Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation

doi:10.1007/s42967-020-00060-y

摘要/Abstract

摘要： The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-difusion equation. The time variable has been discretized by a second-order fnite diference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confrm the theoretical results.

关键词: Spectral method, Finite diference method, Fractional advection-difusion equation, Galerkin weak form, Unconditional stability

Abstract: The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-difusion equation. The time variable has been discretized by a second-order fnite diference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confrm the theoretical results.

Key words: Spectral method, Finite diference method, Fractional advection-difusion equation, Galerkin weak form, Unconditional stability

中图分类号:

Mostafa Abbaszadeh, Hanieh Amjadian. Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 653-669.

参考文献

1. Abbaszadeh, M.:Error estimate of second-order fnite diference scheme for solving the Riesz space distributed-order difusion equation. Appl. Math. Lett. 88, 179-185 (2019)
2. Abbaszadeh, M., Dehghan, M.:An improved meshless method for solving two-dimensional distributed order time-fractional difusion-wave equation with error estimate. Numer. Algor. 75(1), 173-211 (2017)
3. Abbaszadeh, M., Dehghan, M.:Numerical and analytical investigations for neutral delay fractional damped difusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method. Appl. Numer. Math. 145, 488-506 (2019)
4. Abdelkawy, M., Zaky, M., Bhrawy, A., Baleanu, D.:Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model. Rom. Rep. Phys. 67(3), 773-791 (2015)
5. Bhrawy, A.H., Baleanu, D.:A spectral Legendre-Gauss-Lobatto collocation method for a spacefractional advection-difusion equations with variable coefcients. Rep. Math. Phys. 72, 219-233 (2013)
6. Bhrawy, A., Zaky, M.:An improved collocation method for multi-dimensional space-time variableorder fractional Schrödinger equations. Appl. Numer. Math. 111, 197-218 (2017)
7. Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.:Numerical solution of the two-sided space-time fractional telegraph equation via Chebyshev tau approximation. J. Optim. Theory Appl. 174(1), 321-341 (2017)
8. Bu, W., Tang, Y., Wu, Y., Yang, J.:Crank-Nicolson ADI Galerkin fnite element method for twodimensional fractional FitzHugh-Nagumo monodomain model. Appl. Math. Comput. 257, 355-364 (2015)
9. Bu, W., Tang, Y., Wu, Y., Yang, J.:Finite diference/fnite element method for two-dimensional space and time fractional Bloch-Torrey equations. J. Comput. Phys. 293, 264-279 (2015)
10. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.:Spectral Methods:Fundamentals in Single Domains. Springer, Berlin (2006)
11. Chen, C.-M., Liu, F., Anh, V., Turner, I.:Numerical simulation for the variable-order Galilei invariant advection difusion equation with a nonlinear source term. Appl. Math. Comput. 217(12), 5729-5742 (2011)
12. Dehghan, M., Abbaszadeh, M.:Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis. Appl. Numer. Math. 119, 51-66 (2017)
13. Dehghan, M., Abbaszadeh, M.:Error estimate of fnite element/fnite diference technique for solution of two-dimensional weakly singular integro-partial diferential equation with space and time fractional derivatives. J. Comput. Appl. Math. 356, 314-328 (2019)
14. Dehghan, M., Sabouri, M.:A spectral element method for solving the Pennes bioheat transfer equation by using triangular and quadrilateral elements. Appl. Math. Model. 36, 6031-6049 (2012)
15. Dehghan, M., Sabouri, M.:A Legendre spectral element method on a large spatial domain to solve the predator-prey system modeling interacting populations. Appl. Math. Model. 37, 1028-1038 (2013)
16. Dehghan, M., Abbaszadeh, M., Deng, W.:Fourth-order numerical method for the space-time tempered fractional difusion-wave equation. Appl. Math. Lett. 73, 120-127 (2017)
17. Deng, K., Chen, M., Sun, T.:A weighted numerical algorithm for two and three dimensional twosided space fractional wave equations. Appl. Math. Comput. 257, 264-273 (2015)
18. Deville, M.O., Fischer, P.F., Fischer, P.F., Mund, E., et al.:High-Order Methods for Incompressible Fluid Flow, vol. 9. Cambridge University Press, Cambridge (2002)
19. Ding, H.:A high-order numerical algorithm for two-dimensional time-space tempered fractional difusion-wave equation. Appl. Numer. Math. 135, 30-46 (2019)
20. Ding, H., Li, C.:High-order algorithms for Riesz derivative and their applications (iii). Fract. Calc. Appl. Anal. 19(1), 19-55 (2016)
21. Ding, H., Li, C.P.:A high-order algorithm for time-Caputo-tempered partial diferential equation with Riesz derivatives in two spatial dimensions. J. Sci. Comput. 80, 81-109 (2019)
22. Ding, H., Li, C., Chen, Y.:High-order algorithms for Riesz derivative and their applications. J. Comput. Phys. 293, 218-237 (2015)
23. Fakhar-Izadi, F., Dehghan, M.:The spectral methods for parabolic Volterra integro-diferential equations. J. Comput. Appl. Math. 235, 4032-4046 (2011)
24. Giraldo, F.X.:Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations. Comput. Math. Appl. 45, 97-121 (2003)
25. Hafez, R.M., Youssri, Y.H.:Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation. Comput. Appl. Math. 37, 5315-5333 (2018)
26. Khader, M.M., Sweilam, N.H.:Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method. Comput. Appl. Math. 33, 739-750 (2014)
27. Li, X., Xu, C.:A space-time spectral method for the time fractional difusion equation. SIAM J. Numer. Anal. 47(3), 2108-2131 (2009)
28. Li, C., Zeng, F.:Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, Boca Raton (2015)
29. Li, C.P., Zeng, F., Liu, F.:Spectral approximations to the fractional integral and derivative. Fract. Calcul. Appl. Anal. 15, 383-406 (2012)
30. Li, H., Cao, J., Li, C.:High-order approximation to Caputo derivatives and Caputo-type advectiondifusion equations. J. Comput. Appl. Math. 299, 159-175 (2016)
31. Li, M., Huang, C., Ming, W.:Mixed fnite-element method for multi-term time-fractional difusion and difusion-wave equations. Comput. Appl. Math. 37, 2309-2334 (2018)
32. Li, C., Deng, W., Zhao, L.:Well-posedness and numerical algorithm for the tempered fractional diferential equations. Discrete Cont. Dyn. Syst. B 24(4), 1989-2015 (2019)
33. Lin, Y., Xu, C.:Finite diference/spectral approximations for the time-fractional difusion equation. J. Comput. Phys. 225, 1533-1552 (2007)
34. Lin, Y., Li, X., Xu, C.:Finite diference/spectral approximations for the fractional cable equation. Math. Comput. 80, 1369-1396 (2011)
35. Maerschalck, B. D.:Space-time least-squares spectral element method for unsteady fows application and valuation linear and non-linear hyperbolic scalar equations, Master Thesis, Department of Aerospace Engineering at Delft University of Technology (February 28, 2003)
36. Moghaddam, B.P., Tenreiro Machado, J.A., Morgado, M.L.:Numerical approach for a class of distributed order time fractional partial diferential equations. Appl. Numer. Math. 136, 152-162 (2019)
37. Osman, S.A., Langlands, T.A.M.:An implicit Keller Box numerical scheme for the solution of fractional subdifusion equations. Appl. Math. Comput. 348, 609-626 (2019)
38. Pandey, P., Kumar, S., Das, S.:Approximate analytical solution of coupled fractional order reactionadvection-difusion equations. Euro. Phys. J. Plus 134, 364 (2019). https://doi.org/10.1140/epjp/i2019-12727-6
39. Quarteroni, A., Valli, A.:Numerical Approximation of Partial Diferential Equations. Springer-Verlag, New York (1997)
40. Shen, J.:Efcient spectral-Galerkin method i. direct solvers for second-and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489-1505 (2020). https://doi.org/10.1137/0915089
41. Tian, W., Zhou, H., Deng, W.:A class of second order diference approximations for solving space fractional difusion equations. Math. Comput. 84(294), 1703-1727 (2015)
42. Wang, Z., Vong, S.:Compact diference schemes for the modifed anomalous fractional sub-difusion equation and the fractional difusion-wave equation. J. Comput. Phys. 277, 1-15 (2014)
43. Wang, T., Guo, B., Zhang, L.:New conservative diference schemes for a coupled nonlinear Schrödinger system. Appl. Math. Comput. 217, 1604-1619 (2010)
44. Wu, X., Deng, W., Barkai, E.:Tempered fractional Feynman-Kac equation, arXiv preprint arXiv:1602.00071
45. Yuttanan, B., Razzaghi, M.:Legendre wavelets approach for numerical solutions of distributed order fractional diferential equations. Appl. Math. Model. 70, 350-364 (2019)
46. Zaky, M.A., Ameen, I.G.:A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and Volterra-Fredholm integral equations with smooth solutions. Numer. Algor. (2019). https://doi.org/10.1007/s11075-019-00743-5
47. Zaky, M., Baleanu, D., Alzaidy, J., Hashemizadeh, E.:Operational matrix approach for solving the variable-order nonlinear Galilei invariant advection-difusion equation. Adv. Difer. Equ. 2018(1), 102 (2018)
48. Zayernouri, M., Karniadakis, G.E.:Fractional Sturm-Liouville eigen-problems:theory and numerical approximation. J. Comput. Phys. 252, 495-517 (2013)
49. Zayernouri, M., Karniadakis, G.E.:Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. 293, 312-338 (2015)
50. Zeng, F., Ma, H., Zhao, T.:Alternating direction implicit Legendre spectral element method for Schrödinger equations. J. Shanghai Univ. (Nat. Sci. Edition) 60(6), 724-727 (2011)
51. Zhang, G., Huang, C., Li, M.:A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations. Euro. Phys. J. Plus 133, 155 (2018). https://doi.org/10.1140/epjp/i2018-11982-3
52. Zheng, M., Liu, F., Turner, I., Anh, V.:A novel high-order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J. Sci. Comput. 37, A701-A724 (2015)
53. Zhu, W., Kopriva, D.A.:A spectral element method to price European options, I. Single asset with and without jump difusion. J. Sci. Comput. 39, 222-243 (2009)
54. Zhu, W., Kopriva, D.A.:A spectral element approximation to price European options with one asset and stochastic volatility. J. Sci. Comput 42, 426-446 (2010)

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