Abstract
In this paper, we present a fast and efficient numerical method to solve a class of parabolic integro-differential equations with weakly singular kernels, compact difference approach for spatial discretization and alternating direction implicit method in time, combined with second-order fractional quadrature rule suggested by Lubich approximating the integral term. The \(L^2\) stability and convergence are derived. Two numerical examples with known exact solution are given to support the theoretical results.
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References
Pani, A.K., Fairweather, G., Fernandes, R.I.: ADI orthogonal spline collocation methods for parabolic partial integro-differential equations. IMA J. Numer. Anal. 30, 248–276 (2010)
Pipkin, A.C.: Lectures on Viscoelasticity Theory. Springer, Berlin (1972)
Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical Problems in Viscoelasticity. Longman Scientific and Technical, London (1987)
Friedman, A., Shinbrot, M.: Volterra integral equations in Banach space. Trans. Am. Math. Soc. 126, 131–179 (1967)
Heard, M.L.: An abstract parabolic Volterra integrodifferential equation. SIAM J. Math. Anal. 13, 81–105 (1982)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Sloan, I.H., Thomee, V.: Time discretization of an integro-differential equation of parabolic type. SIAM J. Numer. Anal. 23, 1052–1061 (1986)
Yanik, E.G., Fairweather, G.: Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 12, 785–809 (1988)
Lin, Y.P., Thomee, V., Wahlbin, L.B.: Ritz–Volterra projections to finite element spaces and applications to integrodifferential and related equations. SIAM J. Numer. Anal. 2, 1047–1070 (1991)
Xu, D.: On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel, I: smooth initial data. Appl. Math. Comput. 58, 1–27 (1993)
Chen, C.M., Thomée, V., Wahlbin, L.B.: Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 58, 587–602 (1992)
Larsson, S., Thomee, V., Wahlbin, L.B.: Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin methods. Math. Comput. 67, 45–71 (1998)
Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25, 319–327 (1988)
Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integro-differential equation. SIAM J. Numer. Anal. 27, 20–31 (1990)
Xu, D.: Finite element methods for the nonlinear integro-differential equations. Appl. Math. Comput. 58, 241–273 (1993)
Mustapha, K., Mustapha, H.: A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA J. Numer. Anal. 30, 555–578 (2009)
Mclean, W., Thomee, V.: Numerical solution of an evolution equation with a positive type memory term. J. Aust. Math. Soc. Ser. B 35, 23–70 (1993)
Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11, 309–319 (1993)
Chen, H., Xu, D.: A second-order fully discrete difference scheme for a partial integro-differential equation (in Chinese). Math. Numer. Sin. 28, 141–154 (2006)
Chen, H., Xu, D.: A compact difference scheme for an evolution equation with a weakly singular kernel. Numer. Math. Theor. Meth. Appl. 5, 559–572 (2012)
Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of elliptic and parabolic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
Liao, H., Sun, Z.: Maximum error estimates of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 26, 37–60 (2010)
Zhang, Y., Sun, Z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)
Li, L., Xu, D.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471–485 (2013)
Li, L., Xu, D.: Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation. J. Comput. Phys. 236, 157–168 (2013)
Chen, H., Xu, D., Peng, Y.: An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation. Int. J. Comput. Math. 20, 2178–2197 (2015)
Cuesta, E., Palencia, C.: A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces. Appl. Numer. Math. 45, 139–159 (2003)
Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52, 187–199 (1988)
Lubich, C.: On convolution quadrature and Hille-Philips operational calculus. Appl. Numer. Math. 9, 704–719 (1992)
Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)
Zhang, Y., Sun, Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)
Acknowledgements
I am grateful to the referees for many helpful suggestions. The project supported by National Natural Science Foundation of China (Grant No. 11671131), Construct Program of the Key Discipline in Hunan Province.
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Leijie Qiao and Da Xu have equally contributed to this work.
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Qiao, L., Xu, D. Compact Alternating Direction Implicit Scheme for Integro-Differential Equations of Parabolic Type. J Sci Comput 76, 565–582 (2018). https://doi.org/10.1007/s10915-017-0630-5
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DOI: https://doi.org/10.1007/s10915-017-0630-5