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## Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems

 Institute for Groundwater Studies, Faculty of Agricultural and Natural Sciences, University of the Free State, 9301, Bloemfontein, Free State, South Africa

* Corresponding author: A. Allwright

Received  June 2018 Revised  July 2018 Published  March 2019

The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.

Citation: Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020025
##### References:
 [1] A. Allwright and A. Atangana, Augmented upwind numerical schemes for the groundwater transport advection-dispersion equation with local operators, International Journal for Numerical Methods in Fluids, 87 (2018), 437-462. doi: 10.1002/fld.4497. Google Scholar [2] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar [3] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010. Google Scholar [4] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. doi: 10.1029/2000WR900031. Google Scholar [5] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, PhD thesis, University of Nevada, Reno, 1998. doi: 10.1029/2000WR900031. Google Scholar [6] K. Diethelm, N. J. Ford, A. D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Computer methods in applied mechanics and engineering, 194 (2005), 743-773. doi: 10.1016/j.cma.2004.06.006. Google Scholar [7] R. Fazio and A. Jannelli, A finite difference method on quasi-uniform mesh for time-fractional advection-diffusion equations with source term, arXiv preprint arXiv 1801.07160.Google Scholar [8] R. Gnitchogna and A. Atangana, New two step laplace adam-bashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34 (2018), 1739-1758. doi: 10.1002/num.22216. Google Scholar [9] F. Huang and F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, Journal of Applied Mathematics and Computing, 18 (2005), 339-350. doi: 10.1007/BF02936577. Google Scholar [10] Q. Huang, G. Huang and H. Zhan, A finite element solution for the fractional advection–dispersion equation, Advances in Water Resources, 31 (2008), 1578-1589. doi: 10.1016/j.advwatres.2008.07.002. Google Scholar [11] H. Jafari and H. Tajadodi, Numerical solutions of the fractional advection-dispersion equation, Prog. Fract. Differ. Appl, 1 (2015), 37-45. Google Scholar [12] S. Javadi, M. Jani and E. Babolian, A numerical scheme for space-time fractional advection-dispersion equation, arXiv preprint, arXiv: 1512.06629.Google Scholar [13] X. Li and H. Rui, A high-order fully conservative block-centered finite difference method for the time-fractional advection–dispersion equation, Applied Numerical Mathematics, 124 (2018), 89-109. doi: 10.1016/j.apnum.2017.10.004. Google Scholar [14] Z. Li, Z. Liang and Y. Yan, High-order numerical methods for solving time fractional partial differential equations, Journal of Scientific Computing, 71 (2017), 785-803. doi: 10.1007/s10915-016-0319-1. Google Scholar [15] F. Liu, V. V. Anh, I. Turner and P. Zhuang, Time fractional advection-dispersion equation, Journal of Applied Mathematics and Computing, 13 (2003), 233-245. doi: 10.1007/BF02936089. Google Scholar [16] F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Applied Mathematics and Computation, 191 (2007), 12-20. doi: 10.1016/j.amc.2006.08.162. Google Scholar [17] T. Liu and M. Hou, A fast implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions, Advances in Mathematical Physics, 2017 (2017), Art. ID 8716752, 8 pp. doi: 10.1155/2017/8716752. Google Scholar [18] Z. Liu and X. Li, A crank–nicolson difference scheme for the time variable fractional mobile–immobile advection–dispersion equation, Journal of Applied Mathematics and Computing, 56 (2018), 391-410. doi: 10.1007/s12190-016-1079-7. Google Scholar [19] V. E. Lynch, B. A. Carreras, D. del Castillo-Negrete, K. Ferreira-Mejias and H. Hicks, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics, 192 (2003), 406-421. doi: 10.1016/j.jcp.2003.07.008. Google Scholar [20] M. M. Meerschaert, Fractional calculus, anomalous diffusion, and probability, in Fractional Dynamics: Recent Advances, World Scientific, 2012,265–284. Google Scholar [21] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar [22] R. Metzler, W. G. Glöckle and T. F. Nonnenmacher, Fractional model equation for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 211 (1994), 13-24. Google Scholar [23] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar [24] G. Pang, W. Chen and Z. Fu, Space-fractional advection–dispersion equations by the kansa method, Journal of Computational Physics, 293 (2015), 280-296. doi: 10.1016/j.jcp.2014.07.020. Google Scholar [25] Y. Povstenko, Space-time-fractional advection diffusion equation in a plane, in Advances in Modelling and Control of Non-Integer-Order Systems, Springer, 320 (2015), 275–284. Google Scholar [26] Y. Povstenko, Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables, Mathematical Problems in Engineering, 2014 (2014), Art. ID 705364, 7 pp. doi: 10.1155/2014/705364. Google Scholar [27] Q. Rubbab, I. A. Mirza and M. Z. A. Qureshi, Analytical solutions to the fractional advection-diffusion equation with time-dependent pulses on the boundary, AIP Advances, 6 (2016), 075318. doi: 10.1063/1.4960108. Google Scholar [28] W. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144. doi: 10.1063/1.528578. Google Scholar [29] S. Shen, F. Liu, V. Anh, I. Turner and J. Chen, A novel numerical approximation for the space fractional advection–dispersion equation, IMA journal of Applied Mathematics, 79 (2014), 431-444. doi: 10.1093/imamat/hxs073. Google Scholar [30] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics, 228 (2009), 4038-4054. doi: 10.1016/j.jcp.2009.02.011. Google Scholar [31] E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the riemann–liouville derivative, Applied Numerical Mathematics, 90 (2015), 22-37. doi: 10.1016/j.apnum.2014.11.007. Google Scholar [32] L. Su, W. Wang and Q. Xu, Finite difference methods for fractional dispersion equations, Applied Mathematics and Computation, 216 (2010), 3329-3334. doi: 10.1016/j.amc.2010.04.060. Google Scholar [33] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 52.Google Scholar [34] K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection–diffusion equations, Advances in Water Resources, 34 (2011), 810-816. doi: 10.1016/j.advwatres.2010.11.003. Google Scholar [35] W. Wyss, The fractional diffusion equation, Journal of Mathematical Physics, 27 (1986), 2782-2785. doi: 10.1063/1.527251. Google Scholar [36] Y. Yirang, L. Changfeng and S. Tongjun, The second-order upwind finite difference fractional steps method for moving boundary value problem of oil-water percolation, Numerical Methods for Partial Differential Equations, 30 (2014), 1103-1129. doi: 10.1002/num.21859. Google Scholar [37] Y. Yirang, Y. Qing, L. Changfeng and S. Tongjun, Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems, Acta Mathematica Scientia, 37 (2017), 259-279. doi: 10.1016/S0252-9602(16)30129-1. Google Scholar [38] Y. Yuan, The upwind finite difference fractional steps methods for two-phase compressible flow in porous media, Numerical Methods for Partial Differential Equations: An International Journal, 19 (2003), 67-88. doi: 10.1002/num.10036. Google Scholar

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##### References:
 [1] A. Allwright and A. Atangana, Augmented upwind numerical schemes for the groundwater transport advection-dispersion equation with local operators, International Journal for Numerical Methods in Fluids, 87 (2018), 437-462. doi: 10.1002/fld.4497. Google Scholar [2] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar [3] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010. Google Scholar [4] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. doi: 10.1029/2000WR900031. Google Scholar [5] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, PhD thesis, University of Nevada, Reno, 1998. doi: 10.1029/2000WR900031. Google Scholar [6] K. Diethelm, N. J. Ford, A. D. Freed and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Computer methods in applied mechanics and engineering, 194 (2005), 743-773. doi: 10.1016/j.cma.2004.06.006. Google Scholar [7] R. Fazio and A. Jannelli, A finite difference method on quasi-uniform mesh for time-fractional advection-diffusion equations with source term, arXiv preprint arXiv 1801.07160.Google Scholar [8] R. Gnitchogna and A. Atangana, New two step laplace adam-bashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34 (2018), 1739-1758. doi: 10.1002/num.22216. Google Scholar [9] F. Huang and F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, Journal of Applied Mathematics and Computing, 18 (2005), 339-350. doi: 10.1007/BF02936577. Google Scholar [10] Q. Huang, G. Huang and H. Zhan, A finite element solution for the fractional advection–dispersion equation, Advances in Water Resources, 31 (2008), 1578-1589. doi: 10.1016/j.advwatres.2008.07.002. Google Scholar [11] H. Jafari and H. Tajadodi, Numerical solutions of the fractional advection-dispersion equation, Prog. Fract. Differ. Appl, 1 (2015), 37-45. Google Scholar [12] S. Javadi, M. Jani and E. Babolian, A numerical scheme for space-time fractional advection-dispersion equation, arXiv preprint, arXiv: 1512.06629.Google Scholar [13] X. Li and H. Rui, A high-order fully conservative block-centered finite difference method for the time-fractional advection–dispersion equation, Applied Numerical Mathematics, 124 (2018), 89-109. doi: 10.1016/j.apnum.2017.10.004. Google Scholar [14] Z. Li, Z. Liang and Y. Yan, High-order numerical methods for solving time fractional partial differential equations, Journal of Scientific Computing, 71 (2017), 785-803. doi: 10.1007/s10915-016-0319-1. Google Scholar [15] F. Liu, V. V. Anh, I. Turner and P. Zhuang, Time fractional advection-dispersion equation, Journal of Applied Mathematics and Computing, 13 (2003), 233-245. doi: 10.1007/BF02936089. Google Scholar [16] F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Applied Mathematics and Computation, 191 (2007), 12-20. doi: 10.1016/j.amc.2006.08.162. Google Scholar [17] T. Liu and M. Hou, A fast implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions, Advances in Mathematical Physics, 2017 (2017), Art. ID 8716752, 8 pp. doi: 10.1155/2017/8716752. Google Scholar [18] Z. Liu and X. Li, A crank–nicolson difference scheme for the time variable fractional mobile–immobile advection–dispersion equation, Journal of Applied Mathematics and Computing, 56 (2018), 391-410. doi: 10.1007/s12190-016-1079-7. Google Scholar [19] V. E. Lynch, B. A. Carreras, D. del Castillo-Negrete, K. Ferreira-Mejias and H. Hicks, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics, 192 (2003), 406-421. doi: 10.1016/j.jcp.2003.07.008. Google Scholar [20] M. M. Meerschaert, Fractional calculus, anomalous diffusion, and probability, in Fractional Dynamics: Recent Advances, World Scientific, 2012,265–284. Google Scholar [21] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar [22] R. Metzler, W. G. Glöckle and T. F. Nonnenmacher, Fractional model equation for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 211 (1994), 13-24. Google Scholar [23] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. Google Scholar [24] G. Pang, W. Chen and Z. Fu, Space-fractional advection–dispersion equations by the kansa method, Journal of Computational Physics, 293 (2015), 280-296. doi: 10.1016/j.jcp.2014.07.020. Google Scholar [25] Y. Povstenko, Space-time-fractional advection diffusion equation in a plane, in Advances in Modelling and Control of Non-Integer-Order Systems, Springer, 320 (2015), 275–284. Google Scholar [26] Y. Povstenko, Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables, Mathematical Problems in Engineering, 2014 (2014), Art. ID 705364, 7 pp. doi: 10.1155/2014/705364. Google Scholar [27] Q. Rubbab, I. A. Mirza and M. Z. A. Qureshi, Analytical solutions to the fractional advection-diffusion equation with time-dependent pulses on the boundary, AIP Advances, 6 (2016), 075318. doi: 10.1063/1.4960108. Google Scholar [28] W. Schneider and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, 30 (1989), 134-144. doi: 10.1063/1.528578. Google Scholar [29] S. Shen, F. Liu, V. Anh, I. Turner and J. Chen, A novel numerical approximation for the space fractional advection–dispersion equation, IMA journal of Applied Mathematics, 79 (2014), 431-444. doi: 10.1093/imamat/hxs073. Google Scholar [30] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics, 228 (2009), 4038-4054. doi: 10.1016/j.jcp.2009.02.011. Google Scholar [31] E. Sousa and C. Li, A weighted finite difference method for the fractional diffusion equation based on the riemann–liouville derivative, Applied Numerical Mathematics, 90 (2015), 22-37. doi: 10.1016/j.apnum.2014.11.007. Google Scholar [32] L. Su, W. Wang and Q. Xu, Finite difference methods for fractional dispersion equations, Applied Mathematics and Computation, 216 (2010), 3329-3334. doi: 10.1016/j.amc.2010.04.060. Google Scholar [33] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 52.Google Scholar [34] K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection–diffusion equations, Advances in Water Resources, 34 (2011), 810-816. doi: 10.1016/j.advwatres.2010.11.003. Google Scholar [35] W. Wyss, The fractional diffusion equation, Journal of Mathematical Physics, 27 (1986), 2782-2785. doi: 10.1063/1.527251. Google Scholar [36] Y. Yirang, L. Changfeng and S. Tongjun, The second-order upwind finite difference fractional steps method for moving boundary value problem of oil-water percolation, Numerical Methods for Partial Differential Equations, 30 (2014), 1103-1129. doi: 10.1002/num.21859. Google Scholar [37] Y. Yirang, Y. Qing, L. Changfeng and S. Tongjun, Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems, Acta Mathematica Scientia, 37 (2017), 259-279. doi: 10.1016/S0252-9602(16)30129-1. Google Scholar [38] Y. Yuan, The upwind finite difference fractional steps methods for two-phase compressible flow in porous media, Numerical Methods for Partial Differential Equations: An International Journal, 19 (2003), 67-88. doi: 10.1002/num.10036. Google Scholar
Summary of the established stability condition, and corresponding assumption, for each numerical approximation scheme
 Scheme Assumptions Stability condition Upwind (explicit) $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } > \frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unstable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Conditionally stable $\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} +v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m}+\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <\frac{2 \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) }$ Upwind (implicit) $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unconditionally stable / Conditionally stable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} v\frac{0.5 \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } +\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unconditionally stable / Conditionally stable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} \frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unstable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } + v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Conditionally stable / Unstable Weighted upwinddownwind (implicit) $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } +v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}}$ Unconditionally stable / conditionally stable $\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n}  Scheme Assumptions Stability condition Upwind (explicit)$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } > \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Unstable$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Conditionally stable$ \frac{4D_{L}}{ \left( \Delta x \right) ^{2}} +v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m}+\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <\frac{2 \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $Upwind (implicit)$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Unconditionally stable / Conditionally stable$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} v\frac{0.5 \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } +\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Unconditionally stable / Conditionally stable$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Unstable$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } + v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } <\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Conditionally stable / Unstable Weighted upwinddownwind (implicit)$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } +v \left( 2 \theta -1 \right) \frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha }>\frac{2D_{L}}{ \left( \Delta x \right) ^{2}} $Unconditionally stable / conditionally stable$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n}
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