Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

Yan Yang; Yubin Yan; Neville J. Ford

doi:10.1515/cmam-2017-0037

Article

Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

Yan Yang , Yubin Yan and Neville J. Ford

Published/Copyright: September 2, 2017

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Computational Methods in Applied Mathematics Volume 18 Issue 1

Abstract

We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an O⁢(k) convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k denotes the time step size. In this paper, we approximate the Riemann–Liouville fractional derivative by Diethelm’s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate O⁢(k) with nonsmooth initial data for the homogeneous problem when A is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is O⁢(k1+α), 0<α<1, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is O⁢(k1+α), 0<α<1, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Keywords: Fractional Diffusion Problem; Nonsmooth Data; Error Estimates; Laplace Transform

MSC 2010: 26A33; 65M06; 65M12; 65M15; 35R11

A Appendix

In this Appendix, we will give the proof of Lemma 2.2. To do this, we need to introduce the polylogarithm function

Lip⁡(z)=∑j=1∞zjjp.

The polynomial function Lip⁡(z) is well defined for |z|<1 and p∈ℂ. It can be continued analytically to the split complex plane ℂ\[1,+∞); see Flajolet [6]. With z=1, it recovers the Riemann zeta function ς⁢(p)=Lip⁡(1). We also recall an important singular expansion of the function Lip⁡(e-z) (cf. [6, Theorem 1]).

Lemma A.1 ([9, Lemma 3.2]).

For p≠1,2,…, the function Lip⁡(e-z) satisfies the singular expansion

Lip⁡(e-z)∼Γ⁢(1-p)⁢zp-1+∑l=0∞(-1)l⁢ς⁢(p-l)⁢zll! as ⁢z→0,

where ς⁢(z) denotes the Riemann zeta function.

Lemma A.2 ([9, Lemma 3.4]).

Let |z|≤πsin⁡θ with θ∈(π2,5⁢π6) and -1<p<0. Then

Lip⁡(e-z)=Γ⁢(1-p)⁢zp-1+∑l=0∞(-1)l⁢ς⁢(p-l)⁢zll!

converges absolutely.

Proof of Lemma 2.2.

We have, by the weights in (2.4), with ζ=e-z⁢k,

w~⁢(z)=∑j=0∞wj⁢ζj=1Γ⁢(1+α)⁢(ζ-1-2+ζ)⁢∑j=1∞jα⁢ζj

(A.1)=1Γ⁢(1+α)⁢((e-z⁢k)-1-2+e-z⁢k)⁢Li-α⁡(ζ),

where, by Lemma A.2,

(A.2)Li-α⁡(ζ)=Li-α⁡(e-z⁢k)=Γ⁢(1+α)⁢(z⁢k)-α-1+∑l=0∞(-1)l⁢ς⁢(α-l)⁢(z⁢k)ll!.

By (2.9), we have

zkα=(δ⁢(ζ)k)α=w~⁢(ζ)-1⁢(1-ζ)kα=1kα⁢ψ⁢(z⁢k),

where

ψ⁢(z⁢k)=1Γ⁢(1+α)⁢(ez⁢k-1)⁢Li-α⁡(e-z⁢k).

Using [18, Lemma 1], we may write, with Cα=πsin⁡(π⁢α) and z⁢k=ρ⁢ei⁢θ=r+i⁢ϕ,

ψ⁢(z⁢k)=1Γ⁢(1+α)⁢(ez⁢k-1)⁢Li-α⁡(e-z⁢k)=1Cα⁢∫0∞s-α1-e-z⁢k-s⁢1-e-ss⁢𝑑s

=1Cα⁢∫0∞s-α1-e-z⁢k-s⁢1-e-ss⁢𝑑s=1Cα⁢∫0∞s-α1-e-r-i⁢ϕ-s⁢1-e-ss⁢𝑑s

=1Cα⁢∫0∞s-α1-e-r-s⁢(cos⁡ϕ-i⁢sin⁡ϕ)⁢1-e-ss⁢𝑑s

=1Cα⁢∫0∞(s-α-1⁢(1-e-s)⁢(1-e-r-s⁢cos⁡ϕ))-(s-α-1⁢(1-e-s)⁢(e-r-s⁢sin⁡ϕ))⁢i1-2⁢e-r-s⁢cos⁡ϕ+e-2⁢r-2⁢s⁢𝑑s,

which implies that

zkα=Cαkα⁢1A-B⁢i=Cαkα⁢A+B⁢iA2+B2,

where

A=∫0∞(s-α-1⁢(1-e-s)⁢(1-e-r-s⁢cos⁡ϕ))1-2⁢e-r-s⁢cos⁡ϕ+e-2⁢r-2⁢s⁢𝑑s,

B=∫0∞(s-α-1⁢(1-e-s)⁢(e-r-s⁢sin⁡ϕ))1-2⁢e-r-s⁢cos⁡ϕ+e-2⁢r-2⁢s⁢𝑑s.

Therefore

ℜ⁡(zkα)=Cαkα⁢AA2+B2,ℑ⁡(zkα)=Cαkα⁢BA2+B2.

Let us first consider the case for θ=π2. In this case, we have, with r=ρ⁢cos⁡θ=0,ϕ=ρ⁢sin⁡θ=ρ,

ℜ⁡(zkα)=Cαkα⁢(A2+B2)⁢∫0∞s-α-1⁢(1-e-s)⁢(1-e-s⁢cos⁡ρ)1-2⁢e-s⁢cos⁡ρ+e-2⁢s⁢𝑑s.

Note that

1-2⁢e-s⁢cos⁡ρ+e-2⁢s>1-2⁢e-s+e-2⁢s=(1-e-s)2≥0,

and

1-e-s⁢cos⁡ρ>1-e-s>0,

we get ℜ⁡(zkα)≥0 which implies that zkα∈Σθ0 for any θ0∈(π2,π). Now let us choose θ close to π2, θ>π2. By the continuity of zkα with respect to θ, cf. [9, Proof of Lemma 3.6], there exists θ0∈(π2,π) such that

zkα∈Σθ0 for all ⁢z∈Γθ.

Together these estimates complete the proof of Lemma 2.2. ∎

Acknowledgements

The second author thanks the organizers of the Mini-Symposium “Numerical Methods for Fractional Differential Equations” at the conference for the Mathematics of Finite Elements and Applications (MAFELAP), 2016 in Brunel, UK. Some results in this paper were presented in that Mini-Symposium.

References

[1] C.-M. Chen, F. Liu, V. Anh and I. Turner, Numerical methods for solving a two-dimensional variable-order anomalous sub-diffusion equation, Math. Comp. 81 (2012), 345–366. 10.1090/S0025-5718-2011-02447-6Search in Google Scholar

[2] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp. 75 (2006), 673–696. 10.1090/S0025-5718-06-01788-1Search in Google Scholar

[3] M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys. 228 (2009), 7792–7804. 10.1016/j.jcp.2009.07.021Search in Google Scholar

[4] W. Deng, Finite element method for the space and time fractional Fokker–Planck equation, SIAM J. Numer. Anal. 47 (2008), 204–226. 10.1137/080714130Search in Google Scholar

[5] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997), 1–6. Search in Google Scholar

[6] P. Flajolet, Singularity analysis and asymptotics of Bernoulli sums, Theoret. Comput. Sci. 215 (1999), 371–381. 10.1016/S0304-3975(98)00220-5Search in Google Scholar

[7] N. J. Ford, J. Xiao and Y. Yan, A finite element method for time-fractional partial differential equations, Fract. Calc. Appl. Anal. 14 (2011), 454–474. 10.2478/s13540-011-0028-2Search in Google Scholar

[8] B. Jin, R. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), 445–466. 10.1137/120873984Search in Google Scholar

[9] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36 (2016), 197–221. 10.1093/imanum/dru063Search in Google Scholar

[10] B. Jin, B. Li and Z. Zhou, An analysis of the Crank–Nikolson method for subdiffusion, IMA J. Numer. Anal. (2017), 10.1093/imanum/drx019. 10.1093/imanum/drx019Search in Google Scholar

[11] B. Jin, B. Li and Z. Zhou, Correction of higher-order BDF convolution quadrature for fractional evolution equations, preprint (2017), https://arxiv.org/abs/1607.06948v2. Search in Google Scholar

[12] T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), 719–736. 10.1016/j.jcp.2004.11.025Search in Google Scholar

[13] C. Li and H. Ding, Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model. 38 (2014), 3802–3821. 10.1016/j.apm.2013.12.002Search in Google Scholar

[14] Z. Li, Z. Liang and Y. Yan, High-order numerical methods for solving time fractional partial differential equations, J. Sci. Comput. 71 (2017), 785–803. 10.1007/s10915-016-0319-1Search in Google Scholar

[15] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), 1533–1552. 10.1016/j.jcp.2007.02.001Search in Google Scholar

[16] C. Lubich, I. H. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp. 65 (1996), no. 213, 1–17. 10.1090/S0025-5718-96-00677-1Search in Google Scholar

[17] W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a fractional diffusion equation, Numer. Algorithms 52 (2009), 69–88. 10.1007/s11075-008-9258-8Search in Google Scholar

[18] W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, J. Comput. Phys. 293 (2015), 201–217. 10.1016/j.jcp.2014.08.050Search in Google Scholar

[19] K. Mustapha, An implicit finite difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements, IMA J. Numer. Anal. 31 (2011), 719–739. 10.1093/imanum/drp057Search in Google Scholar

[20] K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal. 32 (2012), 906–925. 10.1093/imanum/drr027Search in Google Scholar

[21] K. Mustapha and W. McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal. 51 (2013), 491–515. 10.1137/120880719Search in Google Scholar

[22] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[23] M. Stynes, E. O’riordan and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), 1057–1079. 10.1137/16M1082329Search in Google Scholar

[24] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer, Berlin, 2006. Search in Google Scholar

[25] Y.-N. Zhang, Z.-Z. Sun and H.-L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys. 265 (2014), no. 4, 195–210. 10.1016/j.jcp.2014.02.008Search in Google Scholar

Received: 2017-4-27

Revised: 2017-8-17

Accepted: 2017-8-18

Published Online: 2017-9-2

Published in Print: 2018-1-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/cmam-2017-0037

Keywords for this article

Fractional Diffusion Problem; Nonsmooth Data; Error Estimates; Laplace Transform