Numerical Algorithms

, Volume 80, Issue 3, pp 849–877

# A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

• Haixiang Zhang
• Xuehua Yang
• Da Xu
Original Paper

## Abstract

In the present work, orthogonal spline collocation (OSC) method with convergence order O(τ3−α + hr+ 1) is proposed for the two-dimensional (2D) fourth-order fractional reaction-diffusion equation, where τ, h, r, and α are the time-step size, space size, polynomial degree of space, and the order of the time-fractional derivative (0 < α < 1), respectively. The method is based on applying a high-order finite difference method (FDM) to approximate the time Caputo fractional derivative and employing OSC method to approximate the spatial fourth-order derivative. Using the argument developed recently by Lv and Xu (SIAM J. Sci. Comput. 38, A2699–A2724, 2016) and mathematical induction method, the optimal error estimates of proposed fully discrete OSC method are proved in detail. Then, the theoretical analysis is validated by a number of numerical experiments. To the best of our knowledge, this is the first proof on the error estimates of high-order numerical method with convergence order O(τ3−α + hr+ 1) for the 2D fourth-order fractional equation.

## Keywords

Fourth-order fractional equation Orthogonal spline collocation Finite difference method Error estimate

## Mathematics Subject Classification (2010)

65M12 65M06 65M70 35S10

## Notes

### Acknowledgements

The authors thank the anonymous reviewers for their constructive comments and suggestions and Professor Graeme Fairweather for stimulating discussions and for his constant encouragement and support.

### Funding information

The work is supported by the National Natural Science Foundation of China (11701168, 11601144, 11626096), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (16K026,YB2016B033), China Postdoctoral Science Foundation (2016M600964), Science Challenge Project (TZ2016002).

## References

1. 1.
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
2. 2.
Sun, Z.Z., Wu, X.N.: A fully difference scheme for a diffusion-wave system. Appl. Numer. Math. 2, 193–209 (2006)
3. 3.
Gao, G., Sun, Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
4. 4.
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
5. 5.
Li, C.P., Wu, R.F., Ding, H.F.: High-order approximation to Caputo derivative and Caputo-type advection-diffusion equations. Commun. Appl. Ind. Math 6(2), e-536 (2014)
6. 6.
Cao, J., Li, C., Chen, Y.Q.: High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (ii). Fract. Calc. Appl. Anal. 18, 735–761 (2015)
7. 7.
Li, H., Cao, J., Li, C.: High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J. Comput. Appl. Math. 299, 159–175 (2016)
8. 8.
Lv, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)
9. 9.
Li, Z.Q., Liang, Z.Q., Yan, Y.B.: High-order numerical methods for solving time fractional partial differential equations. J. Sci. Comput. 71, 785–803 (2017)
10. 10.
Li, Z.Q., Yan, Y.B., Ford, N.J.: Error estimates of a high order numerical method for solving linear fractional differential equations. Appl. Numer. Math. 114, 201–220 (2016)
11. 11.
Yan, Y.B., Pal, K., Ford, N.J.: Higher order numerical methods for solving fractional differential equations. BIT Numer. Math. 54, 555–584 (2014)
12. 12.
Dehghan, M., Abbaszadeh, M., Mohebbib, A.: Error estimate for the numerical solution of fractional reaction-subdiffusion process based on a meshless method. J. Comput. Appl. Math. 280, 14–36 (2015)
13. 13.
Dehghan, M., Fakhar-Izadi, F.: The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves. Math. Comput. Model. 53, 1865–1877 (2011)
14. 14.
Dehghan, M., Abbaszadeh, M.: Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction-diffusion system with and without cross-diffusion. Comput. Methods Appl. Mech. Eng. 300, 770–797 (2016)
15. 15.
Dehghan, M., Abbaszadeh, M.: Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs. Appl. Anal. 96, 936–969 (2017)
16. 16.
Dehghan, M., Abbaszadeh, M.: Element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi-type equation with robin boundary condition. Comput. Math. Appl. 73, 1270–1285 (2017)
17. 17.
Oldhan, K.B., Spainer, J.: The Fractional Calculus. Academic Press, New York (1974)Google Scholar
18. 18.
Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53, 1336–1339 (1996)
19. 19.
Ji, C.C., Sun, Z.Z., Hao, Z.P.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J. Sci. Comput. 66, 1148–1174 (2015)
20. 20.
Hu, X.L., Zhang, L.M.: On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, 5019–5034 (2012)
21. 21.
Hu, X.L., Zhang, L.M.: A compact finite difference scheme for the fourth-order fractional diffusion-wave system. Comput. Phys. Commun. 230, 1645–1650 (2011)
22. 22.
Guo, J., Li, C.P., Ding, H.F.: Finite difference methods for time subdiffusion equation with space fourth order. Commun. Appl. Math. Comput. 28, 96–108 (2014). in Chinese
23. 23.
Vong, S., Wang, Z.: Compact finite difference scheme for the fourth-order fractional subdiffusion system. Adv. Appl. Math. Mech. 6, 419–435 (2014)
24. 24.
Zhang, P., Pu, H.: A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation. Numer. Algor. (2017)
25. 25.
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)
26. 26.
Liu, Y., Fang, Z. C., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)
27. 27.
Liu, Y., Du, Y.W., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction diffusion problem. Comput. Math. Appl. 70, 573–591 (2015)
28. 28.
Liu, Y., Du, Y.W., Li, H., Li, J.C., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction diffusion problem with time-fractional derivative. Comput. Math. Appl. 70, 2474–2492 (2015)
29. 29.
Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92, 1496–1518 (2014)
30. 30.
Cao, J., Xu, C., Wang, Z.: A high order finite difference/spectral approximations to the time fractional diffusion equations. Adv. Mater. Res. 875, 781–785 (2014)
31. 31.
Li, B., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables. SIAM J. Numer. Anal. 35, 453–477 (1998)
32. 32.
Fairweather, G., Gladwell, I.: Algorithms for almost block diagonal linear systems. SIAM Rev. 46, 49–58 (2004)
33. 33.
Bialecki, B.: Convergence analysis of orthogonal spline collocation for elliptic boundary value problems. SIAM J. Numer. Anal. 35, 617–631 (1998)
34. 34.
Percell, P., Wheeler, M.F.: A C 1 finite element collocation method for elliptic equations. SIAM J. Numer. Anal. 17, 605–622 (1980)
35. 35.
Greenwell-Yanik, C.E., Fairweather, G.: Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables. SIAM J. Numer. Anal. 23, 282–296 (1986)
36. 36.
Jiang, Y.J., Ma, J.T.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285–3290 (2011)
37. 37.
Zhao, Y.M., Chen, P., Bu, W.P., Liu, X.T., Tang, Y.F.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70, 407–428 (2017)
38. 38.
Huang, J.F., Tang, Y.F., Vázquez, L., Yang, J.Y.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algor. 64, 707–720 (2013)
39. 39.
Manickam, A.V., Moudgalya, K.M., Pani, A.K.: Second order splitting and orthogonal cubic spline collocation methods for Kuramoto-Sivashinsky equation. Comput. Math. Appl. 35, 5–25 (1998)
40. 40.
Yan, Y., Fairweather, G.: Orthogonal spline collocation methods for some partial integro-differential equations. SIAM J. Numer. Anal. 29, 755–768 (1992)
41. 41.
Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456–467 (2013)