Numerical Algorithms

, Volume 77, Issue 3, pp 675–690

# Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation

Original Paper

## Abstract

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)3−α ), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(h k+1 + (Δt)3−α ), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis.

### Keywords

Fractional Cattaneo equation Time fractional derivative Local discontinuous Galerkin method Stability

## Notes

### Acknowledgments

This work is supported by the High-Level Personal Foundation of Henan University of Technology (2013BS041), Plan For Scientific Innovation Talent of Henan University of Technology (2013CXRC12), and the National Natural Science Foundation of China (11461072, 11426090), Foundation of Henan Educational Committee (15A110018), and China Postdoctoral Science Foundation funded project (2015M572115).

### References

1. 1.
Basu, T.S., Wang, H.: A fast second-order finite difference method for space-fractional diffusion equations. Int. J. Numer. Anal. Model. 9, 658–666 (2012)
2. 2.
Carella, A.R., Dorao, C.A.: Least-squares spectral method for the solution of a fractional advection-dispersion equation. J. Comput. Phys. 232, 33–45 (2013)
3. 3.
Chen, W., Ye, L., Sun, H.: Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, 1614–1620 (2010)
4. 4.
Chen, W., Zhang, J., Zhang, J.: A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures. Fract. Calc. Appl. Anal. 16, 76–92 (2013)
5. 5.
Cui, M.R.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)
6. 6.
Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A: Math. Gen. 30, 7277–7289 (1997)
7. 7.
Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)
8. 8.
Ding, H.F., Li, C.P.: Mixed spline function method for reaction-subdiffusion equation. J. Comput. Phys. 242, 103–123 (2013)
9. 9.
Du, R., Cao, W.R., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34, 2998–3007 (2010)
10. 10.
Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)
11. 11.
Fix, G., Roop, J.: Least squares finite element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)
12. 12.
Huang, C., Yu, X., Wang, C., Li, Z., An, N.: A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation. Appl. Math. Comput. 264, 483–492 (2015)
13. 13.
Gao, G.H., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)
14. 14.
He, J.H., Wu, X.H.: Variational iteration method: New development and applications. Comput. Math. Appl. 54, 881–894 (2007)
15. 15.
Hilfer, R. (ed.): Applications of fractional calculus in physics. World Scientific, Singapore (2000)
16. 16.
Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)
17. 17.
Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)
18. 18.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol. 204. Elsevier, Amsterdam, The Netherlands (2006)
19. 19.
Kiryakova, V.: Generalized fractional calculus and applications. Longman & Wiley, Harlow-N. York (1994)
20. 20.
Kiryakova, V.: All the special functions are fractional differintegrals of elementary functions. J. Phys. A: Math. Gen. 30, 5085–5103 (1997)
21. 21.
Kiryakova, V.: Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal. 2, 445–462 (1999)
22. 22.
Kiryakova, V.: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 241–259 (2000)
23. 23.
Kiryakova, V.: The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus. Comput. Math. Appl. 59, 1885–1895 (2010)
24. 24.
Klages, R., Radons, G., Sokolov, I.M. (eds.): Anomalous transport: Foundations and applications. Elsevier, Amsterdam, The Netherlands (2008)Google Scholar
25. 25.
Kosztolowicz, T., Lewandowska, K.D.: Hyperbolic subdiffusive impedance. J. Phys. A: Math. Theor. 42, 055004 (2009)
26. 26.
Lewandowskaw, K.D.: Application of generalized Cattaneo equation to model subdiffusion impedance. Acta. Phys. Pol. B 39, 1211–1220 (2008)Google Scholar
27. 27.
Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)
28. 28.
Li, X.J., Xu, C.J.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)
29. 29.
Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)
30. 30.
Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)
31. 31.
Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
32. 32.
Li, C.P., Zeng, F.H.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149–179 (2013)
33. 33.
Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64, 2990–3007 (2012)
34. 34.
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion. J. Comput. Appl. Math. 172, 65–77 (2004)
35. 35.
Metzler, R., Nonnenmacher, T.F.: Fractional diffusion, waiting-time distributions, and Cattaneo-type equations. Phys. Rev. E. 57, 6409–6414 (1998)
36. 36.
Momani, S., Odibat, Z.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput. Math. Appl. 54, 910–919 (2007)
37. 37.
Podlubny, I.: Fractional differential equations, vol. 198. Academic Press, Calif, USA (1999)
38. 38.
Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y.Q., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys. 228, 3137–3153 (2009)
39. 39.
Povstenko, Y.: Fractional thermoelasticity. Springer, New York (2015)
40. 40.
Sun, H.G., Chen, W., Sze, K.Y.: A semi-discrete finite element method for a class of time-fractional diffusion equations. Philos. Trans. R. Soc. A 371, 1471–2962 (2013)
41. 41.
Shao, L., Feng, X., He, Y.: The local discontinuous Galerkin finite element method for Burger’s equation. Math. Comput. Modell. 54, 2943–2954 (2011)
42. 42.
Wang, H., Wang, K.X., Sircar, T.: A direct O(N l o g 2 N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)
43. 43.
Wang, K., Wang, H.: A fast characteristic finite difference method for fractional advection-diffusion equations. Adv. Water Resour. 34, 810–816 (2011)
44. 44.
Wang, L., Zhou, X., Wei, X.: Heat conduction. Springer, Berlin (2008)
45. 45.
Xu, Y., Shu, C.-W.: Local discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Numer. Anal. 46, 1998–2021 (2008)
46. 46.
Yang, Q.Q., Turner, I., Liu, F., Ilic, M.: Novel numerical methods for solving the timespace fractional diffusion equation in two dimensions. SIAM J. Sci. Comput. 33, 1159–1180 (2011)
47. 47.
Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)
48. 48.
Zhang, Q., Shu, C.-W.: Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data. Numer. Math. 126, 703–740 (2014)
49. 49.
Zhang, Q.: Third order explicit Runge-Kutta discontinuous Galerkin method for linear conservation law with inflow boundary condition. J. Sci. Comput. 46, 294–313 (2011)
50. 50.
Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)
51. 51.
Zhang, X., Tang, B., He, Y.: Homotopy analysis method for higher-order fractional integro-differential equations. Comput. Math. Appl. 62, 3194–3203 (2011)
52. 52.
Zhao, X., Sun, Z.Z.: Compact crank-nicolson schemes for a class of fractional cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 747–771 (2015)
53. 53.
Zheng, Y.Y., Li, C.P., Zhao, Z.G.: A note on the finite element method for the space fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2010)
54. 54.
Wei, L.L., He, Y.N.: Analysis of the fractional Kawahara equation using an implicit fully discrete local discontinuous Galerkin method. Numer. Methods Partial Differ. Eq. 29, 1441–1458 (2013)
55. 55.
Wei, L.L., He, Y.N.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)