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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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Volume 20, Issue 5

Issues

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Neville J. Ford / Yubin Yan
Published Online: 2017-10-31 | DOI: https://doi.org/10.1515/fca-2017-0058

Abstract

In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

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

Key Words and Phrases: time fractional partial differential equations; finite difference method; convolution quadrature; error estimates

Dedicated to the 20th anniversary of FCAA and 65th anniversary of Professor Virginia Kiryakova Also, to the Memory of Christopher TH Baker (1939-2017)

1 Introduction

In this paper, we review an approach to looking for higher order time discretisation methods for solving the following time fractional partial differential equations, with 0 < α < 1, 0CDtαu(t)+Au(t)=f(t),for0<tT,withu(0)=u0,(1.1)

where 0CDtαu(t) denotes the Caputo fractional derivative defined by 0CDtαu(t)=1Γ(1α)0t(ts)αu(s)ds,

and u(s) = u / s and A is a selfadjoint positive definite second order elliptic partial differential operator in Ω ⊂ ℝd, d = 1, 2, 3, with 𝓓(A) = H01(Ω)H2(Ω),whereH01(Ω),H2(Ω) denote the standard Sobolev spaces. For example, we may choose A = −Δ subject to the Dirichlet boundary condition, where Δ denotes the Laplacian.

The assumption that A is positive definite implies that A generates an analytic semigroup, so that for some π/2 < θ0 < π and with C = Cθ0 we have the resolvent estimate, see Lubich et al. [33], Thomée [48], (zI+A)1C|z|1forzΣθ0={z0:|argz|<θ0}.(1.2)

In our analysis, we will choose θ > π/2 close to π/2 such that θ < θ0 which implies that zα ∈ Σθ0 for any z ∈ Σθ since arg (zα) = α θ < θ < θ0 for 0 < α < 1. Hence there exists a constant C which depends only on θ and α such that, see Jin et al. [21, (2.3)], (zαI+A)1C|z|α,zΣθ={z0:|argz|<θ}.(1.3)

Under the assumptions that the solutions of (1.1) are sufficiently smooth, for example uC2[0, T], there are many time discretisation schemes in the literature, for example, Lubich’s fractional multistep methods [7], [30], [55], [65], [61], [62], [64], the L1 scheme and its modification [28], [26], [18], [29], [47], the spectral method [58] [63], [4], [68], [57], nonpolynomial collocation method [14], discontinuous Galerkin method [39], etc.

However, the solutions of (1.1) are not smooth in general. The regularity of the solution of (1.1) is restrictive. For example, for the homogeneous equation with initial data u0L2(Ω), we have the following stability estimate [44] 0CDtαuL2Ctαu0L2.

That is, the αth order Caputo derivative is already unbounded when t → 0. Hence, the C2-regularity assumption generally does not hold for (1.1). Numerical experiments indicate that the convergence rate actually does not hold uniformly in t even for smooth data u0 [20]. Stynes et al. [46] showed that the solution u of (1.1) behaves as tα, 0 < α < 1 which is not in C2[0, T], see also Stynes [45]. Hence error estimates of the time discretisation schemes of (1.1) obtained under the assumptions that the solutions of (1.1) are sufficiently smooth do not hold when the solutions of (1.1) are not smooth, for example uC2[0, T]. In other words, the optimal convergence rates of the time discretisation schemes cannot be achieved when the solutions are not smooth. Therefore it is interesting to consider error estimates of time discretisation schemes for solving (1.1) when the solutions of (1.1) are not smooth. There are several alternative approaches to construct higher order time discretisation schemes for solving (1.1) when the solutions of (1.1) are not smooth.

  1. Use non-uniform meshes and approximate the solution near the singular point t = 0 by using finer meshes, see, [36], [39], [43], [56], [66].

  2. Use some nonpolynomial (or singular) basis functions or collocation spectral methods to capture the singularity of the solutions of (1.1), see [1], [5], [13], [14], [34], [27], [59], [63], [67].

  3. Separate the solution into two parts: smooth and nonsmooth parts. The nonsmooth part of the solution has the form u(t) = tλl, l = 0, 1, 2, …, s with 0 ≤ λ0 < λ1 < … < λs. Approximate the nonsmooth part by introducing some starting weights such that the time discretisation scheme holds exactly for the nonsmooth part u(t)= tλl, l = 0, 1, 2, …, s with 0 ≤ λ0 < λ1 < … < λs, see [30], [60], [62].

  4. Introduce a Correction to the starting steps of the time discretisation schemes for solving (1.1) to capture the singularity of the solution of (1.1), see [33], [22], [23], [50], [52], [53].

Approach 4 is the main focus of this paper. Approach 3 is most closely related to it and we make the following remarks: Let 0 < t0 < t1 < … < tN = T be a partition of [0, T] and τ the time step size. To construct a time discretisation scheme for solving (1.1), we first need to approximate the Caputo fractional derivative 0CDtαu(t) at t = tn, n = 1, 2, …, N. Note that the Caputo fractional derivative has the following relation with the Riemann-Liouville fractional derivative, 0CDtαu(t)=0RDtα(u(t)u(0)), 0 < α < 1, where 0RDtαu(t) denotes the Riemann-Liouville fractional derivative defined by 0RDtαu(t)=1Γ(1α)ddt0t(ts)αu(s)ds.

We may consider the approximation of the Riemann-Liouville fractional derivative instead. Based on Lubich’s convolution quadrature [30], [31], [32], the Riemann-Liouville fractional derivative 0RDtαu(t) can be approximated by, with 0 < α < 1, 0RDtαu(t)ταj=0nwnju(tj).(1.4)

Here wj, j = 0, 1, 2, … are the weights generated by the characteristic polynomial δ(ζ) where, [23, (2.2)] δ(ζ)α=j=0wjζjwithδ(ζ)=j=1p1j(1ζ)j,p=1,2,,6.(1.5)

The approximation method (1.4) generated by (1.5) is also called the backward difference method with order p (BDFp) where p = 1, 2, … 6, [30], [31], [32]. When p = 1, it is also called the Grünwald-Letnikov approximation method [42].

If the solution u is sufficiently smooth and has many vanishing derivatives at 0, then we have, [30], [31], [32], 0RDtαu(tn)=ταj=0nwnju(tj)+O(τp),p=1,2,,6.(1.6)

However the solutions of (1.1) has low regularity [44]. We only obtain first-order accuracy when solving (1.1) by using the approximation scheme (1.6) with p ≥ 2. This has been observed numerically [7], [20].

To obtain the convergence rate O(τp), p = 2, …, 6, for the approximation scheme (1.6) when u(t) has low regularity, one may use the starting weights to correct the approximate scheme (1.6). More precisely, we may have, for n ≥ 1, 0RDtαu(tn)=ταj=0nwnju(tj)+j=0swn,ju(tj)+O(τp),p=2,,6,(1.7)

for some suitable starting weights wn, j, j = 0, 1, …, s with some s ∈ ℤ+.

To determine the starting weights, we assume that (1.7) holds exactly for u(t) = tλl, l = 0, 1, …, s, where 0 ≤ λ0 < λ1 < … < λs are some numbers depending on 0 < α < 1 and p. Hence we may obtain the starting weights by solving the following linear system, with u(t) = tλl, l = 0, 1, …, s, 0RDtαu(tn)=ταj=0nwnju(tj)+j=0swn,ju(tj),p=2,,6.(1.8)

For each fixed n ≥ 1, ns, we need to solve the linear system (1.8) to determine the starting weights wn, j, j = 0, 1, …, s in order to calculate the approximate solutions of the considered fractional differential equations. This idea works well for solving the following nonlinear fractional differential equation, with 0 < α < 1, 0CDtαy(t)=f(t,y(t)),withy(0)=y0,(1.9)

since the solution of (1.9) is well understood and it can be expressed as the following form, for sufficiently smooth f, u(t)=c0tλ0+c1tλ1++cstλs+ψ(t),

for some 0 = λ0 < λ1 < … < λs. Here ψ(t) is some smooth function and ci ∈ ℝ, i = 0, 1, …, s, s ∈ ℤ+. Diethelm et al. [12] and Weilbeer [49] discussed the implementation of this idea in detail for solving (1.9), see also [3], [54], [6], [58], [61].

More recently, Zeng et al. [65] extended this idea to construct the higher order time discretisation scheme for solving time fractional partial differential equation (1.1). Under the assumption that the solution u(t, x) of (1.1) satisfies u(t,x)u(0,x)=j=0scj(x)tγj+cs+1(x)tγs+1+,(1.10)

for some 0 < γ0 < γ1 < … < γs, s ∈ ℤ+, Zeng et al. [65] constructed a second order numerical method for solving (1.1) by using starting weights and the weighted shifted Grünwald-Letnikov formula for both smooth and nonsmooth solutions. However Jin et al. [23] mentioned that the assumption (1.10) requires imposing certain compatibility conditions on the source term f which may not be satisfied for the general f in (1.1).

Without using the assumption (1.10) and the starting weights, we may also obtain higher order time discretisation schemes by using Approach 4, i.e., by correcting the starting steps of the standard time discretisation schemes when the solutions of (1.1) are not smooth. This idea was first used in Lubich et al. [33] to construct a second order time discretisation scheme for solving diffusion wave equation, see (3.14) and (3.15) in Section 3 below. They proved the error estimates by using the Laplace transform method. Recently Jin et al. [22], [23] and Yan et al. [50], [52], [53] used the similar ideas as in Lubich et al [33] to construct higher order time discretisation methods for solving (1.1). In this paper, we will review some recent results for constructing the time discretisation scheme for solving (1.1) by using Approach 4 when the solutions of (1.1) are not smooth.

The paper is organized as follows. In Section 2, we review the corrected Lubich’s fractional multistep methods for solving time fractional partial differential equations. In Section 3, we review the corrected L1 scheme (or Diethelm’s finite difference method) and its modifications for solving time fractional partial differential equations. In Section 4, we consider the discontinuous Galerkin method for solving time fractional partial differential equations. Finally, we propose some open problems and issues for further work.

Throughout, the notations C and c, with or without a subscript, denote generic constants, which may differ at different occurrences, but are always independent of the step size k.

2 Correction of the Lubich fractional multistep methods

To obtain a higher order time discretisation method for solving (1.1), Jin et al. [23] introduced some corrected backward difference formula (BDF) for solving (1.1).

To see this, we first note that (1.1) is equivalent to the following form 0RDtα(u(t)u(0))+Au(t)=f(t),for0<tT,withu(0)=u0.(2.1)

Let Unu(tn) denote the approximate solution of u(tn). We define the following time discretisation scheme for solving (2.1), with n ≥ 1, ταj=0nwnj(UjU0)+AUn=f(tn),withU0=u0.(2.2)

Jin et al. [23] showed that ║Unu(tn)║ has only first order accuracy even if the initial data u0 and the source term f are sufficiently smooth.

To restore the pth accuracy with p = 2, 3, …, 6, Jin et al. [23] corrected the BDFp at the starting p − 1 steps by (as usual, the summation disappears if the upper index is smaller than the lower one) ταj=0nwnj(UjU0)+AUn=f(tn)+an(p)(Au0+f(0))+τll=1p2bl,n(p)f(l)(0),1np1,(2.3) ταj=0nwnj(UjU0)+AUn=f(tn),pnN,(2.4) U0=u0,(2.5)

where an(p)andbl,n(p) are coefficients determined in Tables 1 and 2, respectively. They are constructed so as to improve the accuracy of the overall scheme to O(τp) for general initial data u0 ∈ 𝓓 (A) and a possibly incompatible right-hand side f.

Table 1

The coefficients aj(p) in (2.3)-(2.5) given in [23, Table 1]

Table 2

The coefficients bl,j(p) in (2.3)-(2.5) given in [23, Table 2]

Theorem 2.1

([23, Theorem 2.2] Let u(tn) and Un be the solutions of (2.1) and (2.3)-(2.5), respectively. Then we have, with p = 1, 2, …, 6, Unu(tn)Cτp(tnαpf(0)+Au0+l=1p1tnα+lpf(l)(0)+0tn(tns)α1f(p)(s)ds).

Remark 2.1

([23, Remark 2.3]) For any fixed tn > 0, by Theorem 2.1, the corrected BDFp method (2.3)-(2.5) has convergence rate O(τp), p = 1, 2, …, 6. To get a uniform rate O(τp), one may need the following compatibility conditions [32] f(0)+Au(0)=0,andf(l)(0)=0,l=1,2,,p1.(2.6)

When these conditions do not hold, the error estimates may deteriorate as t → 0 and we cannot obtain the uniform rate O(τp), p = 1, 2, …, 6.

Remark 2.2

([23, Remark 2.4]) When the initial value u0H = L2(Ω), we may obtain similar error estimates as in Theorem 2.1. More precisely, we have Unu(tn)Cτp(tnpf(0)+u0+l=1p1tnα+lpf(l)(0)+0tn(tns)α1f(p)(s)ds).

We see that if the initial value u0 is nonsmooth, i.e., u0H, then in the error estimates, tnαp reduces to tnp, in other words, tn loses αth power if u0 is nonsmooth.

3 Correction of L1 scheme and its modification

3.1 Correction of the L1 scheme

Another approach to approximate the fractional derivative is by using the L1 scheme (or the Diethelm finite difference method). The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].

Let us first consider how to obtain the L1 scheme by the direct approximation of the derivative in the definition of the Caputo fractional derivative. Note that, with 0 < α < 1, 0CDtαu(tn)=1Γ(1α)0tn(tns)αu(s)ds=1Γ(1α)k=0n1tktk+1(tns)αu(s)ds,

where u(s)=duds.

Approximating u (s) on [tk, tk+1] by using the finite difference quotient (u(tk+1) − u(tk))/τ, we obtain the following so-called L1 scheme 0CDtαu(tn)1Γ(1α)0tn(tns)αu(tk+1)u(tk)τds=ταj=0nwnj,nu(tj),(3.1)

where wj, n, j = 0, 1, 2, …, n are given by Γ(2α)wj,n=1,forj=0,2j1α+(j1)1α+(j+1)1α,forj=1,2,,n1,(j1)1αj1α,forj=n.

Lin et al. [28] and Sun et al. [47] independently proved that the truncation error of the L1 scheme is O(τ2−α) for a sufficiently smooth function u.

We next consider how to obtain the L1 scheme by approximating the Hadamard finite-part integral. Noting that Riemann-Liouville fractional derivative can be expressed by the Hadamard finite-part integral, Diethelm [9], [10] approximated the Hadamard finite-part integral by using the piecewise interpolation polynomials and obtained the approximate schemes to the Riemann-Liouville fractional derivatives. More precisely, we have, with 0 < α < 1, 0RDtαu(t)=1Γ(α)0t(ts)α1u(s)ds,(3.2)

where the integral ∮ is in the Hadamard sense [9], [10]. At t = tn, we have 0RDtαu(tn)=1Γ(α)k=0n1tktk+1(tns)α1u(s)ds.

Approximating u(s) on [tk, tk+1] by using the linear interpolation polynomial P1(s)=stk+1tktk+1u(tk)+stktk+1tku(tk+1),

we get 0CDtαu(tn)=0RDtα(u(tn)u(0))1Γ(α)k=0n1tktk+1(tns)α1P1(s)ds0RDtαu(0)=ταj=0nwnj,nu(tj),(3.3)

where the weights wj, n, j = 0, 1, …, n, are exactly the same as the weights defined in (3.1). Hence we conclude that, when using the piecewise linear interpolation polynomial to approximate the Hadamard finite-part integral, the scheme (3.3) for approximating the Riemann-Liouville fractional derivative is the same as the L1 scheme (3.1) for approximating the Caputo fractional derivative.

We now define the L1 scheme for solving (2.1). From now on, we only consider the homogeneous case with f = 0. The inhomogeneous case can be discussed similarly by using the argument in Jin et al. [20], [22], [23].

Let u(t) − u0 = V(t). Then solving (1.1) with f = 0 is equivalent to find V(t) such that, noting that 0CDtα u0 = 0, 0CDtαV(t)+AV(t)=Au0,withV(0)=0.(3.4)

It proves more convenient to consider the error estimates of the time discretisation scheme of (3.4), see [33].

Let VjV(tj) be the approximate solution of V(tj) in (3.4). We define the following L1 scheme for solving (3.4), ταj=0nwnj,nVj+AVn=Au0,n1,with V0=0,(3.5)

or ταj=1nwnj,nVj+AVn=Au0,n1.(3.6)

For any fixed n ≥ 1, we observe that wj,n, j = 0, 1, …, n − 1 only depend on j = 0, 1, 2, …, n − 1. For example, we have w0,n = 1/Γ(2 − α) for any n ≥ 1, w1,n = 1/Γ(2 − α) ((−2)11−α + (1 − 1)1−α + (1 + 1)1−α) for any n ≥ 2, …. Therefore, it is reasonable to write wj,n, j = 0, 1, 2, …, n − 1 with wj since wj,n, j = 0, 1, 2, …, n − 1 are independent of n. In other words, we denote w0 = w0,n, w1 = w1,n, w2 = w2,n, …, wn−1 = wn−1,n for any fixed n ≥ 1. Further we define wj for all j = 0, 1, 2, … as follows: Γ(2α)wj=1,for j=0,2j1α+(j1)1α+(j+1)1α,for j=1,2,.(3.7)

Using the notations wj, j = 0, 1, 2, … defined in (3.7), the time discretisation scheme (3.6) can now be written as ταj=1nwnjVj+AVn=Au0,n1.(3.8)

Theorem 3.1

([21, Theorem 3.16]). Let V(tn) and Vn be the solutions of (3.4) and (3.8), respectively. Let u0H = L2(Ω). Then we have, with 0 < α < 1, V(tn)VnCτtn1u0.(3.9)

To obtain a higher order method for solving (1.4), we correct the starting steps of the L1 scheme as used in Jin et al. [23]. Let c0 = 1/2, we correct the L1 scheme for solving (3.4) as follows ταj=1nwnjVj+AVn=(Au0)(1+c0), for n=1,(3.10) ταj=1nwnjVj+AVn=Au0, for n2,(3.11) V0=0,(3.12)

where the weights wnj, j = 1, 2, …, n are given by (3.7).

The idea of introducing the correction term in the first step n = 1 in (3.10) comes from the idea in Lubich et al. [33] where the authors introduced a modified scheme to construct a higher order time discretisation scheme for solving an evolution equation with a positive-type memory term. To see this, let us first note that (3.4) is equivalent to the following form, with 0 < α < 1, V(t)+0RDtα(AV(t))=0RDtα(Au0),withV(0)=0,(3.13)

where 0RDtα V(t) denotes the Riemann-Liouville fractional integral. To obtain a higher order time discretisation scheme for solving (3.13), following the idea in Lubich et al. [33], we may introduce the following modified time discretisation scheme to approximate (3.13), Vn+qnc(AV)=qnc(Au0),with V0=0,(3.14)

where qnc(φ) is the modification of the quadrature formula approximating the Riemann-Liouville fractional integral 0RDtα φ, defined by qnc(φ)=kαk=1nβnjφj+c0βn1φ0,with c0=1/2.(3.15)

Here β0, β1, … are generated by some function β^(ζ)=j=0βjζj.

Lemma 3.1

([52]) Assume that β^(ζ)=(j=0βjζj)1, where wj, j = 0, 1, 2, … are defined in (3.7). Then the modified L1 scheme (3.10)-(3.12) is equivalent to (3.14).

Remark 3.1

From Lemma 3.1, we note that the correction on the first step n = 1 in (3.10)-(3.12) is equivalent to the correction in (3.14). Therefore we see that the corrected L1 scheme (3.10)-(3.12) is actually equivalent to the modified scheme (3.14) which has been used to improve the convergence rate of the time discretisation scheme for solving an evolution equation with a positive-type memory term in Lubich et al. [33].

Theorem 3.2

([52]) Let V(tn) and Vn be the solutions of (3.4) and (3.10)-(3.12), respectively. Let u0H = L2(Ω). We have V(tn)VnCτ2αtnα2u0.

3.2 Numerical examples

In this subsection, we will consider the experimentally determined convergence rates of the L1 and the corrected L1 schemes for smooth and nonsmooth data. We only show the numerical results for the homogeneous problem here. Similarly we may consider the numerical results for the inhomogeneous problem with f ≠ 0 as discussed in Jin et al. [20], [23]. Let us consider 0CDtαu(x,t)uxx=0,0<x<1,t>0,(3.16) u(0,t)=u(1,t)=0,(3.17) u(x,0)=u0(x),(3.18)

where u0(x) = x(1 − x) or u0(x) = χ(0,1/2).

Let 0 < t0 < t1 < … tN = T be the time partition and τ the time step size. Let Nh be a positive integer. Let 0 = x0 < x1 < x2 < … xNh = 1 be the space partitions and h the space step size. We will use finite element methods to consider the spatial discretisation.

We first consider the scheme (3.8) and the convergence rate was proved to be O(τ) for both smooth and nonsmooth data in [21].

To observe this convergence order, we first calculate the reference exact solution uref(t) at T = 1 with href = 2−6 and τref = 2−10. We then use h = 2−6 and τ = kappa * τref with kappa = [22, 23, 24, 25, 26] to obtain the approximate solutions u(t) at t = 1. We choose the smooth and nonsmooth initial data (a) u0 = x(1 − x) and the nonsmooth data (b) u0 = χ(0,1/2) we obtain the following results which are consistent with the Table 1 in [21]. In Table 3, we observe that the convergence rate indeed is almost O(τ) for the different α ∈ (0, 1) in both smooth and nonsmooth data cases.

Table 3

Time convergence rates with the different α for the L1 scheme (3.8)

We next consider the corrected L1 scheme (3.10)-(3.12). By Theorem 3.2, the convergence rate of the corrected L1 scheme (3.10)-(3.12) is O(τ2−α) for smooth and nonsmooth data. To observe this convergence rate, we first calculate the reference exact solution uref(t) at T = 1 with href = 2−6 and τref = 2−10. We then use h = 2−6 and τ = kappa * τref with kappa = [22, 23, 24,25, 26] to obtain the approximate solutions u(t) at t = 1. We choose the smooth and nonsmooth initial data (a) u0 = sin (2πx) and (b) u0 = χ(0,1/2) we obtain the following results.

We found that in Table 4 the corrected L1 scheme has the better accuracy than the L1 scheme and the errors are about 1e − 05 or 1e − 04 for all α ∈ (0, 1). The errors of the L1 scheme are only 1e − 03. For the convergence rates, when α < 1/2, we observe that, in Table 4, the convergence rates are almost 2 which is better than the theoretical results 2 − α. However when α > 1/2, the convergence rates are almost 2 − α as we expected.

Table 4

Time convergence rates with the different α for the modified L1 scheme (3.10)-(3.12)

3.3 Correction of the modified L1 scheme

In this subsection we will introduce a corrected higher order time discretisation scheme (which we call the modified L1 scheme) for solving (3.4) and show the error estimates for the nonsmooth data [50]. Let us first introduce a higher order scheme (the modified L1 scheme) to approximate the Riemann-Liouville derivative by using Diethelm’s approach [9], [10].

Lemma 3.2

Let 0 < α < 1 and assume that uC3 [0, T]. Let n ≥ 5, then we have 0RDtαu(tn)=ταj=0nwj,nu(tnj)+O(τ3α),

where Γ(3α)wj,n=(1+α2)21α,j=0,(2+α2)31α(3+32α)21α,j=1,(3+α2)41α(6+32α)31α+(3+32α)21α,j=2,

and, with j = 3, 4, …, n − 2, Γ(3α)wj,n=(j+2)2α(1α2)(j+2)1α3(j+1)2α+(332α)(j+1)1α+3j2α(33α2)j1α(j1)2α+(1α2)(j1)1α,j=3,4,n2,

and Γ(3α)wn1,n=(42n2α)n1α+(3n6+3α2)(n1)1α(n+α23)(n2)1α,Γ(3α)wn,n=(n3+3α2)n1α+(1α)(2α)nα(n2+α2)(n1)1α.

Proof

Note that the Riemann-Liouville fractional derivative 0RDtα u(t), 0 < α < 1 may be written as 0RDtαu(t)=1Γ(1α)ddt0t(ts)αu(s)ds=1Γ(α)0t(ts)α1u(s)ds,

where the integral ∮ must be interpreted as a Hadamard finite-part integral, see e.g., [9], [10].

At t = tn, n = 5, 6, …, N, we have 0RDtαu(tn)=1Γ(α)0tn(tns)α1u(s)ds=tnαΓ(α)01wα1u(tntnw)dw=tnαΓ(α)j=1nwj1wjwα1u(tntnw)dw.

Denote g(w) = u(tntnw) and approximate g(w) by the piecewise quadratic interpolation polynomial g2(w) defined on the nodes wl = ln, l = 0, 1, 2, …, n by g2(w)=(ww1)(ww2)(w0w1)(w0w2)g(w0)+(ww0)(ww2)(w1w0)(w1w2)g(w1)+(ww0)(ww1)(w2w0)(w2w1)g(w2),for w[w0,w1],

and g2(w)=(wwj1)(wwj)(wj2wj1)(wj2wj)g(wj2)+(wwj2)(wwj)(wj1wj2)(wj1wj)g(wj1)+(wwj2)(wwj1)(wjwj2)(wjwj1)g(wj),for w[wj1,wj],j=2,3,,n.

More precisely, on [wj−1, wj], j = 2, 3, …, n, g(w) is approximated by the quadratic interpolation polynomial g2(w) determined by g(wj−2), g(wj−1), g(wj), and on [w0, w1], g(w) is approximated by the quadratic interpolation polynomial g2 (w) determined by g(w0), g(w1), g(w2).

By [10, Theorem 2.4], we have the following error estimates 0RDtαu(tn)=tnαΓ(α)01wα1g(w)dw=tnαΓ(α)01wα1g2(w)dw+O(k3α).(3.19)

We next calculate the integral on the right side of (3.19). Note that 01wα1g2(w)dw=0w1wα1g2(w)dw+j=2nwj1wjwα1g2(w)dw,(3.20)

where only the first integral on the right side of (3.20) is in the sense of the Hadamard finite-part integral and the other integrals on the right side of (3.20) are in the usual sense. By using the definition of the Hadamard finite-part integral [9], [10], we have 0w1wα1g2(w)dw=0w1wα1(g2(0)+0w1g2(y)dy)dw=w1ααg2(0)+0w1wα1(0w1g2(y)dy)dw,

where g2(y)=(n2/2)(2y(w1+w2))g(0)n2(2yw2)g(w1)+(n2/2)(2yw1)g(w2).

Hence, 0w1wα1g2(w)dw=1(α)(α+1)(α+2)nα×((2α/2)g(0)+(α)(3α)g(w1)+(α/2)g(w2)).

For the other integrals in (3.20), we have, with j = 2, 3, …, n, wj1wjwα1g2(w)dw=(n2/2)g(wj2)wj1wjwα1(wwj1)(wwj)dwn2g(wj1)wj1wjwα1(wwj2)(wwj)dw+(n2/2)g(wj)wj1wjwα1(wwj2)(wwj1)dw=1(α)(α+1)(α+2)nα(12Ejg(wj2)Fjg(wj1)+12Gjg(wj)),

where Ej=2jα+2(α+2)jα+12(j1)α+2(α+2)(j1)α+1,Fj=2jα+22(α+2)jα+12(j1)α+2+(α+1)(α+2)(j1)α,Gj=2jα+23(α+2)jα+12(j1)α+2+(α+2)(j1)α+1+2(α+1)(α+2)jα.

Thus we get 01wα1g2(w)dw=j=0nαjng(wj),

where (α)(α+1)(α+2)nααjn=212α+12E2,forj=0,(α)(3α)+12E3F2,forj=1,12α+12E4F3+12G2,forj=2,12Ej+2Fj+1+12Gj,forj=3,,n2,Fn+12Gn1,forj=n1,12Gn.forj=n.

Therefore, we have 0RDtαu(tn)=tnαΓ(α)01wα1g2(w)dw+O(k3α)=tnαΓ(α)j=0nαjng(wj)+O(k3α)=kαj=0n(α)(α+1)(α+2)nααjnΓ(3α)g(wj)+O(k3α)=kαj=0nwjnu(tnj)+O(k3α),

where, with j = 0,1,2,…,n, n = 5, 6,…,N, Γ(3α)wj,n=(α)(α+1)(α+2)nααjn.

More precisely, for j = 0, we have Γ(3α)w0,n=2α2+12E2=(1+α2)21α.

For j = 1, we have Γ(3α)w1,n=(α)(3α)+12E3F2=(2+α2)31α(3+32α)21α.

For j = 2, we have Γ(3α)w2,n=12α+12E4F3+12G2=(3+α2)41α(6+32α)31α+(3+32α)21α.

For j = 3,4,…, n−2, Γ(3α)wj,n=12Ej+2Fj+1+12Gj=(j+2)2α(1α2)(j+2)1α3(j+1)2α+(332α)(j+1)1α+3j2α(33α2)j1α(j1)2α+(1α2)(j1)1α.

For j = n−1, we have Γ(3α)wj,n=(42n2α)n1α+(3n6+3α2)(n1)1α(n+α23)(n2)1α,

For j = n, we have Γ(3α)wn,n=(n3+3α2)n1α+(1α)(2α)nα(n2+α2)(n1)1α.

Together, these estimates complete the proof of Lemma 3.2. □

We remark that for any fixed n ≥ 5, wj,n,j = 0,1,…,n−2 are independent on n and only wn−1, n and wn, n depend on n. For example, we have w0,n=(1+α2)21α for any n5,w1,n=(2+α2)31α(3+32α)21α for any n ≥ 5, …. Based on this observation, we may define wj, j = 0, 1, 2, …, n−1 which we need in the numerical scheme (3.22)-(3.25) as follows. For j = 0, 1, 2, we define w0 = w0,n, w1 = w1,n, w2 = w2,n with n ≥ 5. For wj, j = 3, 4, …, n−2, we define wj = wj, n, j = 3, 4, …, n−2, where Γ(3α)wj,n=12Ej+2Fj+1+12Gj=(j+2)2α(1α2)(j+2)1α3(j+1)2α+(332α)(j+1)1α+3j2α(33α2)j1α(j1)2α+(1α2)(j1)1α.

For wj, jn−1, we also define wj by Γ(3α)wj=12Ej+2Fj+1+12Gj=(j+2)2α(1α2)(j+2)1α3(j+1)2α+(332α)(j+1)1α+3j2α(33α2)j1α(j1)2α+(1α2)(j1)1α.

Now for any fixed n ≥ 5, we have defined all wj, j = 0, 1, 2, …, n−2, n−1, n, …. We remark that in our numerical method (3.22)-(3.25), we only need the weights wj, j = 0, 1, 2, …, n−1, where wj = wj, n, j = 0, 1, 2, …, n−2, but wn−1wn−1, n. We shall see that in the proof of Theorem 3.3, we need to define all wj, j = 0, 1, 2, …, n−2, n−1, n, … in order to use the discrete Laplace transform. For wj, j = 0, 1, 2, … defined above, we may introduce the characteristic function δ (ζ) δ(ζ)α=j=0wjζj.(3.21)

We now introduce the following corrected higher order time discretisation scheme for solving (2.1), with c0 = 11/12, c1 = −5/12, ταj=1nwnjVj+AVn=(Au0)(1+c0),n=1,(3.22) ταj=1nwnjVj+AVn=(Au0)(1+c1),n=2,(3.23) ταj=1nwnjVj+AVn=Au0,n=3,4,N,(3.24) V0=0,(3.25)

where wj, j = 0, 1, 2, … are determined by (3.21) below.

Remark 3.2

Note that the weights wj, j = 0, 1, 2, … in (3.22)-(3.25) are deduced based on Lemma 3.2 for the approximation of Riemann-Liouville fractional derivative 0RDtαu(tn) with n ≥ 5. Therefore one needs to choose at least N ≥ 5 for the numerical method (3.22)-(3.25).

Theorem 3.3

([50]) Let c0 = 11/12, c1 = −5/12 and let V(tn) and Vn be the solutions of (3.4) and (3.22)-(3.25), respectively. Let u0L2 (Ω). Then there exists a positive constant C such that VnV(tn)Ck3αtnα3u0,n5.

4 Correction of the discontinuous Galerkin method in McLean and Mustapha [35]

In this section, we will consider the discontinuous Galerkin method for solving (1.1) introduced by McLean and Mustapha [35]. We first obtain the same time discretisation scheme as in McLean and Mustapha [35] by approximating the Riemann-Liouville fractional derivative with the L1 method (or Diethelm’s finite difference method). We then introduce a higher order numerical method for solving (1.1) based on this scheme.

4.1 The discontinuous Galerkin scheme in McLean and Mustapha

[35]. In this subsection, we obtain the same time discretisation scheme as in McLean and Mustapha [35] by approximating the Riemann-Liouville fractional derivative by using the L1 method. We first write (1.1) as the following equivalent form ut+0RDt1αAu=0,for0<tT,withu(0)=u0,(4.1)

where ut denotes the time derivative and 0RDtαu(t) denotes the Riemann-Liouville fractional derivative.

At t = tn, we approximate the time derivative by using the backward Euler method ut(tn)=(u(tn)u(tn1))/τ+O(τ),asτ0.

To approximate the Riemann-Liouville fractional derivative 0RDt1α A u (tn), we shall use the following Diethelm finite difference method [9], with uC2[0, T; 𝓓 (A)] 0RDt1αAu(tn)=τα1j=0nwnjAu(tj)+O(τ1+α),asτ0,(4.2)

where wj, j = 0, 1, 2, …, n−1 are given again by (3.7) and wn satisfies Γ(1+α)wn=(n1)αnα+αnα1.(4.3)

With Unu(tn), we define the following time discretisation problem for solving (4.1) UnUn1+ταj=0nwnjAUj=0,n1,withU0=u0,(4.4)

where wj, j = 0, 1, 2, …, n−1 are given by (3.7) and wn is corrected as Γ(1+α)wn=2nα+(n1)α+(n+1)α.(4.5)

Now all wj, j = 0, 1, 2, …, n are defined by (3.7). The reason for correcting wn is that we shall use the discrete Laplace transform for (w0, w1, w2, …, wn, …) in the proof of the error estimates.

Theorem 4.1

([53]) Let u (tn) and Un be the solutions of (4.1) and (4.4), respectively. Let u0L2 (Ω). Then we have, with 0 < α < 1, u(tn)UnC(ταtnα+τtn1)u0.(4.6)

We note that the convergence rate of the time discretisation scheme (4.4) is only O(τα), 0 < α < 1 for smooth and nonsmooth initial data. To derive a time discretisation method for solving (4.1) with the convergence rate O(τ) for smooth and nonsmooth initial data u0, we will approximate 0RDt1α A u (tn) by 0RDt1αAu(tn)τα1j=1nwnjAu(tj),

where we ignore the term A u(t0) in (4.2). More precisely, we choose wn = 0 in the summation j=0nwnjAu(tj) in (4.2). It is easy to show that 0RDt1αAu(tn)=τα1j=1nwnjAu(tj)+O(τ),as τ0.(4.7)

To see this, by (4.2), it suffices to show that, for the fixed tn = n τ, τα1wn=O(τ),asτ0.(4.8)

In fact, let tn be fixed, for example, assume that tn = 1, n = 1/τ, we have Γ(1+α)τα1wn=τα1(αnα1+(n1)αnα)=αtnα1+(n1)ατα1nατα1=tnα1(α+(n1)αnα1nαnα1)=tnα1(α+(1/τ1)α(1/τ)α11τ)=tnα1(α+(1τ)α1τ)=tnα1(α+(1τα+O(τ2)1τ)=tnα1O(τ)=O(τ),asτ0,

which implies (4.8) and therefore (4.7) follows.

Now we may define the time discretisation scheme for solving (4.1), with U0 = u0, UnUn1+kαj=1nwnjAUj=0,n1,(4.9)

where wj, j = 0, 1, 2, …, n−1, n ≥ 1 are given by Γ(1+α)wj=1,forj=0,2jα+(j1)α+(j+1)α,forj=1,2,,n1.(4.10)

The time stepping method (4.9) was first introduced in McLean and Mustapha [35] for solving (4.1) by using the piecewise discontinuous Galerkin method, see other related works for the discontinuous Galerkin method [8], [37], [38], [40].

Theorem 4.2

([35]) Let u (tn) and Un be the solutions of (4.1) and (4.9), respectively. Let u0L2 (Ω). Then we have, with 0 < α < 1, u(tn)UnCτtn1u0.(4.11)

4.2 Correction of the discontinuous Galerkin method of McLean and Mustapha [35]

In this subsection, we shall introduce a modified piecewise constant discontinuous Galerkin method for solving (4.1). We show that this method has the convergence rate O(τ1+α) for smooth and nonsmooth initial data u0.

Following the idea in Lubich et al. [33], we approximate the time derivative ut (tn) by using a second order backward difference method ut(tn)=32u(tn)2u(tn1)+12u(tn2)τ+O(τ2),as τ0.

We define the following finite difference method for solving (4.1), with Unu(tn) and c0 = 1/2, D¯Un+τα1(j=1nwnjAUj+c0wn1Au0)=0,n2,(4.12) D¯Un+τα1(j=1nwnjAUj+c0wn1Au0)=0,n=1,(4.13) U0=u0,U1=u0,(4.14)

where D¯Un=32Un2Un1+12Un2τ,n1,

and wj, j = 0, 1, 2, …, n−1 are defined by (4.10). Here we use a modification term c0wn−1 A u0 as in Lubich et al. [33, (1.18)].

Theorem 4.3

([53]) Let u (tn) and Un be the solutions of (4.1) and (4.12)-(4.14), respectively. Let u0L2 (Ω). Then we have, with 0 < α < 1, u(tn)UnCτ1+αtn1αu0,n1.(4.15)

4.3 Numerical examples

In this subsection, we will consider the experimentally determined convergence rates of the different numerical methods discussed in Section 4 for solving (4.1).

Let us consider the following time fractional partial differential equation in the one dimensional case: 0CDtαu(x,t)uxx=0,0<x<1,0<tT,u(0,t)=u(1,t)=0,u(x,0)=u0(x).

Let 0 < t0 < t1 < … < tN = T be the time partition of [0, T] with T = 1 and τ the time step size. Let Nh be a positive integer. Let 0 = x0 < x1 < x2 < … < xNh = 1 be the space partition and h the space step size. The space is discretised by using the standard linear finite element method.

We first consider the scheme (4.4): the convergence rate was proved to be O(τα) for both smooth and nonsmooth data in Theorem 4.1. To observe this convergence rate, we first calculate the reference solution uref(t) at T = 1 with href = 2−6 and τref = 2−10. We then use h = 2−6 and τ = kappa * τref with kappa = [22, 23, 24,25, 26] to obtain the approximate solution at u(T) with T = 1. Let eτ denote the error of u(T) at T = 1 with the time step size τ and the fixed space step size h = 2−6. By Theorem 4.1, we have ∥eτ∥ ≤ C τα. Thus the convergence rate α is determined experimentally by αlog2(e2τeτ).

Choosing the smooth and nonsmooth initial data (a) u0 = sin (π x) and (b) u0 = χ[0, 1/2], we observe, in Table 5, that the experimentally determined convergence rate is indeed almost O(kα) for the different α ∈ (0, 1) in both smooth and nonsmooth data cases.

Table 5

Time convergence rates with the different α ∈ (0, 1) for the numerical method (4.4)

We next consider the numerical method (4.9) proposed by McLean and Mustapha [35] which has the convergence rate O(τ) for both smooth and nonsmooth data. Using the same notations and the same initial data as in Table 5, we found, in Table 6, that the experimentally determined convergence rate of this method is indeed approximately 1.

Table 6

Time convergence rates with the different α ∈ (0, 1) for the numerical method (4.9)

Finally we consider the improved numerical method (3.22)-(3.22) which has the convergence rate O(τ1+α) for both smooth and nonsmooth data. Using the same notations and the same initial data as in Tables 5 and 6, we found, in Table 7, that the experimentally determined convergence rate is approximately τ1+α (actually the experimentally determined convergence rate is better than 1+ α) as we expected.

Table 7

Time convergence rates with the different α ∈ (0, 1) for the numerical methods (3.22)-(3.24)

5 Open problems and issues for further work

In this paper, we have reviewed an approach for constructing the higher order time discretisation schemes for solving (1.1). We propose some open problems and issues for further works.

  • To use the starting weights to construct higher order time discretisation schemes for solving (1.1), one needs to assume that the solution u(t, x) has the series expansion form (1.10) which implies that the source term f should satisfy some compatibility conditions. We may need to ask: which types of compatibility conditions should f satisfy? For which types of f in (1.1) may we use the starting weights techniques to improve the convergence rates of the numerical methods for solving (1.1)?

  • To approximate the Hadamard finite-part integral with piecewise p-th order polynomials with p ≥ 3 to obtain the approximate scheme to the Riemann-Liouville fractional derivative with the convergence order p+1−α. Can one use such a scheme to construct the corrected higher order time discretisation method for solving (1.1)?

  • To construct the corrected higher order numerical methods for solving (1.1) by using the higher order approximate schemes to the Riemann-Liouville fractional derivatives developed in Li et al. [2], [25].

  • To construct the corrected higher order time discretisation method for solving u(t)u(0)+0RDtα(Au(t))=0,(5.1)

    by approximating the Riemann-Liouville integral 0RDtα(Au(t)) with the Lubich convolution quadratures discussed in Jin et al. [23].

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About the article

Received: 2017-05-24

Revised: 2017-08-31

Published Online: 2017-10-31

Published in Print: 2017-10-26


Citation Information: Fractional Calculus and Applied Analysis, Volume 20, Issue 5, Pages 1076–1105, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2017-0058.

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