A new fractional finite volume method for solving the fractional diffusion equation
The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-si...
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| Published in | Applied mathematical modelling Vol. 38; no. 15-16; pp. 3871 - 3878 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.08.2014
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| Subjects | |
| Online Access | Get full text |
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| Abstract | The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.
Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions. |
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| AbstractList | The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.
Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions. The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space-time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space-time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered. Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grunwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank-Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions. |
| Author | Burrage, K. Turner, I. Zhuang, P. Liu, F. Anh, V. |
| Author_xml | – sequence: 1 givenname: F. surname: Liu fullname: Liu, F. email: f.liu@qut.edu.au organization: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia – sequence: 2 givenname: P. surname: Zhuang fullname: Zhuang, P. organization: School of Mathematical Sciences, Xiamen University, Xiamen 361005, China – sequence: 3 givenname: I. surname: Turner fullname: Turner, I. organization: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia – sequence: 4 givenname: K. surname: Burrage fullname: Burrage, K. organization: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia – sequence: 5 givenname: V. surname: Anh fullname: Anh, V. organization: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia |
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| Cites_doi | 10.1016/S0370-1573(02)00331-9 10.1029/2000WR900032 10.1016/j.cam.2003.09.028 10.1016/j.cam.2009.02.013 10.1016/j.camwa.2009.08.071 10.1016/j.camwa.2007.11.012 10.1016/j.apnum.2005.02.008 10.1016/j.camwa.2009.08.004 10.1007/s11075-010-9393-x 10.1029/2004WR003818 10.1007/s10543-014-0484-2 10.1007/s11118-011-9243-z 10.2478/s11534-013-0317-y 10.1016/S0167-7322(99)00143-9 10.1016/j.camwa.2011.02.045 10.1016/j.cam.2005.06.005 10.1016/j.amc.2006.08.162 10.1016/j.cam.2004.01.033 10.1029/2006WR004912 10.1093/imamat/hxn033 10.1016/j.jmaa.2013.02.046 10.1029/2000WR900031 10.1016/j.jcp.2012.10.018 10.1016/j.advwatres.2009.01.008 10.1137/060673114 10.1007/s11538-007-9220-2 |
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| Keywords | Finite volume method Fractional diffusion equation Two-sided space fractional derivative Nonlinear source term Space–time dependent variable coefficient |
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| References | Baeumer, Kovály, Meerschaert (b0055) 2008; 55 Liu, Zhuang, Anh, Turner, Burrage (b0070) 2007; 191 Shen, Liu, Anh (b0080) 2011; 56 Leonenko, Meerschaert, Sikorskii (b0030) 2013; 403 Chen, Ye, Sun (b0100) 2010; 59 Benson, Wheatcraft, Meerschaert (b0010) 2000; 36 Zhang, Benson, Reeves (b0050) 2009; 32 Meerschaert, Tadjeran (b0060) 2004; 172 Meerschaert, Tadjeran (b0065) 2006; 56 Roop (b0110) 2006; 193 Zhang, Crawford, Deeks, Shutler, Bengough, Young (b0130) 2005; 41 Liu, Yang, Burrage (b0095) 2009; 231 H. Hejazi, T. Moroney, F. Liu, A finite volume method for solving the time–space fractional advection–dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14–17, Hohai University, Nanjing, China (MS11, Paper ID 038). Kochubei (b0035) 2012; 37 A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-inspace reaction–diffusion equations, J. Comp. Phys., 2013, in press. Metzler, Klafter (b0020) 2000; 86 Li, Zhao, Chen (b0120) 2011; 62 Fu, Chen, Yang (b0105) 2013; 235 S. Shen, F. Liu, V. Anh, I. Turner, A second-order accuracy numerical approximation for the Riesz space fractional advection–dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14–17, 2012, Hohai University, Nanjing, China. Zhang, Benson, Meerschaert, Labolle (b0045) 2007; 43 Meerschaert, Sikorskii (b0025) 2012; 43 Zheng, Li, Zhao (b0115) 2010; 59 Baeumer, Kovály, Meerschaert (b0140) 2007; 69 Liu, Anh, Turner (b0015) 2004; 16 Zhuang, Liu, Anh, Turner (b0090) 2008; 46 Zaslavsky (b0040) 2002; 371 Shen, Liu, Anh, Turner (b0075) 2008; 73 Benson, Wheatcraft, Meerschaert (b0005) 2000; 36 Zhang (10.1016/j.apm.2013.10.007_b0045) 2007; 43 Benson (10.1016/j.apm.2013.10.007_b0010) 2000; 36 Liu (10.1016/j.apm.2013.10.007_b0015) 2004; 16 Zheng (10.1016/j.apm.2013.10.007_b0115) 2010; 59 Zhuang (10.1016/j.apm.2013.10.007_b0090) 2008; 46 Meerschaert (10.1016/j.apm.2013.10.007_b0060) 2004; 172 Li (10.1016/j.apm.2013.10.007_b0120) 2011; 62 Roop (10.1016/j.apm.2013.10.007_b0110) 2006; 193 Baeumer (10.1016/j.apm.2013.10.007_b0140) 2007; 69 10.1016/j.apm.2013.10.007_b0135 Zaslavsky (10.1016/j.apm.2013.10.007_b0040) 2002; 371 Baeumer (10.1016/j.apm.2013.10.007_b0055) 2008; 55 Fu (10.1016/j.apm.2013.10.007_b0105) 2013; 235 Benson (10.1016/j.apm.2013.10.007_b0005) 2000; 36 Kochubei (10.1016/j.apm.2013.10.007_b0035) 2012; 37 Metzler (10.1016/j.apm.2013.10.007_b0020) 2000; 86 Leonenko (10.1016/j.apm.2013.10.007_b0030) 2013; 403 Meerschaert (10.1016/j.apm.2013.10.007_b0065) 2006; 56 Meerschaert (10.1016/j.apm.2013.10.007_b0025) 2012; 43 Liu (10.1016/j.apm.2013.10.007_b0070) 2007; 191 Shen (10.1016/j.apm.2013.10.007_b0080) 2011; 56 Chen (10.1016/j.apm.2013.10.007_b0100) 2010; 59 Shen (10.1016/j.apm.2013.10.007_b0075) 2008; 73 10.1016/j.apm.2013.10.007_b0125 Liu (10.1016/j.apm.2013.10.007_b0095) 2009; 231 Zhang (10.1016/j.apm.2013.10.007_b0130) 2005; 41 Zhang (10.1016/j.apm.2013.10.007_b0050) 2009; 32 10.1016/j.apm.2013.10.007_b0085 |
| References_xml | – reference: H. Hejazi, T. Moroney, F. Liu, A finite volume method for solving the time–space fractional advection–dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14–17, Hohai University, Nanjing, China (MS11, Paper ID 038). – volume: 59 start-page: 1614 year: 2010 end-page: 1620 ident: b0100 article-title: Fractional diffusion equations by the Kansa method publication-title: Comput. Math. Appl. – volume: 36 start-page: 1403 year: 2000 end-page: 1412 ident: b0005 article-title: Application of a fractional advection–dispersion equation publication-title: Water Resour. Res. – volume: 371 start-page: 461 year: 2002 end-page: 580 ident: b0040 article-title: Chaos, fractional kinetics, and anomalous transport publication-title: Phys. Rep. – volume: 193 start-page: 243 year: 2006 end-page: 268 ident: b0110 article-title: Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in publication-title: J. Comput. Appl. Math. – volume: 231 start-page: 160 year: 2009 end-page: 176 ident: b0095 article-title: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term publication-title: J. Comput. Appl. Math. – volume: 86 start-page: 219 year: 2000 end-page: 228 ident: b0020 article-title: The fractional Fokker–Planck equation: dispersive transport in an external force field publication-title: J. Mol. Liq. – volume: 41 start-page: 1 year: 2005 end-page: 10 ident: b0130 article-title: A mass balance based numerical method for the fractional advection–dispersion equation: theory and application publication-title: Water Resour. Res. – volume: 36 start-page: 1413 year: 2000 end-page: 1423 ident: b0010 article-title: The fractional-order governing equation of levy motion publication-title: Water Resour. Res. – volume: 62 start-page: 855 year: 2011 end-page: 875 ident: b0120 article-title: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion publication-title: Comput. Math. Appl. – volume: 73 start-page: 850 year: 2008 end-page: 872 ident: b0075 article-title: The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation publication-title: IMA J. Appl. Math. – volume: 69 start-page: 2281 year: 2007 end-page: 2297 ident: b0140 article-title: Fractional reproduction-dispersal equations and heavy tail dispersal kernels publication-title: Bull. Math. Biol. – volume: 172 start-page: 65 year: 2004 end-page: 77 ident: b0060 article-title: Finite difference approximations for fractional advection–dispersion flow equations publication-title: J. Comput. Appl. Math. – volume: 55 start-page: 2212 year: 2008 end-page: 2226 ident: b0055 article-title: Numerical solutions for fractional reaction–diffusion equations publication-title: Comput. Math. Appl. – volume: 191 start-page: 12 year: 2007 end-page: 21 ident: b0070 article-title: Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation publication-title: Appl. Math. Comput. – volume: 46 start-page: 1079 year: 2008 end-page: 1095 ident: b0090 article-title: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation publication-title: SIAM J. Numer. Anal. – reference: S. Shen, F. Liu, V. Anh, I. Turner, A second-order accuracy numerical approximation for the Riesz space fractional advection–dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14–17, 2012, Hohai University, Nanjing, China. – volume: 32 start-page: 561 year: 2009 end-page: 581 ident: b0050 article-title: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications publication-title: Adv. Water Resour. – volume: 56 start-page: 383 year: 2011 end-page: 403 ident: b0080 article-title: Numerical approximations and solution techniques for the space–time Riesz–Caputo fractional advection–diffusion equation publication-title: Numer. Algorithms – volume: 56 start-page: 80 year: 2006 end-page: 90 ident: b0065 article-title: Finite difference approximations for two-sided space-fractional partial differential equations publication-title: Appl. Numer. Math. – volume: 59 start-page: 1718 year: 2010 end-page: 1726 ident: b0115 article-title: A note on the finite element method for the space-fractional advection diffusion equation publication-title: Comput. Math. Appl. – volume: 43 year: 2012 ident: b0025 article-title: Stochastic models for fractional calculus publication-title: De Gruyter Studies in Mathematics – volume: 235 start-page: 52 year: 2013 end-page: 66 ident: b0105 article-title: Boundary particle method for Laplace transformed time fractional diffusion equations publication-title: J. Comput. Phys. – volume: 37 start-page: 1 year: 2012 end-page: 30 ident: b0035 article-title: Fractional-parabolic systems publication-title: Potential Anal. – reference: A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-inspace reaction–diffusion equations, J. Comp. Phys., 2013, in press. – volume: 403 start-page: 532 year: 2013 end-page: 546 ident: b0030 article-title: Fractional Pearson diffusions publication-title: J. Math. Anal. Appl. – volume: 16 start-page: 209 year: 2004 end-page: 219 ident: b0015 article-title: Numerical solution of the space fractional Fokker–Planck equation publication-title: J. Comput. Appl. Math. – volume: 43 start-page: W05439 year: 2007 ident: b0045 article-title: Space-fractional advection–dispersion equations with variable parameters: diverse formulas, numerical solutions, and application to the MADE-site data publication-title: Water Resour. Res. – volume: 371 start-page: 461 year: 2002 ident: 10.1016/j.apm.2013.10.007_b0040 article-title: Chaos, fractional kinetics, and anomalous transport publication-title: Phys. Rep. doi: 10.1016/S0370-1573(02)00331-9 – volume: 36 start-page: 1413 issue: 6 year: 2000 ident: 10.1016/j.apm.2013.10.007_b0010 article-title: The fractional-order governing equation of levy motion publication-title: Water Resour. Res. doi: 10.1029/2000WR900032 – volume: 16 start-page: 209 issue: 6 year: 2004 ident: 10.1016/j.apm.2013.10.007_b0015 article-title: Numerical solution of the space fractional Fokker–Planck equation publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2003.09.028 – volume: 231 start-page: 160 issue: 1 year: 2009 ident: 10.1016/j.apm.2013.10.007_b0095 article-title: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.02.013 – volume: 59 start-page: 1718 year: 2010 ident: 10.1016/j.apm.2013.10.007_b0115 article-title: A note on the finite element method for the space-fractional advection diffusion equation publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2009.08.071 – volume: 43 year: 2012 ident: 10.1016/j.apm.2013.10.007_b0025 article-title: Stochastic models for fractional calculus – volume: 55 start-page: 2212 year: 2008 ident: 10.1016/j.apm.2013.10.007_b0055 article-title: Numerical solutions for fractional reaction–diffusion equations publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2007.11.012 – volume: 56 start-page: 80 issue: 1 year: 2006 ident: 10.1016/j.apm.2013.10.007_b0065 article-title: Finite difference approximations for two-sided space-fractional partial differential equations publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2005.02.008 – ident: 10.1016/j.apm.2013.10.007_b0085 – volume: 59 start-page: 1614 year: 2010 ident: 10.1016/j.apm.2013.10.007_b0100 article-title: Fractional diffusion equations by the Kansa method publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2009.08.004 – volume: 56 start-page: 383 year: 2011 ident: 10.1016/j.apm.2013.10.007_b0080 article-title: Numerical approximations and solution techniques for the space–time Riesz–Caputo fractional advection–diffusion equation publication-title: Numer. Algorithms doi: 10.1007/s11075-010-9393-x – volume: 41 start-page: 1 year: 2005 ident: 10.1016/j.apm.2013.10.007_b0130 article-title: A mass balance based numerical method for the fractional advection–dispersion equation: theory and application publication-title: Water Resour. Res. doi: 10.1029/2004WR003818 – ident: 10.1016/j.apm.2013.10.007_b0125 doi: 10.1007/s10543-014-0484-2 – volume: 37 start-page: 1 issue: 1 year: 2012 ident: 10.1016/j.apm.2013.10.007_b0035 article-title: Fractional-parabolic systems publication-title: Potential Anal. doi: 10.1007/s11118-011-9243-z – ident: 10.1016/j.apm.2013.10.007_b0135 doi: 10.2478/s11534-013-0317-y – volume: 86 start-page: 219 year: 2000 ident: 10.1016/j.apm.2013.10.007_b0020 article-title: The fractional Fokker–Planck equation: dispersive transport in an external force field publication-title: J. Mol. Liq. doi: 10.1016/S0167-7322(99)00143-9 – volume: 62 start-page: 855 year: 2011 ident: 10.1016/j.apm.2013.10.007_b0120 article-title: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2011.02.045 – volume: 193 start-page: 243 issue: 1 year: 2006 ident: 10.1016/j.apm.2013.10.007_b0110 article-title: Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in R2 publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2005.06.005 – volume: 191 start-page: 12 year: 2007 ident: 10.1016/j.apm.2013.10.007_b0070 article-title: Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2006.08.162 – volume: 172 start-page: 65 year: 2004 ident: 10.1016/j.apm.2013.10.007_b0060 article-title: Finite difference approximations for fractional advection–dispersion flow equations publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2004.01.033 – volume: 43 start-page: W05439 year: 2007 ident: 10.1016/j.apm.2013.10.007_b0045 article-title: Space-fractional advection–dispersion equations with variable parameters: diverse formulas, numerical solutions, and application to the MADE-site data publication-title: Water Resour. Res. doi: 10.1029/2006WR004912 – volume: 73 start-page: 850 year: 2008 ident: 10.1016/j.apm.2013.10.007_b0075 article-title: The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation publication-title: IMA J. Appl. Math. doi: 10.1093/imamat/hxn033 – volume: 403 start-page: 532 issue: 2 year: 2013 ident: 10.1016/j.apm.2013.10.007_b0030 article-title: Fractional Pearson diffusions publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2013.02.046 – volume: 36 start-page: 1403 issue: 6 year: 2000 ident: 10.1016/j.apm.2013.10.007_b0005 article-title: Application of a fractional advection–dispersion equation publication-title: Water Resour. Res. doi: 10.1029/2000WR900031 – volume: 235 start-page: 52 year: 2013 ident: 10.1016/j.apm.2013.10.007_b0105 article-title: Boundary particle method for Laplace transformed time fractional diffusion equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2012.10.018 – volume: 32 start-page: 561 year: 2009 ident: 10.1016/j.apm.2013.10.007_b0050 article-title: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications publication-title: Adv. Water Resour. doi: 10.1016/j.advwatres.2009.01.008 – volume: 46 start-page: 1079 issue: 2 year: 2008 ident: 10.1016/j.apm.2013.10.007_b0090 article-title: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation publication-title: SIAM J. Numer. Anal. doi: 10.1137/060673114 – volume: 69 start-page: 2281 year: 2007 ident: 10.1016/j.apm.2013.10.007_b0140 article-title: Fractional reproduction-dispersal equations and heavy tail dispersal kernels publication-title: Bull. Math. Biol. doi: 10.1007/s11538-007-9220-2 |
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| SubjectTerms | Coefficients Dependent variables Diffusion Dirichlet problem Finite volume method Fractional diffusion equation Mathematical analysis Mathematical models Nonlinear source term Space–time dependent variable coefficient Three dimensional Two-sided space fractional derivative |
| Title | A new fractional finite volume method for solving the fractional diffusion equation |
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