Discontinuous spectral element methods for time-and space-fractional advection equations

SIAM Journal on Scientific Computing

Volumn 36, Issue 4, 2014, Pages B684-B707

  Zayernouri, Mohsen ^a   Karniadakis, George Em ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Fractional basis functions; Fractional pdes; Jacobi polyfractonomials; Spectral convergence

Indexed keywords

ADVECTION; CONVERGENCE OF NUMERICAL METHODS; GALERKIN METHODS; INTEGRATION; POLYNOMIALS; TENSORS; TESTING;

ADVECTION EQUATIONS; BASE FUNCTION; FRACTIONAL BASE FUNCTION; FRACTIONAL PDES; JACOBI POLYFRACTONOMIALS; SPECTRAL CONVERGENCE; SPECTRAL METHODS; SPECTRAL/HP ELEMENT METHOD; STURM-LIOUVILLE PROBLEM; TIME-INTEGRATION;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84986922240     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/130940967     Document Type: Article

References (43)

1. R. Askey and J. Fitch, Integral representations for Jacobi polynomials and some applications, J. Math. Anal. Appl., 26 (1969), pp. 411-437.
2. E. Barkai, R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132.
3. L. Blank, Numerical Treatment of Differential Equations of Fractional Order, Citeseer, 1996.
4. J. Cao and C. Xu, A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comput. Phys., 238 (2013), pp. 154-168.
5. A. R. Carella, Spectral Finite Element Methods for solving Fractional Differential Equations with applications in Anomalous Transport, Ph.D. thesis, Norwegian University of Science and Technology, 2012.
6. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continum Mechanics, Springer-Verlag, Berlin, 1998.
7. W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech., 18 (1964), pp. 44-64.
8. W. H. Deng and J. S. Hesthaven, Local discontinuous Galerkin methods for fractional diffusion equations, ESAIM: Math. Model. Numer. Anal., 47 (2013), pp. 1845-1864.
9. K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), pp. 229-248.
10. G. J. Fix and J. P. Roop, Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48 (2004), pp. 1017-1033.
11. B. Gustafsson, H. O. Kreiss, and J. Oliger, Time-Dependent Problems and Difference Methods, Pure and Appl. Methods 67, Jhon Wiley, New York, 1995.
12. E. Hanert, A comparison of three Eulerian numerical methods for fractional-order transport models, Environ. Fluid Mech., 10 (2010), pp. 7-20.
13. B. Henry and S. Wearne, Fractional reaction-diffusion, Phys. A, 276 (2000), pp. 448-455.
14. J. S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge Moriogr. Appl. Comput. Math. 21, Cambridge University Press, Cambridge, UK, 2007.
15. I. M. Jaimoukha and E. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), pp. 227-251.
16. G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD, 2nded., Oxford University Press, New York, 2005.
17. J. J. Keller, Propagation of simple non-linear waves in gas filled tubes with friction, Z.Angew. Math. Phys., 32 (1981), pp. 170-181.
18. M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 2535-2542.
19. M. M. Khader and A. S. Hendy, The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, Internat. J. Pure Appl. Math., 74 (2012), pp. 287-297.
20. V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Engrg., 124 (2002), pp. 803-806.
21. T. Langlands and B. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719-736.
22. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math. 31, Cambridge University Press, Cambridge, UK, 2002.
23. X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108-2131.
24. X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016-1051.
25. Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533-1552.
26. W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms, 52 (2009), pp. 69-88.
27. M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, DeGruyter Stud. Math. 43 de Gruyter, Berlin, 2012.
28. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 1993.
29. T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21 (1999), pp. 1401-1418.
30. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
31. E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176 (2006), pp. 1-6.
32. W. H. Reed and T. R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM, 1973.
33. J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in r2, J. Comput. Appl. Math., 193 (2006), pp. 243-268.
34. Y. Saad, Numerical Solution of Large Lyapunov Equations, Research Institute for Advanced Computer Science, NASA Ames Research Center, 1989.
35. N. Sugimoto, Burgers equation with a fractional derivative: Hereditary effects on nonlinear acoustic waves, J. Fluid Mech, 225 (1991), pp. 631-653.
36. N. Sugimoto and T. Kakutani, Generalized Burgers' equation for nonlinear viscoelastic waves, Wave Motion, 7 (1985), pp. 447-458.
37. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), pp. 193-209.
38. E. L. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), pp. 87-90.
39. Q. Xu and J. S. Hesthaven, Discontinuous Galerkin Method for Fractional Convection-Diffusion Equations, arXiv:1304.6047, 2013.
40. M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximations, J. Comput. Phys., 47 (2013), pp. 2108-2131.
41. M. Zayernouri and G. E. Karniadakis, Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257 (2014), pp. 460-480.
42. M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A40-A62.
43. O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, London, 2005.