Some comments on using fractional derivative operators in modeling non-local diffusion processes

T. Namba; P. Rybka; V. R. Voller

doi:10.1016/j.cam.2020.113040

Some comments on using fractional derivative operators in modeling non-local diffusion processes

T. Namba, P. Rybka, V. R. Voller

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5 Scopus citations

Abstract

We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

Original language	English (US)
Article number	113040
Journal	Journal of Computational and Applied Mathematics
Volume	381
DOIs	https://doi.org/10.1016/j.cam.2020.113040
State	Published - Jan 1 2021

Bibliographical note

Funding Information:
The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics—Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland , through the Grant No. 2017/26/M/ST1/00700 . All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper.

Funding Information:
The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics?Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland, through the Grant No. 2017/26/M/ST1/00700. All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper.

Publisher Copyright:
© 2020 Elsevier B.V.

Keywords

Caputo fractional derivative
Fractional diffusion operator
Riemann–Liouville fractional derivative

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10.1016/j.cam.2020.113040

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title = "Some comments on using fractional derivative operators in modeling non-local diffusion processes",

abstract = "We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.",

keywords = "Caputo fractional derivative, Fractional diffusion operator, Riemann–Liouville fractional derivative",

author = "T. Namba and P. Rybka and Voller, {V. R.}",

note = "Funding Information: The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics—Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland , through the Grant No. 2017/26/M/ST1/00700 . All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper. Funding Information: The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics?Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland, through the Grant No. 2017/26/M/ST1/00700. All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper. Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

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AU - Rybka, P.

AU - Voller, V. R.

N1 - Funding Information: The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics—Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland , through the Grant No. 2017/26/M/ST1/00700 . All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper. Funding Information: The authors thank BIRS in Banff for creating a stimulating environment at the workshop Advanced Developments for Surface and Interface Dynamics?Analysis and Computation, where this paper was initiated. VV notes the support of the James L. Record Chair, from the Department of Civil, Environmental and Geo- Engineering, University of Minnesota. A part of the research for this paper was done at the University of Warsaw, where PR and VV enjoyed a partial support of the National Science Centre, Poland, through the Grant No. 2017/26/M/ST1/00700. All authors acknowledge stimulating discussion with Adam Kubica and Katarzyna Ryszewska. The authors thank the referee for the remarks, which helped to improve this paper. Publisher Copyright: © 2020 Elsevier B.V.

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N2 - We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

AB - We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

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Some comments on using fractional derivative operators in modeling non-local diffusion processes

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