ORIGINAL PAPER

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations

Expand
  • 1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran;
    2 EPFL-SB-MATHICES-MCSS, École Polytechnique Fédéral de Lausanne, 1015 Lausanne, Switzerland

Received date: 2019-07-22

  Revised date: 2020-03-10

  Online published: 2020-09-11

Abstract

For two-dimensional (2D) time fractional difusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a fnite diference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the efectiveness and accuracy of the method and confrm the analysis.

Cite this article

Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven . A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations[J]. Communications on Applied Mathematics and Computation, 2020 , 2(4) : 689 -709 . DOI: 10.1007/s42967-020-00065-7

References

1. Basu, T.S., Wang, H.:A fast second-order fnite diference method for space-fractional difusion equations. Int. J. Numer. Anal. Model. 9, 658-666 (2012)
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. Cockburn, B., Dong, B.:An analysis of the minimal dissipation local discontinuous Galerkin method for convection-difusion problems. J. Sci. Comput. 32, 233-262 (2007)
4. Cockburn, B., Kanschat, G., Perugia, I., Schotzau, D.:Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264-285 (2002)
5. Cockburn, B., Shu, C.W.:The local discontinuous Galerkin method for time-dependent convection-difusion systems. SIAM J. Numer. Anal. 35, 2440-2463 (1998)
6. Cui, M.R.:Compact fnite diference method for the fractional difusion equation. J. Comput. Phys. 228, 7792-7804 (2009)
7. Deng, W.H., Hesthaven, J.S.:Local discontinuous Galerkin methods for fractional difusion equations. Math. Model. Numer. Anal. 47, 1845-1864 (2013)
8. Deng, W.H., Hesthaven, J.S.:Local discontinuous Galerkin methods for fractional ordinary diferential equations. BIT Numer. Math. 55, 967-985 (2015)
9. Ding, H.F., Li, C.P.:Mixed spline function method for reaction-subdifusion equation. J. Comput. Phys. 242, 103-123 (2013)
10. Dong, B., Shu, C.W.:Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems. SIAM J. Numer. Anal. 47, 3240-3268 (2009)
11. Du, R., Cao, W.R., Sun, Z.Z.:A compact diference scheme for the fractional difusion-wave equation. Appl. Math. Model. 34, 2998-3007 (2010)
12. Ervin, V.J., Heuer, N., Roop, J.P.:Numerical approximation of a time dependent, nonlinear, spacefractional difusion equation. SIAM J. Numer. Anal. 45, 572-591 (2007)
13. Eshaghi, J., Kazem, S., Adibi, H.:The local discontinuous Galerkin method for 2D nonlinear time-fractional advection-difusion equations. Eng. Comput. 35, 1317-1332 (2019). https://doi.org/10.1007/s00366-018-0665-8
14. Fix, G., Roop, J.:Least squares fnite element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017-1033 (2004)
15. Gao, G.H., Sun, Z.Z.:A compact fnite diference scheme for the fractional sub-difusion equations. J. Comput. Phys. 230, 586-595 (2011)
16. Gao, G.H., Sun, Z., Zhang, H.:A new fractional numerical diferentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33-50 (2014)
17. He, J.H., Wu, X.H.:Variational iteration method:new development and applications. Comput. Math. Appl. 54, 881-894 (2007)
18. Huang, C., An, N., Yu, X.:A fully discrete direct discontinuous Galerkin method for the fractional difusion-wave equation. Appl. Anal. 97, 659-675 (2018)
19. Jiang, Y., Ma, J.:High-order fnite element methods for time-fractional partial diferential equations. J. Comput. Appl. Math. 235, 3285-3290 (2011)
20. Jin, B., Lazarov, R., Liu, Y., Zhou, Z.:The Galerkin fnite element method for a multi-term timefractional difusion equation. J. Comput. Phys. 281, 825-843 (2015)
21. Karaa, S., Mustapha, K., Pani, Amiya K.:Finite volume element method for two-dimensional fractional subdifusion problems. IMA J. Numer. Anal. 37, 945-964 (2017)
22. Kochubei, A., Luchko, Y. (eds.) Handbook of Fractional Calculus with Applications, vol. 1. Basic Theory. Walter de Gruyter GmbH, Berlin (2019)
23. Li, C.Z., Chen, Y.:Numerical approximation of nonlinear fractional diferential equations with subdifusion and superdifusion. Comput. Math. Appl. 62, 855-875 (2011)
24. Li, X.J., Xu, C.J.:A space-time spectral method for the time fractional difusion equation. SIAM J. Numer. Anal. 47, 2108-2131 (2009)
25. Li, C.P., Zeng, F.H.:The fnite diference methods for fractional ordinary diferential equations. Numer. Funct. Anal. Optim. 34, 149-179 (2013)
26. Lin, Y.M., Xu, C.J.:Finite diference/spectral approximations for the time-fractional difusion equation. J. Comput. Phys. 225, 1533-1552 (2007)
27. Liu, Y., Shu, C.W., Zhang, M.:Superconvergence of energy-conserving discontinuous Galerkin methods for linear hyperbolic equations. Commun. Appl. Math. Comput. 1, 101-116 (2019)
28. 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)
29. Meerschaert, M.M., Tadjeran, C.:Finite diference approximations for fractional advection-dispersion. J. Comput. Appl. Math. 172, 65-77 (2004)
30. Metzler, R., Klafter, J.:The random walk's guide to anomalous difusion:a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000)
31. Miller, K.S., Ross, B.:An Introduction to the Fractional Calculus and Fractional Diferential Equations. Wiley, New York (1993)
32. Mokhtari, R., Mostajeran, F.:A high order formula to approximate the Caputo fractional derivative. Commun. Appl. Math. Comput. 2, 1-29 (2020)
33. Oldham, K., Spanier, J.:The Fractional Calculus:Theory and Applications of Diferentiation and Integration of Arbitray Order. Academic Press, New York (1974)
34. Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y.Q., Jara, B.M.V.:Matrix approach to discrete fractional calculus II:partial fractional diferential equations. J. Comput. Phys. 228, 3137-3153 (2009)
35. Qiu, L., Deng, W., Hesthaven, J.S.:Nodal discontinuous Galerkin methods for fractional difusion equations on 2D domain with triangular meshes. J. Comput. Phys. 298, 678-694 (2015)
36. Roop, J.P.:Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J. Comput. Appl. Math. 193, 243-268 (2006)
37. Wang, H., Zhang, Q., Wang, S., Shu, C.W.:Local discontinuous Galerkin methods with explicitimplicit-null time discretizations for solving nonlinear difusion problems. Sci. China Math. 63(1), 183-204 (2020). https://doi.org/10.1007/s11425-018-9524-x
38. Wang, H., Zhang, Q., Shu, C.W.:Third order implicit-explicit Runge-Kutta local discontinuous Galerkin methods with suitable boundary treatment for convection-difusion problems with Dirichlet boundary conditions. J. Comput. Appl. Math. 342, 164-179 (2018)
39. Xu, Q., Zheng, Z.:Discontinuous Galerkin method for time fractional difusion equation. J. Inf. Comput. Sci. 11, 3253-3264 (2013)
40. Yang, Q.Q., Turner, I., Liu, F., Ilic, M.:Novel numerical methods for solving the time space fractional difusion equation in two dimensions. SIAM J. Sci. Comput. 33, 1159-1180 (2011)
41. Yeganeh, S., Mokhtari, R., Fouladi, S.:Using an LDG method for solving an inverse source problem of the time-fractional difusion equation. Iran. J. Numer. Anal. Optim. 7, 115-135 (2017)
42. Yeganeh, S., Mokhtari, R., Hesthaven, J.S.:Space-dependent source determination in a time-fractional difusion equation using a local discontinuous Galerkin method. BIT Numer. Math. 57, 685-707 (2017)
43. Zhang, X., Tang, B., He, Y.:Homotopy analysis method for higher-order fractional integro-diferential equations. Comput. Math. Appl. 62, 3194-3203 (2011)
44. Zhao, Y., Chen, P., Bu, W., Liu, X., Tang, Y.:Two mixed fnite element methods for time-fractional difusion equations. J. Sci. Comput. 70, 407-428 (2017)
45. Zhuang, P., Liu, F.:Finite diference approximation for two-dimensional time fractional difusion equation. J. Algorithm Comput. Tech. 1, 1-15 (2007)
Outlines

/