Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

Yan Yang; Yubin Yan; Neville J. Ford

doi:10.1515/cmam-2017-0037

Published by De Gruyter September 2, 2017

Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

Yan Yang , Yubin Yan and Neville J. Ford

From the journal Computational Methods in Applied Mathematics

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

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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.

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Received: 2017-4-27

Revised: 2017-8-17

Accepted: 2017-8-18

Published Online: 2017-9-2

Published in Print: 2018-1-1

Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

Abstract

A Appendix

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

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

Proof of Lemma 2.2.

Acknowledgements

References

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