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Three-Point Combined Compact Alternating Direction Implicit Difference Schemes for Two-Dimensional Time-Fractional Advection-Diffusion Equations

Published online by Cambridge University Press:  22 January 2015

Guang-Hua Gao  and
Hai-Wei Sun
Guang-Hua Gao
Affiliation:
College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210046, P.R. China
Hai-Wei Sun*
Affiliation:
Department of Mathematics, University of Macau, Macao
*
*Email addresses: gaoguanghua1107@163.com (G. Gao), hsun@umac.mo (H. Sun)
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Abstract

This paper is devoted to the discussion of numerical methods for solving two-dimensional time-fractional advection-diffusion equations. Two different three-point combined compact alternating direction implicit (CC-ADI) schemes are proposed and then, the original schemes for solving the two-dimensional problems are divided into two separate one-dimensional cases. Local truncation errors are analyzed and the unconditional stabilities of the obtained schemes are investigated by Fourier analysis method. Numerical experiments show the effectiveness and the spatial higher-order accuracy of the proposed methods.

Keywords

65M0665M1265N0665N12Time-fractional advection-diffusion equationscombined compact difference (CCD) schemeADIstabilityFourier analysis
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 487 - 509
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Uchaikin, V., Fractional Derivatives for Physicists and Engineers, Higher Education Press, Beijing, 2013.CrossRefGoogle Scholar
[2]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[3]Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier Science and Technology, Boston, 2006.Google Scholar
[4]Yuste, S. and Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), 1862–1874.CrossRefGoogle Scholar
[5]Langlands, T. and Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), 719–736.CrossRefGoogle Scholar
[6]Chen, C., Liu, F., Turner, I. and Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886–897.CrossRefGoogle Scholar
[7]Sun, Z.Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209.CrossRefGoogle Scholar
[8]Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792–7804.CrossRefGoogle Scholar
[9]Gao, G.H. and Sun, Z.Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), 586–595.CrossRefGoogle Scholar
[10]Zhang, Y., Sun, Z.Z. and Wu, H., Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation, SIAM J. Numer. Anal., 49 (2011), 2302–2322.CrossRefGoogle Scholar
[11]Mohebbi, A. and Abbaszadeh, M., Compact finite difference scheme for the solution of time fractional advection-dispersion equation, Numer. Algor., 63 (2013), 431–452.CrossRefGoogle Scholar
[12]Du, R., Cao, W. and Sun, Z.Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998–3007.CrossRefGoogle Scholar
[13]Hu, X. and Zhang, L., A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Comput. Phys. Comm., 182 (2011), 1645–1650.CrossRefGoogle Scholar
[14]Ren, J. and Sun, Z.Z., Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions, J. Sci. Comput., 56 (2013), 381–408.CrossRefGoogle Scholar
[15]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552.Google Scholar
[16]Brunner, H., Ling, L. and Yamamoto, M., Numerical simulations of 2D fractional subdiffusion problems, J. Comput. Phys., 229 (2010), 6613–6622.CrossRefGoogle Scholar
[17]Chen, C., Liu, F., Turner, I. and Anh, V., Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor., 54 (2010), 1–21.CrossRefGoogle Scholar
[18]Zhang, Y. and Sun, Z.Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), 8713–8728.Google Scholar
[19]Cui, M., Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comput. Phys., 231 (2012), 2621–2633.CrossRefGoogle Scholar
[20]Cui, M., Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation, Numer. Algor., 62 (2013), 383–409.CrossRefGoogle Scholar
[21]Zhang, Y., Sun, Z.Z. and Zhao, X., Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50 (2012), 1535–1555.Google Scholar
[22]Zhang, Y. and Sun, Z.Z., Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput., 59 (2014), 104–128.CrossRefGoogle Scholar
[23]Chu, P. and Fan, C., A three-point combined compact difference scheme, J. Comput. Phys., 140 (1998), 370–399.CrossRefGoogle Scholar
[24]Mousa, M., Abadeer, A. and Abbas, M., Combined compact finite difference treatment of Burgers’ equation, Int. J. Pure Appl. Math., 75 (2012), 169–184.Google Scholar
[25]Chen, W. and Chen, J., Combined compact difference method for solving the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 68 (2012), 1234–1256.CrossRefGoogle Scholar
[26]Takahashi, F., Implementation of a high-order combined compact difference scheme in problems of thermally driven convection and dynamo in rotating spherical shells, Geophys. As-trophys. Fluid Dyn., 106 (2012), 231–249.CrossRefGoogle Scholar
[27]Sun, H. and Li, L., A CCD-ADI method for unsteady convection-diffusion equations, Com-put. Phys. Comm., 185 (2014), 790–797.CrossRefGoogle Scholar
[28]Chu, P. and Fan, C., A three-point sixth-order nonuniform combined compact difference scheme, J. Comput. Phys., 148 (1999), 663–674.CrossRefGoogle Scholar
[29]Chu, P. and Fan, C., A three-point sixth-order staggered combined compact difference scheme, Math. Comput. Modelling, 32 (2000), 323–340.CrossRefGoogle Scholar
[30]Nihei, T. and Ishii, K., A fast solver of the shallow water equations on a sphere using a combined compact difference scheme, J. Comput. Phys., 187 (2003), 639–659.CrossRefGoogle Scholar
[31]Sengupta, T., Vijay, V. and Bhaumik, S., Further improvement and analysis of CCD scheme: Dissipation discretization and de-aliasing properties, J. Comput. Phys., 228 (2009), 6150–6168.CrossRefGoogle Scholar
[32]Gao, G.H. and Sun, H., Three-point combined compact difference schemes for time-fractional advection-diffusion equations, submitted to J. Comput. Phys., (2013).Google Scholar