Hostname: page-component-6bf8c574d5-w79xw Total loading time: 0 Render date: 2025-02-17T17:12:57.153Z Has data issue: false hasContentIssue false
  • English
  • Français

A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel

Published online by Cambridge University Press:  28 May 2015

Hongbin Chen  and
Da Xu
Hongbin Chen*
Affiliation:
Institute of Mathematics and Physics, College of Science, Central South University of Forestry and Technology, Changsha, 410004, Hunan, China
Da Xu*
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular kernel. The integral term is treated by means of the second order convolution quadrature suggested by Lubich. The stability and convergence are proved by the energy method. A numerical experiment is reported to verify the theoretical predictions.

Keywords

45K0565M0665M1265M15Evolution equationweakly singular kernelcompact difference schemestabilityconvergencenumerical experiment
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 4 , November 2012 , pp. 559 - 572
Copyright
Copyright © Global Science Press Limited 2012

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]Chen, C., Thomée, V and Wahlbin, L. B., Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp., 58, (1992), pp. 587–602.CrossRefGoogle Scholar
[2]Chen, H., Chen, C. and Xu, D., A second-order fully discrete difference scheme for a partial integro-differential equation (in Chinese), Math. Numer. Sinica, 28, (2006), pp. 141–154.Google Scholar
[3]Cui, M., Compact finite difference methodfor the fractional diffusion equation, J. Comp. Phys., 228, (2009), pp. 7792–7804.CrossRefGoogle Scholar
[4]Feng, X.-F., Li, Z.-L. and Qiao, Z.-H., High order compact finite difference schemes for The Helmholtz equation With Discontinuous Coefficients, J. Comp. Math., 29 (2011), pp. 324–340.CrossRefGoogle Scholar
[5]Fujita, Y., Integro-differential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), pp. 319–327.Google Scholar
[6]Fujita, Y., Integro-differential equation which interpolates the heat equation and the wave equation (II), Osaka J. Math., 27 (1990), pp. 797–804.Google Scholar
[7] Y.-Huang, Q., Time discretization scheme for an integro-differential equation of parabolic type, J. Comp. Math., 12 (1994), pp. 259–263.Google Scholar
[8]Jain, M. K., Jain, R. K. and Mohanty, R. K., A fourth order difference method for the one-dimensional general quasilinear parabolic partial differential equation, Numer. Methods Partial Differential Eqs., 6 (1990), pp. 311–319.CrossRefGoogle Scholar
[9]Li, X.-J. and Xu, C.-J., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016–1051.Google Scholar
[10]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comp. Phys., 225 (2007), pp. 1533–1552.CrossRefGoogle Scholar
[11]Lopez-Marcos, J. C., A difference scheme for a nonlinear partial integro-differential equation, SIAM. J. Numer. Anal., 27 (1990), pp. 20–31.CrossRefGoogle Scholar
[12]Lubich, C., Discretized fractional calculus, SIAM. J. Math. Anal., 17 (1986), pp. 704–719.CrossRefGoogle Scholar
[13]Lubich, C., Sloan, I. H. and Thomée, V., Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp., 65 (1996), pp. 1–17.CrossRefGoogle Scholar
[14]Mclean, W. and Thomée, V., Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser., B35, (1993), pp. 23–70.CrossRefGoogle Scholar
[15]Mclean, W. and Mustapha, K., A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105 (2006), pp. 481–510.CrossRefGoogle Scholar
[16]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[17]Sanz-Serna, J. M., A Numerical method for a partial integro-differential equation, SIAM. J. Numer. Anal., 25 (1988), pp. 319–327.CrossRefGoogle Scholar
[18]Sloan, I. H. and Thomee, V., Time discretization of an integro-differential equation of parabolic type, SIAM. J. Numer. Anal., 23 (1986), pp. 1052–1061.CrossRefGoogle Scholar
[19]Sun., Z., An unconditionally stable and O(τ 2 + h 4) order L$$ convergent difference scheme for linear parabolic equations with variable coefficients, Numer. Methods Partial Differential Eqs., 17 (2001), pp. 619–631.CrossRefGoogle Scholar
[20]Sun, Z., On the compact difference schemes for heat equation with Neumann boundary conditions, Numer. Methods Partial Differential Eqs., 25 (2009), pp. 1320–1341.CrossRefGoogle Scholar
[21]Xu, D., The global behavior of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO, 34 (1997), pp. 71–104.Google Scholar
[22]Zhang, Y., Sun, Z. and Wu, H., Error estimates of Crank-Nicolson type difference schemes for the sub-diffusion equation, SIAM. J. Numer. Anal., 49 (2011), pp. 2302–2322.CrossRefGoogle Scholar