5 (May 20, 2019): e0216207. http://dx.doi.org/10.1371/journal.pone.0216207.

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30

Castiglione, P., A. Mazzino, and P. Muratore-Ginanneschi. "Numerical study of strong anomalous diffusion." Physica A: Statistical Mechanics and its Applications 280, no. 1-2 (May 2000): 60–68. http://dx.doi.org/10.1016/s0378-4371(99)00618-4.

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31

Hellman, Fredrik, Tim Keil, and Axel Målqvist. "Numerical Upscaling of Perturbed Diffusion Problems." SIAM Journal on Scientific Computing 42, no. 4 (January 2020): A2014—A2036. http://dx.doi.org/10.1137/19m1278211.

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32

Tóth, Gábor, Xing Meng, Tamas I. Gombosi, and Aaron J. Ridley. "Reducing numerical diffusion in magnetospheric simulations." Journal of Geophysical Research: Space Physics 116, A7 (July 2011): n/a. http://dx.doi.org/10.1029/2010ja016370.

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33

Bychuk, Oleg V., and Ben O’Shaughnessy. "Anomalous surface diffusion: A numerical study." Journal of Chemical Physics 101, no. 1 (July 1994): 772–80. http://dx.doi.org/10.1063/1.468132.

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34

Dantzig, J. A., W. J. Boettinger, J. A. Warren, G. B. McFadden, S. R. Coriell, and R. F. Sekerka. "Numerical modeling of diffusion-induced deformation." Metallurgical and Materials Transactions A 37, no. 9 (September 2006): 2701–14. http://dx.doi.org/10.1007/bf02586104.

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35

Truc, O., J. P. Ollivier, and L. O. Nilsson. "Numerical simulation of multi-species diffusion." Materials and Structures 33, no. 9 (November 2000): 566–73. http://dx.doi.org/10.1007/bf02480537.

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36

Lee, J. K. W., and A. A. Aldama. "Multipath diffusion: A general numerical model." Computers & Geosciences 18, no. 5 (June 1992): 531–55. http://dx.doi.org/10.1016/0098-3004(92)90093-7.

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37

Muhamediyeva, D. K., A. Yu Nurumova, and S. Yu Muminov. "Numerical modeling of cross-diffusion processes." E3S Web of Conferences 401 (2023): 05060. http://dx.doi.org/10.1051/e3sconf/202340105060.

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The paper studied the processes represented by a system of equations of a particular product of the parabolic type. For the solution of mutual diffusion problems, evaluations were obtained according to the environmental parameters and the dimensions of the space. Computational schemes, algorithms, and software packages in programming environments were developed, and the obtained solutions were visualized to solve the studied problems.

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38

Wang, Pengfei, Min Zhao, Hengguo Yu, Chuanjun Dai, Nan Wang, and Beibei Wang. "Nonlinear Dynamics of a Toxin-Phytoplankton-Zooplankton System with Self- and Cross-Diffusion." Discrete Dynamics in Nature and Society 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/4893451.

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A nonlinear system describing the interaction between toxin-producing phytoplankton and zooplankton was investigated analytically and numerically, where the system was represented by a couple of reaction-diffusion equations. We analyzed the effect of self- and cross-diffusion on the system. Some conditions for the local and global stability of the equilibrium were obtained based on the theoretical analysis. Furthermore, we found that the equilibrium lost its stability via Turing instability and patterns formation then occurred. In particular, the analysis indicated that cross-diffusion can play an important role in pattern formation. Subsequently, we performed a series of numerical simulations to further study the dynamics of the system, which demonstrated the rich dynamics induced by diffusion in the system. In addition, the numerical simulations indicated that the direction of cross-diffusion can influence the spatial distribution of the population and the population density. The numerical results agreed with the theoretical analysis. We hope that these results will prove useful in the study of toxic plankton systems.

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39

Pozorski, J., and A. Kajzer. "Density diffusion in low Mach number flows." Journal of Physics: Conference Series 2367, no. 1 (November 1, 2022): 012027. http://dx.doi.org/10.1088/1742-6596/2367/1/012027.

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Abstract In the realm of compressible viscous flow modelling, we briefly revisit the debate on a possible inconsistency of the Navier-Stokes (NS) equations. Then, we recall a recent proposal from the literature, put forward by M. Svärd. One of its features is the mass diffusive term in the continuity equation. The presence of density diffusion in the Svärd model reduces dispersive numerical errors when simple centred 2nd order, numerical diffusion free, spatial schemes are used, as confirmed in the simulations of a doubly-periodic shear layer at Ma = 0.05 and Re = 104. Further reduction of the dispersive errors at the spatial discretisation level is possible by more sophisticated approximation techniques.

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40

Lin, Dan-Ling, Yucang Wang, and William W. Guo. "Modelling Gas Diffusion from Breaking Coal Samples with the Discrete Element Method." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/582963.

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Particle scale diffusion is implemented in the discrete element code, Esys-Particle. We focus on the question of how to calibrate the particle scale diffusion coefficient. For the regular 2D packing, theoretical relation between micro- and macrodiffusion coefficients is derived. This relation is then verified in several numerical tests where the macroscopic diffusion coefficient is determined numerically based on the half-time of a desorption scheme. To further test the coupled model, we simulate the diffusion and desorption in the circular sample. The numerical results match the analytical solution very well. An example of gas diffusion and desorption during sample crushing and fragmenting is given at the last. The current approach is the first step towards a realistic and comprehensive modelling of coal and gas outbursts.

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41

Aziz, Imran, and Imran Khan. "Numerical Solution of Diffusion and Reaction–Diffusion Partial Integro-Differential Equations." International Journal of Computational Methods 15, no. 06 (September 2018): 1850047. http://dx.doi.org/10.1142/s0219876218500470.

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In this paper, a collocation method based on Haar wavelet is developed for numerical solution of diffusion and reaction–diffusion partial integro-differential equations. The equations are parabolic partial integro-differential equations and we consider both one-dimensional and two-dimensional cases. Such equations have applications in several practical problems including population dynamics. An important advantage of the proposed method is that it can be applied to both linear as well as nonlinear problems with slide modification. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency and robustness of the proposed method.

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42

Ko’shayevich, Khojaev Ismatullo. "Numerical Method for Calculating Axisymmetric Turbulent Jets of Reacting Gases During Diffusion Combustion." Journal of Advanced Research in Dynamical and Control Systems 12, SP7 (July 25, 2020): 2061–74. http://dx.doi.org/10.5373/jardcs/v12sp7/20202324.

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43

Abundo, Mario. "First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes." Open Systems & Information Dynamics 16, no. 04 (December 2009): 325–50. http://dx.doi.org/10.1142/s1230161209000256.

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For a, b > 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over I = (-b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position 0, it is reflected rightward to δ with probability p > 0 and leftward to -δ with probability 1 - p, where δ > 0. Closed analytical expressions are found for the mean exit time from the interval (-b, a), and for the probability of exit through the right end a, in the limit δ → 0+, generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.

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Carpenter, J. R., T. Sommer, and A. Wüest. "Stability of a Double-Diffusive Interface in the Diffusive Convection Regime." Journal of Physical Oceanography 42, no. 5 (May 1, 2012): 840–54. http://dx.doi.org/10.1175/jpo-d-11-0118.1.

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Abstract In this paper, the authors explore the conditions under which a double-diffusive interface may become unstable. Focus is placed on the case of a cold, freshwater layer above a warm, salty layer [i.e., the diffusive convection (DC) regime]. The “diffusive interface” between these layers will develop gravitationally unstable boundary layers due to the more rapid diffusion of heat (the destabilizing component) relative to salt. Previous studies have assumed that a purely convective-type instability of these boundary layers is what drives convection in this system and that this may be parameterized by a boundary layer Rayleigh number. The authors test this theory by conducting both a linear stability analysis and direct numerical simulations of a diffusive interface. Their linear stability analysis reveals that the transition to instability always occurs as an oscillating diffusive convection mode and at boundary layer Rayleigh numbers much smaller than previously thought. However, these findings are based on making a quasi-steady assumption for the growth of the interfaces by molecular diffusion. When diffusing interfaces are modeled (using direct numerical simulations), the authors observe that the time dependence is significant in determining the instability of the boundary layers and that the breakdown is due to a purely convective-type instability. Their findings therefore demonstrate that the relevant instability in a DC staircase is purely convective.

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ÜNAL, Osman, and Nuri AKKAŞ. "An Innovative Approach for Numerical Solution of the Unsteady Convection-Dominated Flow Problems." Karadeniz Fen Bilimleri Dergisi 12, no. 2 (December 15, 2022): 1069–80. http://dx.doi.org/10.31466/kfbd.1165640.

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In this study, convection-diffusion equation is solved numerically using four different space discretization methods namely first-order upwinding, second-order central difference, cubic (partially upwinded) and cubic-TVD (Total Variation Diminishing) techniques. All methods are compared with the analytical solution. The first-order method is not close to the analytical solution due to the numerical dispersion. The higher-order techniques reduce numerical dispersion. However, they cause another numerical error, unphysical oscillation. This study proposes an innovative approach on cubic-TVD method to eliminate undesired oscillations. Proposed model decreases numerical errors significantly compared to previously developed techniques. Moreover, numerical results of presented model quite close to the analytical solution. Finally, all Matlab codes of numerical and analytical solutions for convection-diffusion equation are added to Appendix in order to facilitate other researchers’ work.

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Liu, Yuanrong, Weimin Chen, Jing Zhong, Ming Chen, and Lijun Zhang. "Application of numerical inverse method in calculation of composition-dependent interdiffusion coefficients in finite diffusion couples." Metallurgical and Materials Engineering 23, no. 3 (September 30, 2017): 197–211. http://dx.doi.org/10.30544/308.

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The previously developed numerical inverse method was applied to determine the composition-dependent interdiffusion coefficients in single-phase finite diffusion couples. The numerical inverse method was first validated in a fictitious binary finite diffusion couple by pre-assuming four standard sets of interdiffusion coefficients. After that, the numerical inverse method was then adopted in a ternary Al-Cu-Ni finite diffusion couple. Based on the measured composition profiles, the ternary interdiffusion coefficients along the entire diffusion path of the target ternary diffusion couple were obtained by using the numerical inverse approach. The comprehensive comparisons between the computations and the experiments indicate that the numerical inverse method is also applicable to high-throughput determination of the composition-dependent interdiffusion coefficients in finite diffusion couples.

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Tupper, P. F., and Xin Yang. "A paradox of state-dependent diffusion and how to resolve it." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2148 (September 5, 2012): 3864–81. http://dx.doi.org/10.1098/rspa.2012.0259.

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Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region, the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients.

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Kloeden, P. E., E. Platen, H. Schurz, and M. Sørensen. "On effects of discretization on estimators of drift parameters for diffusion processes." Journal of Applied Probability 33, no. 4 (December 1996): 1061–76. http://dx.doi.org/10.2307/3214986.

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In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

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Kloeden, P. E., E. Platen, H. Schurz, and M. Sørensen. "On effects of discretization on estimators of drift parameters for diffusion processes." Journal of Applied Probability 33, no. 04 (December 1996): 1061–76. http://dx.doi.org/10.1017/s0021900200100488.

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In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

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Tsuji, Shinji, and Sadatoshi Koroyasu. "Numerical Analysis of Multiphase Diffusion in Homogenization Process of Diffusion Couples." Journal of the Japan Institute of Metals 61, no. 1 (1997): 8–17. http://dx.doi.org/10.2320/jinstmet1952.61.1_8.

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