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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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