High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

doi:10.1007/s42967-019-00058-1

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (4): 613-640.doi: 10.1007/s42967-019-00058-1

• ORIGINAL PAPER • 上一篇

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

Min Zhang¹, Yang Liu², Hong Li²

1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian Province, China;
2 School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

收稿日期:2019-10-21 修回日期:2019-12-09 发布日期:2020-09-11
通讯作者: Yang Liu E-mail:mathliuyang@imu.edu.cn
基金资助:
This work is supported by the National Natural Science Foundation of China (11661058, 11761053), the Natural Science Foundation of Inner Mongolia (2017MS0107), and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07).

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

Min Zhang¹, Yang Liu², Hong Li²

1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian Province, China;
2 School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received:2019-10-21 Revised:2019-12-09 Published:2020-09-11
Contact: Yang Liu E-mail:mathliuyang@imu.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China (11661058, 11761053), the Natural Science Foundation of Inner Mongolia (2017MS0107), and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07).

摘要/Abstract

摘要： In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional difusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with O(h^k+1 + △t²) is derived, where k ≥ 0 denotes the index of the basis function. Extensive numerical results with Q^k(k=0, 1, 2, 3) elements are provided to confrm our theoretical results, which also show that the secondorder convergence rate in time is not impacted by the changed parameter θ.

关键词: Two-dimensional nonlinear fractional difusion equation, High-order LDG method, Second-order θ scheme, Stability and error estimate

Abstract: In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional difusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with O(h^k+1 + △t²) is derived, where k ≥ 0 denotes the index of the basis function. Extensive numerical results with Q^k(k=0, 1, 2, 3) elements are provided to confrm our theoretical results, which also show that the secondorder convergence rate in time is not impacted by the changed parameter θ.

Key words: Two-dimensional nonlinear fractional difusion equation, High-order LDG method, Second-order θ scheme, Stability and error estimate

中图分类号:

65M60
65N30

Min Zhang, Yang Liu, Hong Li. High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 613-640.

参考文献

1. Baleanu D., Diethelm K., Scalas E., Trujillo J.J.:Fractional Calculus:Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientifc Publishing, New York (2012)
2. Cockburn, B., Shu, C.W.:The local discontinuous Galerkin fnite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440-2463 (1998)
3. Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.:Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264-285 (2001)
4. Dehghan, M., Abbaszadeh, M., Mohebbi, A.:Analysis of a meshless method for the time fractional difusion wave equation. Numer. Algor. 73(2), 445-476 (2016)
5. Deng, W.H., Hesthaven, J.S.:Local discontinuous Galerkin methods for fractional difusion equations. ESAIM Math. Model. Numer. Anal. 47, 1845-1864 (2013)
6. Du, Y.W., Liu, Y., Li, H., Fang, Z.C., He, S.:Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial diferential equation. J. Comput. Phys. 344, 108-126 (2017)
7. Feng, L.B., Zhuang, P., Liu, F.W., Turner, I., Gu, Y.T.:Finite element method for space-time fractional difusion equation. Numer. Algor. 72, 749-767 (2016)
8. Gao, F., Qiu, J., Zhang, Q.:Local discontinuous Galerkin fnite element method and error estimates for one class of Sobolev equation. J. Sci. Comput. 41, 436-460 (2009)
9. Gao, G.H., Sun, H.W., Sun, Z.Z.:Stability and convergence of fnite diference schemes for a class of time-fractional sub-difusion equations based on certain superconvergence. J. Comput. Phys. 280, 510-528 (2015)
10. Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.:Fast evaluation of the Caputo fractional derivative and its applications to fractional difusion equations. Commun. Comput. Phys. 21(3), 650-678 (2017)
11. Jin, B.T., Lazarov, R., Liu, Y.K., Zhou, Z.:The Galerkin fnite element method for a multi-term timefractional difusion equation. J. Comput. Phys. 281, 825-843 (2015)
12. Li C.P., Wang Z.:The discontinuous Galerkin fnite element method for Caputo-type nonlinear conservation law. Math. Comput. Simul. 169, 51-73 (2020)
13. Li, C., Liu, S.M.:Local discontinuous Galerkin scheme for space fractional Allen-Cahn equation. Commun. Appl. Math. Comput. 2(1), 73-91 (2020)
14. Li, C.P., Cai, M.:Theory and Numerical Approximation of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
15. Li, C.P., Ding, H.F.:Higher order fnite diference method for the reaction and anomalous-difusion equation. Appl. Math. Model. 38(15/16), 3802-3821 (2014)
16. Li, C.P., Wang, Z.:The local discontinuous Galerkin fnite element methods for Caputo-type partial diferential equations:numerical analysis. Appl. Numer. Math. 140, 1-22 (2019)
17. Li, J.C., Huang, Y.Q., Lin, Y.P.:Developing fnite element methods for Maxwells equations in a Cole-Cole dispersive medium. SIAM J. Sci. Comput. 33(6), 3153-3174 (2011)
18. Lin, Z., Liu, F.W., Wang, D.D., Gu, Y.T.:Reproducing kernel particle method for two-dimensional time-space fractional difusion equations in irregular domains. Eng. Anal. Bound. Elem. 97, 131-143 (2018)
19. Liu, F.W., Zhuang, P., Turner, I., Burrage, K., Anh, V.:A new fractional fnite volume method for solving the fractional difusion equation. Appl. Math. Model. 38, 3871-3878 (2014)
20. Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.:A two-grid mixed fnite element method for a nonlinear fourth-order reaction-difusion problem with time-fractional derivative. Comput. Math. Appl. 70(10), 2474-2492 (2015)
21. Liu, Y., Du, Y.W., Li, H., Wang, J.F.:A two-grid fnite element approximation for a nonlinear time fractional cable equation. Nonlinear Dyn. 85, 2535-2548 (2016)
22. Liu, Y., Zhang, M., Li, H., Li, J.C.:High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdifusion equation. Comput. Math. Appl. 73(6), 1298-1314 (2017)
23. Liu, Z., Cheng, A., Li, X., Wang, H.:A fast solution technique for fnite element discretization of the space-time fractional difusion equation. Appl. Numer. Math. 119, 146-163 (2017)
24. Liu, N., Liu, Y., Li, H., Wang, J.F.:Time second-order fnite diference/fnite element algorithm for nonlinear time-fractional difusion problem with fourth-order derivative term. Comput. Math. Appl. 75(10), 3521-3536 (2018)
25. Liu, Y., Du, Y.W., Li, H., Liu, F.W., Wang, Y.J.:Some second-order θ schemes combined with fnite element method for nonlinear fractional cable equation. Numer. Algor. 80(2), 533-555 (2019). https://doi.org/10.1007/s11075-018-0496-0
26. Machado, J.A.T.:A probabilistic interpretation of the fractional-order diferentiation. Fract. Calc. Appl. Anal. 6(1), 73-80 (2003)
27. Mao, Z., Karniadakis, G.E.:Fractional Burgers equation with nonlinear non-locality:spectral vanishing viscosity and local discontinuous Galerkin methods. J. Comput. Phys. 336, 143-163 (2017)
28. Meerschaert, M.M., Tadjeran, C.:Finite diference approximations for fractional advection-dispersion fow equations. J. Comput. Appl. Math. 172, 65-77 (2004)
29. Moghaddam, P., Machado, J.A.T.:A computational approach for the solution of a class of variableorder fractional integro-diferential equations with weakly singular kernels. Fract. Calc. Appl. Anal. 20(4), 1023-1042 (2017)
30. Mustapha, K., McLean, M.:Piecewise-linear, discontinuous Galerkin method for a fractional difusion equation. Numer. Algor. 56, 159-184 (2011)
31. Podlubny, I.:Fractional Diferential Equations:an Introduction to Fractional Derivatives, Fractional Diferential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, New York (1998)
32. Shi, D.Y., Yang, H.:A new approach of superconvergence analysis for two-dimensional time fractional difusion equation. Comput. Math. Appl. 75(8), 3012-3023 (2018)
33. Sun, H., Sun, Z.Z., Gao, G.H.:Some temporal second order diference schemes for fractional wave equations. Numer. Methods Partial Difer. Equ. 32(3), 970-1001 (2016)
34. Tian, W.Y., Zhou, H., Deng, W.H.:A class of second order diference approximations for solving space fractional difusion equations. Math. Comput. 84, 1703-1727 (2015)
35. Wang, Z., Vong, S.W.:Compact diference schemes for the modifed anomalous fractional subdifusion equation and the fractional difusion-wave equation. J. Comput. Phys. 277, 1-15 (2014)
36. Wang, Y.J., Liu, Y., Li, H., Wang, J.F.:Finite element method combined with second-order time discrete scheme for nonlinear fractional cable equation. Eur. Phys. J. Plus. 131, 61 (2016). https://doi.org/10.1140/epjp/i2016-16061-3
37. Xu, Y., Shu, C.W.:Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1-46 (2010)
38. Yin B.L., Liu Y., Li H., Zhang Z.M.:Two families of novel second-order fractional numerical formulas and their applications to fractional diferential equations, arXiv:1906.01242 (2019)
39. Yin B.L., Liu Y., Li H.:A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl. Math. Comput. 368, 124799 (2020)
40. Yin, B.L., Liu, Y., Li, H., He, S.:Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions. J. Comput. Phys. 379, 351-372 (2019)
41. Yuste, S.B., Quintana-Murillo, J.:Fast, accurate and robust adaptive fnite diference methods for fractional difusion equations. Numer. Algor. 71(1), 207-228 (2016)
42. Zeng, F.H., Liu, F.W., Li, C.P., Burrage, K., Turner, I., Anh, V.:A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-difusion equation. SIAM J. Numer. Anal. 52, 2599-2622 (2014)
43. Zhao, Y., Bu, W., Huang, J., Liu, D.Y., Tang, Y.:Finite element method for two-dimensional spacefractional advection-dispersion equations. Appl. Math. Comput. 257, 553-565 (2015)
44. Zhao, Z., Zheng, Y., Guo, P.:A Galerkin fnite element method for a class of time-space fractional differential equation with nonsmooth data. J. Sci. Comput. 70, 386-406 (2017)
45. Zheng Y., Zhao Z.:The time discontinuous space-time fnite element method for fractional difusion-wave equation. Appl. Numer. Math. https://doi.org/10.1016/j.apnum.2019.09.007 (2019)
46. Zheng, Y.Y., Li, C.P., Zhao, Z.G.:A note on the fnite element method for the space-fractional advection difusion equation. Comput. Math. Appl. 59(5), 1718-1726 (2010)
47. Zheng, M., Liu, F.W., Anh, V., Turner, I.:A high-order spectral method for the multi-term time-fractional difusion equations. Appl. Math. Model. 40(7), 4970-4985 (2016)

[1]	Rémi Abgrall. The Notion of Conservation for Residual Distribution Schemes (or Fluctuation Splitting Schemes), with Some Applications[J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 341-368.
[2]	Daniel J. Frean, Jennifer K. Ryan. Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 461-486.
[3]	Bangti Jin, Zhi Zhou. Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 163-177.
[4]	Yonggui Yan, Jiwei Zhang, Chunxiong Zheng. Numerical Computations of Nonlocal Schrödinger Equations on the Real Line[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 241-260.
[5]	Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian. A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 31-55.
[6]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[7]	Yong Liu, Chi-Wang Shu, Mengping Zhang. Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 101-116.
[8]	Jayesh Badwaik, Praveen Chandrashekar, Christian Klingenberg. Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 541-579.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

编辑推荐

Metrics

本文评价