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Lattice Boltzmann method for the fractional advection-diffusion equation

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Lattice Boltzmann method for the fractional advection-diffusion equation. / Zhou, J. G.; Haygarth, Philip Matthew; Withers, P. J. A. et al.

In: Physical Review E, Vol. 93, No. 4, 043310, 13.04.2016.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Zhou, JG, Haygarth, PM, Withers, PJA, Macleod, CJA, Falloon, PD, Beven, KJ, Ockenden, MC, Forber, KJ, Hollaway, MJ, Evans, R, Collins, AL, Hiscock, KM, Wearing, CL, Kahana, R & Villamizar, M 2016, 'Lattice Boltzmann method for the fractional advection-diffusion equation', Physical Review E, vol. 93, no. 4, 043310. https://doi.org/10.1103/PhysRevE.93.043310, https://doi.org/10.1103/PhysRevE.93.043310

APA

Zhou, J. G., Haygarth, P. M., Withers, P. J. A., Macleod, C. J. A., Falloon, P. D., Beven, K. J., Ockenden, M. C., Forber, K. J., Hollaway, M. J., Evans, R., Collins, A. L., Hiscock, K. M., Wearing, C. L., Kahana, R., & Villamizar, M. (2016). Lattice Boltzmann method for the fractional advection-diffusion equation. Physical Review E, 93(4), [043310]. https://doi.org/10.1103/PhysRevE.93.043310, https://doi.org/10.1103/PhysRevE.93.043310

Vancouver

Zhou JG, Haygarth PM, Withers PJA, Macleod CJA, Falloon PD, Beven KJ et al. Lattice Boltzmann method for the fractional advection-diffusion equation. Physical Review E. 2016 Apr 13;93(4):043310. doi: 10.1103/PhysRevE.93.043310, 10.1103/PhysRevE.93.043310

Author

Zhou, J. G. ; Haygarth, Philip Matthew ; Withers, P. J. A. et al. / Lattice Boltzmann method for the fractional advection-diffusion equation. In: Physical Review E. 2016 ; Vol. 93, No. 4.

Bibtex

@article{0598fd8221ec4eafae19e770ec759e40,
title = "Lattice Boltzmann method for the fractional advection-diffusion equation",
abstract = "Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.",
keywords = "ANOMALOUS DIFFUSION, SOLUTE TRANSPORT, SIMULATION, LAPLACIAN, DELIVERY, DYNAMICS, MODEL",
author = "Zhou, {J. G.} and Haygarth, {Philip Matthew} and Withers, {P. J. A.} and C.J.A Macleod and Falloon, {Peter D} and Beven, {Keith John} and Ockenden, {Mary Catherine} and Forber, {Kirsty Jessica} and Hollaway, {Michael John} and R. Evans and Collins, {A. L.} and Hiscock, {Kevin M} and Wearing, {Catherine Louise} and Ron Kahana and Martha Villamizar",
year = "2016",
month = apr,
day = "13",
doi = "10.1103/PhysRevE.93.043310",
language = "English",
volume = "93",
journal = "Physical Review E",
issn = "1539-3755",
publisher = "American Physical Society",
number = "4",

}

RIS

TY - JOUR

T1 - Lattice Boltzmann method for the fractional advection-diffusion equation

AU - Zhou, J. G.

AU - Haygarth, Philip Matthew

AU - Withers, P. J. A.

AU - Macleod, C.J.A

AU - Falloon, Peter D

AU - Beven, Keith John

AU - Ockenden, Mary Catherine

AU - Forber, Kirsty Jessica

AU - Hollaway, Michael John

AU - Evans, R.

AU - Collins, A. L.

AU - Hiscock, Kevin M

AU - Wearing, Catherine Louise

AU - Kahana, Ron

AU - Villamizar, Martha

PY - 2016/4/13

Y1 - 2016/4/13

N2 - Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

AB - Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

KW - ANOMALOUS DIFFUSION

KW - SOLUTE TRANSPORT

KW - SIMULATION

KW - LAPLACIAN

KW - DELIVERY

KW - DYNAMICS

KW - MODEL

U2 - 10.1103/PhysRevE.93.043310

DO - 10.1103/PhysRevE.93.043310

M3 - Journal article

VL - 93

JO - Physical Review E

JF - Physical Review E

SN - 1539-3755

IS - 4

M1 - 043310

ER -