July  2021, 14(7): 2471-2485. doi: 10.3934/dcdss.2021056

Finite element method for two-dimensional linear advection equations based on spline method

College of Science, Dalian Maritime University, Dalian, 116026, China

* Corresponding author: Kai Qu

Received  February 2020 Revised  October 2020 Published  July 2021 Early access  May 2021

Fund Project: The first author is supported by National Natural Science Foundation of China grant 11801053, the Fundamental Research Funds for the Central Universities (3132019176, 3132019323)

A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.

Citation: Kai Qu, Qi Dong, Chanjie Li, Feiyu Zhang. Finite element method for two-dimensional linear advection equations based on spline method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2471-2485. doi: 10.3934/dcdss.2021056
References:
[1]

W. Bu, X. Liu, Y. Tang and Y. Jiang, Finite element multigrid method for multi-term time fractional advection-diffusion equations, Int. J. Model. Simul. Sci. Comput., 6 (2015), 154001. doi: 10.1142/S1793962315400012.  Google Scholar

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A. R. Carella and C. A. Dorao, Least-squares spectral method for the solution of a fractional advection-dispersion equation, J. Comput. Phys., 232 (2013), 33-45.  doi: 10.1016/j.jcp.2012.04.050.  Google Scholar

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M. DonatelliM. Mazza and S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equation, J. Comput. Phys., 307 (2016), 262-279.  doi: 10.1016/j.jcp.2015.11.061.  Google Scholar

[4]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection-dispersion equation, Numer. Meth. Part. Diff. Equ., 22 (2006), 558-576.  doi: 10.1002/num.20112.  Google Scholar

[5]

P. FrolkovičD. Logashenko and C. Wehner, Flux-based level-set method for two-phase flows on unstructured grids, Comput. Vis. Sci., 18 (2016), 31-52.  doi: 10.1007/s00791-016-0269-z.  Google Scholar

[6]

P. Frolkovič, Application of level set method for groundwater flow with moving boundary, Adv. Water. Resour., 47 (2012), 56-66.  doi: 10.1016/j.advwatres.2012.06.013.  Google Scholar

[7]

P. FrolkovičK. Mikula and J. Urbán, Semi-implicit finite volume level set method for advective motion of interfaces in normal direction, Appl. Num. Math., 95 (2015), 214-228.  doi: 10.1016/j.apnum.2014.05.011.  Google Scholar

[8]

P. Frolkovič and K. Mikula, Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids, Applied Mathematics and Computation, 329 (2018), 129-142.  doi: 10.1016/j.amc.2018.01.065.  Google Scholar

[9]

A. Golbabai and K. Sayevand, Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain, Math. Comput. Model., 53 (2011), 1708-1718.  doi: 10.1016/j.mcm.2010.12.046.  Google Scholar

[10]

S. Gross and A. Reusken, Numerical Methods for Two-Phase Incompressible Flows, Springer, New York, 2011. doi: 10.1007/978-3-642-19686-7.  Google Scholar

[11]

H. HejaziT. Moroney and F. Liu, Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, J. Comput. Appl. Math., 255 (2014), 684-697.  doi: 10.1016/j.cam.2013.06.039.  Google Scholar

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F. Huang and F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput., 19 (2005), 233-245.   Google Scholar

[13]

C. E. KeesI. AkkermanM. W. Farthing and Y. Bazilevs, A conservative level set method suitable for variable-order approximations and unstructured meshes, J. Comput. Phys., 230 (2011), 4536-4558.  doi: 10.1016/j.jcp.2011.02.030.  Google Scholar

[14]

X.-L. LinM. K. Ng and H.-W. Sun, A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J. Comput. Phys., 336 (2017), 69-86.  doi: 10.1016/j.jcp.2017.02.008.  Google Scholar

[15]

F. LiuV. V. AnhI. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089.  Google Scholar

[16]

F. LiuP. ZhuangV. AnhI. Turner and K. Burra, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12-20.  doi: 10.1016/j.amc.2006.08.162.  Google Scholar

[17]

Q. LiuF. LiuI. Turner and V. Anh, Approximation of the L$\ddot{e}$vy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys., 222 (2007), 57-70.  doi: 10.1016/j.jcp.2006.06.005.  Google Scholar

[18]

R. L. Magin and C. Ingo, Entropy and information in a fractional order model of anomalous diffusion, IFAC Proc., 45 (2012), 428-433.  doi: 10.3182/20120711-3-BE-2027.00063.  Google Scholar

[19]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

[20]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[21]

K. MikulaM. Ohlberger and J. Urbán, Inflow-implicit/outflow-explicit finite volume methods for solving advection equations, Appl. Numer. Math., 85 (2014), 16-37.  doi: 10.1016/j.apnum.2014.06.002.  Google Scholar

[22]

K. Mikula and M. Ohlberger, A new level set method for motion in normal direction based on a semi-implicit forward-backward diffusion approach, SIAM J. Sci. Comp., 32 (2010), 1527-1544.  doi: 10.1137/09075946X.  Google Scholar

[23]

S. T. Mohyud-Din, T. Akram, M. Abbas, A. I. Ismail and N. H. M. Ali, A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection-diffusion equation, Adv. Differ. Equ., (2018), 109. doi: 10.1186/s13662-018-1537-7.  Google Scholar

[24]

S. Momani and Z. Odibat, Numerical solutions of the space-time fractional advection-dispersion equation, Numer. Meth. Part. Differ. Equat., 24 (2008), 1416-1429.  doi: 10.1002/num.20324.  Google Scholar

[25]

R. A. Mundewadirk and S. Kumbinarasaiah, Numerical solution of Abel's integral equations using Hermite wavelet, Applied Mathematics and Nonlinear Sciences, 4 (2019), 169-180.  doi: 10.2478/AMNS.2019.1.00017.  Google Scholar

[26]

Y. Povstenko and T. Kyrylych, Two approaches to obtaining the space-time fractional advection-diffusion Equation, Entropy, 19 (2017), 297. doi: 10.3390/e19070297.  Google Scholar

[27]

S. Arshad, D. Baleanu, J. Huang, M. M. Al Qurashi, Y. Tang adn Y. Zhao, Finite difference method for time-space fractional advection-diffusion equations with riesz derivative, Entropy, 20 (2018), 321. doi: 10.3390/e20050321.  Google Scholar

[28]

N. K. Tripathi, S. Das, S. H. Ong, H. Jafari and M. A. Qurashi, Solution of higher order nonlinear time-fractional reaction diffusion equation, Entropy, 18 (2016), 329. doi: 10.3390/e18090329.  Google Scholar

[29]

Y. WangS. Simakhina and M. Sussman, A hybrid level set-volume constraint method for incompressible two-phase flow, J. Comp. Phys., 231 (2012), 6438-6471.  doi: 10.1016/j.jcp.2012.06.014.  Google Scholar

[30]

A. Yokus and S. Gülbahar, Numerical solutions with linearization techniques of the fractional Harry Dym equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 35-42.  doi: 10.2478/AMNS.2019.1.00004.  Google Scholar

[31]

Q. Zhang, Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis, Applied Mathematics and Nonlinear Sciences, 4 (2019), 433-444.  doi: 10.2478/AMNS.2019.2.00041.  Google Scholar

[32]

G. H. Zheng and T. Wei, Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. Comput. Appl. Math., 233 (2010), 2631-2640.  doi: 10.1016/j.cam.2009.11.009.  Google Scholar

show all references

References:
[1]

W. Bu, X. Liu, Y. Tang and Y. Jiang, Finite element multigrid method for multi-term time fractional advection-diffusion equations, Int. J. Model. Simul. Sci. Comput., 6 (2015), 154001. doi: 10.1142/S1793962315400012.  Google Scholar

[2]

A. R. Carella and C. A. Dorao, Least-squares spectral method for the solution of a fractional advection-dispersion equation, J. Comput. Phys., 232 (2013), 33-45.  doi: 10.1016/j.jcp.2012.04.050.  Google Scholar

[3]

M. DonatelliM. Mazza and S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equation, J. Comput. Phys., 307 (2016), 262-279.  doi: 10.1016/j.jcp.2015.11.061.  Google Scholar

[4]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection-dispersion equation, Numer. Meth. Part. Diff. Equ., 22 (2006), 558-576.  doi: 10.1002/num.20112.  Google Scholar

[5]

P. FrolkovičD. Logashenko and C. Wehner, Flux-based level-set method for two-phase flows on unstructured grids, Comput. Vis. Sci., 18 (2016), 31-52.  doi: 10.1007/s00791-016-0269-z.  Google Scholar

[6]

P. Frolkovič, Application of level set method for groundwater flow with moving boundary, Adv. Water. Resour., 47 (2012), 56-66.  doi: 10.1016/j.advwatres.2012.06.013.  Google Scholar

[7]

P. FrolkovičK. Mikula and J. Urbán, Semi-implicit finite volume level set method for advective motion of interfaces in normal direction, Appl. Num. Math., 95 (2015), 214-228.  doi: 10.1016/j.apnum.2014.05.011.  Google Scholar

[8]

P. Frolkovič and K. Mikula, Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids, Applied Mathematics and Computation, 329 (2018), 129-142.  doi: 10.1016/j.amc.2018.01.065.  Google Scholar

[9]

A. Golbabai and K. Sayevand, Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain, Math. Comput. Model., 53 (2011), 1708-1718.  doi: 10.1016/j.mcm.2010.12.046.  Google Scholar

[10]

S. Gross and A. Reusken, Numerical Methods for Two-Phase Incompressible Flows, Springer, New York, 2011. doi: 10.1007/978-3-642-19686-7.  Google Scholar

[11]

H. HejaziT. Moroney and F. Liu, Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, J. Comput. Appl. Math., 255 (2014), 684-697.  doi: 10.1016/j.cam.2013.06.039.  Google Scholar

[12]

F. Huang and F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput., 19 (2005), 233-245.   Google Scholar

[13]

C. E. KeesI. AkkermanM. W. Farthing and Y. Bazilevs, A conservative level set method suitable for variable-order approximations and unstructured meshes, J. Comput. Phys., 230 (2011), 4536-4558.  doi: 10.1016/j.jcp.2011.02.030.  Google Scholar

[14]

X.-L. LinM. K. Ng and H.-W. Sun, A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J. Comput. Phys., 336 (2017), 69-86.  doi: 10.1016/j.jcp.2017.02.008.  Google Scholar

[15]

F. LiuV. V. AnhI. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089.  Google Scholar

[16]

F. LiuP. ZhuangV. AnhI. Turner and K. Burra, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12-20.  doi: 10.1016/j.amc.2006.08.162.  Google Scholar

[17]

Q. LiuF. LiuI. Turner and V. Anh, Approximation of the L$\ddot{e}$vy-Feller advection-dispersion process by random walk and finite difference method, J. Comput. Phys., 222 (2007), 57-70.  doi: 10.1016/j.jcp.2006.06.005.  Google Scholar

[18]

R. L. Magin and C. Ingo, Entropy and information in a fractional order model of anomalous diffusion, IFAC Proc., 45 (2012), 428-433.  doi: 10.3182/20120711-3-BE-2027.00063.  Google Scholar

[19]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

[20]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[21]

K. MikulaM. Ohlberger and J. Urbán, Inflow-implicit/outflow-explicit finite volume methods for solving advection equations, Appl. Numer. Math., 85 (2014), 16-37.  doi: 10.1016/j.apnum.2014.06.002.  Google Scholar

[22]

K. Mikula and M. Ohlberger, A new level set method for motion in normal direction based on a semi-implicit forward-backward diffusion approach, SIAM J. Sci. Comp., 32 (2010), 1527-1544.  doi: 10.1137/09075946X.  Google Scholar

[23]

S. T. Mohyud-Din, T. Akram, M. Abbas, A. I. Ismail and N. H. M. Ali, A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection-diffusion equation, Adv. Differ. Equ., (2018), 109. doi: 10.1186/s13662-018-1537-7.  Google Scholar

[24]

S. Momani and Z. Odibat, Numerical solutions of the space-time fractional advection-dispersion equation, Numer. Meth. Part. Differ. Equat., 24 (2008), 1416-1429.  doi: 10.1002/num.20324.  Google Scholar

[25]

R. A. Mundewadirk and S. Kumbinarasaiah, Numerical solution of Abel's integral equations using Hermite wavelet, Applied Mathematics and Nonlinear Sciences, 4 (2019), 169-180.  doi: 10.2478/AMNS.2019.1.00017.  Google Scholar

[26]

Y. Povstenko and T. Kyrylych, Two approaches to obtaining the space-time fractional advection-diffusion Equation, Entropy, 19 (2017), 297. doi: 10.3390/e19070297.  Google Scholar

[27]

S. Arshad, D. Baleanu, J. Huang, M. M. Al Qurashi, Y. Tang adn Y. Zhao, Finite difference method for time-space fractional advection-diffusion equations with riesz derivative, Entropy, 20 (2018), 321. doi: 10.3390/e20050321.  Google Scholar

[28]

N. K. Tripathi, S. Das, S. H. Ong, H. Jafari and M. A. Qurashi, Solution of higher order nonlinear time-fractional reaction diffusion equation, Entropy, 18 (2016), 329. doi: 10.3390/e18090329.  Google Scholar

[29]

Y. WangS. Simakhina and M. Sussman, A hybrid level set-volume constraint method for incompressible two-phase flow, J. Comp. Phys., 231 (2012), 6438-6471.  doi: 10.1016/j.jcp.2012.06.014.  Google Scholar

[30]

A. Yokus and S. Gülbahar, Numerical solutions with linearization techniques of the fractional Harry Dym equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 35-42.  doi: 10.2478/AMNS.2019.1.00004.  Google Scholar

[31]

Q. Zhang, Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis, Applied Mathematics and Nonlinear Sciences, 4 (2019), 433-444.  doi: 10.2478/AMNS.2019.2.00041.  Google Scholar

[32]

G. H. Zheng and T. Wei, Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. Comput. Appl. Math., 233 (2010), 2631-2640.  doi: 10.1016/j.cam.2009.11.009.  Google Scholar

Figure 1.  Uniform type-2 triangulation, m = 4, n = 4
Figure 2.  A locally supported spline
Figure 3.  (a) Corner B-spline Basis (b)Side B-spline Basis Interior (c)B-spline Basis
Figure 4.  $ \varepsilon = 1 $, (a)$ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $
Figure 5.  $ \varepsilon = 1 $, (a)$ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $
Figure 6.  $ \varepsilon = 1 $, (a)$ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $
Figure 7.  $ \varepsilon = 1 $, (a)$ u(0.7) $, (b)$ \hat{u}(0.7) $, (c)$ \tilde{u}(0.7) $
Figure 8.  $ \varepsilon = 2 $, (a)$ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $
Figure 9.  $ \varepsilon = 2 $, (a)$ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $
Figure 10.  $ \varepsilon = 2 $, (a)$ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $
Figure 11.  $ \varepsilon = 2 $, (a)$ u(0.7) $, (b)$ \hat{u}(0.7) $, (c)$ \tilde{u}(0.7) $
Figure 12.  (a) $ u(0.1) $, (b)$ \hat{u}(0.1) $, (c)$ \tilde{u}(0.1) $
Figure 13.  (a) $ u(0.2) $, (b)$ \hat{u}(0.2) $, (c)$ \tilde{u}(0.2) $
Figure 14.  (a) $ u(0.3) $, (b)$ \hat{u}(0.3) $, (c)$ \tilde{u}(0.3) $
Figure 15.  (a) $ u(0.4) $, (b)$ \hat{u}(0.4) $, (c)$ \tilde{u}(0.4) $
Figure 16.  (a) $ u(0.5) $, (b)$ \hat{u}(0.5) $, (c)$ \tilde{u}(0.5) $
Table 1.  Comparison of numerical and exact solutions of Example 1
Spline method Finite element method
$ \varepsilon=1 $ $ t=0.1 $ 3.774902e-005 6.416033e-005
$ \varepsilon=1 $ $ t=0.3 $ 2.721077e-005 2.347618e-004
$ \varepsilon=1 $ $ t=0.5 $ 6.327942e-005 7.128474e-004
$ \varepsilon=1 $ $ t=0.7 $ 6.323704e-005 2.739811e-004
$ \varepsilon=2 $ $ t=0.1 $ 3.573924e-004 2.159467e-003
$ \varepsilon=2 $ $ t=0.3 $ 5.858324e-004 4.492941e-003
$ \varepsilon=2 $ $ t=0.5 $ 8.340485e-004 6.032486e-003
$ \varepsilon=2 $ $ t=0.7 $ 7.448253e-004 5.265392e-003
Spline method Finite element method
$ \varepsilon=1 $ $ t=0.1 $ 3.774902e-005 6.416033e-005
$ \varepsilon=1 $ $ t=0.3 $ 2.721077e-005 2.347618e-004
$ \varepsilon=1 $ $ t=0.5 $ 6.327942e-005 7.128474e-004
$ \varepsilon=1 $ $ t=0.7 $ 6.323704e-005 2.739811e-004
$ \varepsilon=2 $ $ t=0.1 $ 3.573924e-004 2.159467e-003
$ \varepsilon=2 $ $ t=0.3 $ 5.858324e-004 4.492941e-003
$ \varepsilon=2 $ $ t=0.5 $ 8.340485e-004 6.032486e-003
$ \varepsilon=2 $ $ t=0.7 $ 7.448253e-004 5.265392e-003
Table 2.  Comparison of numerical and exact solutions of Example 2
Spline method Finite element method
$ t=0.1 $ 4.537264e-006 5.276482e-005
$ t=0.2 $ 4.822438e-006 4.653391e-005
$ t=0.3 $ 5.645882e-006 5.283764e-005
$ t=0.4 $ 5.326159e-006 7.563822e-005
$ t=0.5 $ 7.438625e-006 6.435764e-005
Spline method Finite element method
$ t=0.1 $ 4.537264e-006 5.276482e-005
$ t=0.2 $ 4.822438e-006 4.653391e-005
$ t=0.3 $ 5.645882e-006 5.283764e-005
$ t=0.4 $ 5.326159e-006 7.563822e-005
$ t=0.5 $ 7.438625e-006 6.435764e-005
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