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March  2020, 13(3): 443-466. doi: 10.3934/dcdss.2020025

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, 2020, 13 (3) : 443-466. 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. BensonS. 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. DiethelmN. J. FordA. 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. HuangG. 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. LiZ. 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. LiuV. V. AnhI. 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. LiuP. ZhuangV. AnhI. 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. LynchB. A. CarrerasD. del Castillo-NegreteK. 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. MetzlerW. 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. PangW. 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. ShenF. LiuV. AnhI. 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. SuW. 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. YirangL. 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. YirangY. QingL. 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

show all references

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. BensonS. 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. DiethelmN. J. FordA. 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. HuangG. 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. LiZ. 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. LiuV. V. AnhI. 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. LiuP. ZhuangV. AnhI. 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. LynchB. A. CarrerasD. del Castillo-NegreteK. 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. MetzlerW. 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. PangW. 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. ShenF. LiuV. AnhI. 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. SuW. 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. YirangL. 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. YirangY. QingL. 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

Table 1.  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{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2 \delta _{n,m}^{ \alpha } + \left( 2-2cos \phi \right) \beta _{m} \right) $
$ \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{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m} +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
Upwind CrankNicolson $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }>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} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( \delta _{n,m}^{ \alpha } + \beta _{m} \left( 1-cos \phi \right) \right) $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } <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}} $ Conditionally stable
$ \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} <\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( \delta _{n,m}^{ \alpha } + \beta _{m} \left( 1-cos \phi \right) \right) $
$ \frac{4D_{L}}{ \left( \Delta x \right) ^{2}}+\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) }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 1-cos \phi \right) \beta _{m} $
Weighted upwinddownwind (explicit) $ \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}} $ 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} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( \theta +1 \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) $
$ \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}} $ $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) < 2 v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( 1- \theta \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
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{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2 \delta _{n,m}^{ \alpha } + \left( 2-2cos \phi \right) \beta _{m} \right) $
$ \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{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2-2cos \phi \right) \beta _{m} +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
Upwind CrankNicolson $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha }>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} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( \delta _{n,m}^{ \alpha } + \beta _{m} \left( 1-cos \phi \right) \right) $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,n-1}^{ \alpha } <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}} $ Conditionally stable
$ \frac{2D_{L}}{ \left( \Delta x \right) ^{2}} <\frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } $
$ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{n} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( \delta _{n,m}^{ \alpha } + \beta _{m} \left( 1-cos \phi \right) \right) $
$ \frac{4D_{L}}{ \left( \Delta x \right) ^{2}}+\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) }+v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 1-cos \phi \right) \beta _{m} $
Weighted upwinddownwind (explicit) $ \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}} $ 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} <v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( \theta +1 \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) $
$ \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}} $ $ \frac{ \left( \Delta t \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \left( 2+ \beta _{n} \right) < 2 v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \delta _{n,m}^{ \alpha } \left( 1- \theta \right) + 2v\frac{ \left( \Delta x \right) ^{- \alpha }}{ \Gamma \left(2 - \alpha \right) } \beta _{m} \left( 1-cos \phi \right) +\frac{4D_{L}}{ \left( \Delta x \right) ^{2}} $
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