Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T11:23:07.289Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected] (G. Gao), [email protected] (H. Sun)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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