An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements

The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank–Nicolson method and show that the error is of order k2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h2 max(1, log k−1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

[1]  V. Balakrishnan,et al.  Anomalous diffusion in one dimension , 1985 .

[2]  W. Wyss.  The fractional diffusion equation , 1986 .

[3]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[4]  Graeme Fairweather,et al.  Orthogonal spline collocation methods for some partial integrodifferential equations , 1992 .

[5]  Graeme Fairweather,et al.  Spline collocation methods for a class of hyperbolic partial integro-differential equations , 1994 .

[6]  Vidar Thomée,et al.  Discretization with variable time steps of an evolution equation with a positive-type memory term , 1996 .

[7]  Bruce J. West,et al.  Fractional Diffusion Equation , 1999 .

[8]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[9]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[10]  Graeme Fairweather,et al.  An H1-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term , 2002, SIAM J. Numer. Anal..

[11]  C. Palencia,et al.  On the numerical inversion of the Laplace transform of certain holomorphic mappings , 2004 .

[12]  V. Thomée,et al.  Time discretization of an evolution equation via Laplace transforms , 2004 .

[13]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[14]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[15]  Cesar Palencia,et al.  A spectral order method for inverting sectorial Laplace transforms , 2005 .

[16]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

[17]  Eduardo Cuesta,et al.  Convolution quadrature time discretization of fractional diffusion-wave equations , 2006, Math. Comput..

[18]  William McLean,et al.  A second-order accurate numerical method for a fractional wave equation , 2006, Numerische Mathematik.

[19]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

[20]  Christian Lubich,et al.  Adaptive, Fast, and Oblivious Convolution in Evolution Equations with Memory , 2006, SIAM J. Sci. Comput..

[21]  Kassem Mustapha.  Numerical solution for a sub-diffusion equation with a smooth kernel , 2009, J. Comput. Appl. Math..

[22]  William McLean,et al.  Discontinuous Galerkin method for an evolution equation with a memory term of positive type , 2009, Math. Comput..

[23]  V. Thomée,et al.  Numerical solution via Laplace transforms of a fractional order evolution equation , 2010 .

[24]  V. Thomée,et al.  Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation , 2010 .