Convergence outside the initial layer for a numerical method for the time-fractional heat equation Jos´e Luis Gracia, Eugene O’Riordan, and Martin Stynes?? Department of Applied Mathematics, University of Zaragoza, Spain School of Mathematical Sciences, Dublin City University, Ireland Applied and Computational Mathematics Division, Beijing Computational Science Research Center, China.
[email protected] [email protected] [email protected]
Abstract. In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order δ where 0 < δ < 1. It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time t = 0; this reduces the global order of convergence of the finite difference method to O(h2 +τ δ ), where h and τ are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from t = 0. This motivates us to investigate if the finite difference method is more accurate away from t = 0. Numerical experiments with various non-smooth and incompatible initial conditions show that, away from t = 0, one obtains O(h2 + τ ) convergence. Keywords: Time-fractional heat equation, Caputo fractional derivative, initial-boundary value problem, L1 scheme, layer region, smooth and non-smooth data, compatibility conditions.
1
Introduction
In this paper we consider the time-fractional heat equation Dtδ u −
∂2u =0 ∂x2
{prob}
(1a)
{proba}
for (x, t) ∈ Q := (0, π) × (0, 1], with
??
Corresponding Author
u(0, t) = u(π, t) = 0 for t ∈ (0, 1],
(1b) {probb}
u(x, 0) = φ(x) for x ∈ [0, π],
(1c)
{probc}
2
J.L. Gracia, E. O’Riordan and M. Stynes
where 0 < δ < 1 and Dtδ denotes the Caputo fractional derivative, which is defined [1] by Dtδ g(x, t)
{Regu}
1 := Γ (1 − δ)
Z
t
(t − s) s=0
−δ ∂g
∂s
(x, s) ds
¯ for (x, t) ∈ Q.
(2)
For typical solutions u of (1), the temporal derivative ut is unbounded as t → 0; see, e.g., [5, 8]. This weak singularity must be taken into account in any ¯ then this forces the discussion of (1), for if one assumes that ut is bounded on Q, initial condition φ to be identically zero, which implies that the solution u ≡ 0; see [7, Example 2]. In [8], assuming that the data of problem (1) satisfy certain regularity and compatibility conditions, the following estimates for its solution u were proved: ` k ∂ u ∂ u δ−` ∂xk (x, t) ≤ C, for k = 0, 1, 2, 3, 4, ∂t` (x, t) ≤ C(1 + t ) for ` = 1, 2, (3) for all (x, t) ∈ [0, π] × (0, 1]. When problem (1) is approximated on a uniform mesh using the classical L1 approximation [6] for the time-fractional derivative Dtδ u and a standard threepoint scheme for ∂ 2 u/∂x2 , it is proved in [8] that this scheme is O(h2 + τ δ ) convergent nodally, where h, τ are the spatial and temporal mesh widths. But a new sharp estimate for the truncation error of the L1 scheme shows that, while it is only O(1) near t = 0, it is much smaller away from t = 0. Consequently one wonders whether the finite difference method of [8] becomes more accurate away from t = 0. In the present paper we investigate this question by means of numerical experiments and an improved order of convergence is indeed observed: one obtains O(h2 + τ ) convergence on subdomains of Q that ¯ are bounded away from t = 0, in contrast to the O(h2 + τ δ ) convergence in Q that was obtained in [8]. This improved accuracy away from t = 0 is obtained in various examples, even when the initial condition φ of (1c) is non-smooth or is incompatible with the other data of (1). This phenomenon is a remarkable property of the computed solution. It was investigated in [4] in an L2 -norm setting; in the present paper we work in the discrete L∞ norm. The discrete L∞ analysis of the phenomenon will be considered in [3]. The structure of the present paper is as follows. In Section 2 we describe our difference scheme and state a new truncation error estimate for the L1 approximation. Numerical experiments in Section 3 verify the sharpness of this estimate and demonstrate the enhanced accuracy of the computed solution away from t = 0 when the data of (1) are smooth and compatible. In Section 4 we show that this accuracy is still obtained when the initial condition φ is rough or is incompatible with the other data of (1). Notation: In this paper C denotes a generic constant that depends on the data of the boundary value problem (1) but is independent of any mesh used to solve (1) numerically. Note that C can take different values in different places.
{Caputo}
Numerical results for the time-fractional heat equation
2
3
The discretisation and truncation error estimates
{sec:L1}
Let M and N be positive integers. Set h = π/M and xm := mh for m = 0, 1, . . . , M . Set τ = 1/N and tn := nτ for n = 0, 1, . . . , N . Then the mesh is {(xm , tn ) : m = 0, 1, . . . , M, n = 0, 1, . . . , N }. The L1 approximation [6] of the Caputo fractional derivative Dtδ is based on writing Dtδ u(xm , tn ) =
n−1 X Z tk+1 1 ∂u(xm , s) (tn − s)−δ ds , Γ (1 − δ) ∂s s=tk k=0
then approximating ∂u(xm , s)/∂s by (u(xm , tk+1 ) − u(xm , tk ))/τ on each time δ n interval [tk , tk+1 ]. That is, the L1 approximation DN um is given by δ n DN um
n−1 X uk+1 − uk Z tk+1 1 m m (tn − s)−δ ds := Γ (1 − δ) τ s=tk k=0
=
1 Γ (2 − δ)
n−1 X k=0
k uk+1 m − um (tn − tk )1−δ − (tn − tk+1 )1−δ , τ
(4)
{L1}
where unm is the solution computed at (xn , tm ). In our finite difference method, the Caputo fractional derivative Dtδ is approximated by the L1 approximation (4) and the spatial derivative uxx is approximated by the standard formula uxx (xm , tn ) ≈ δx2 unm :=
unm+1 − 2unm + unm−1 . h2
Thus we approximate (1) by the discrete problem δ m DN un − δx2 unm = 0 for 1 ≤ m ≤ M − 1, 1 ≤ n ≤ N,
un0 = unM = 0 for 0 < n ≤ N, u0m = φ(xm ) for 0 ≤ m ≤ M.
{scheme}
(5a) {schemea} (5b) {schemeb} (5c) {schemec}
The discretisation (5) of (1) has been considered by several authors. Under suitable hypotheses on the data of the problem, it is shown in [8] that, taking into considering the weak singularity indicated by the bounds (3), the solution {unm } of (5) satisfies the error estimate max |u(xm , tn ) − unm | ≤ C h2 + τ δ (6) {ConvResult} ¯ (xm ,tn )∈Q
for some constant C, and this estimate is sharp. Remark 1. In all our numerical experiments below we shall take M = N , so the errors associated with the time discretisation will dominate the second-order errors associated with the spatial discretisation. For this reason we ignore the O(h2 ) error in our subsequent discussions and concentrate on the temporal error O(τ β ) for various β > 0.
4
J.L. Gracia, E. O’Riordan and M. Stynes
Using the bounds (3) on the derivatives of u, the truncation error estimate δ DN u(xm , tn ) − Dtδ u(xm , tn ) ≤ Cn−δ , n = 1, 2, . . . , N, (7) was derived in [8]. In [3] this estimate is sharpened as follows: {lem:trunc1}
Lemma 1. Assume that u satisfies (3). Then there exists a constant C such that for each mesh point (xm , tn ) ∈ Q one has δ |DN u(xm , tn ) − Dtδ u(xm , tn )| ≤ Cn−1 .
Remark 2. One can write the truncation error estimate of Lemma 1 as δ |DN u(xm , tn ) − Dtδ u(xm , tn )| ≤ Cn−1 = Cτ t−1 n ,
{trunc2}
(8)
i.e., the truncation error associated with the L1 approximation is first-order if tn ≥ C1 , where C1 is any fixed positive constant in (0, 1]. This estimate motivates us to investigate if the scheme (5) provides better approximations to the solution away from t = 0. Throughout the paper, numerical results will be given in the subdomain ¯ ∗ := [0, π] × [0.1, 1], Q which is a subset of Q that is bounded away from t = 0.
3
Numerical experiments for smooth and compatible initial-boundary conditions
{sec:NumerExpSmooth}
In Sections 3 and 4 we shall consider several test problems with various initial conditions. The errors in the solutions computed by the difference scheme (5) are estimated using the two-mesh principle [2]: on a uniform mesh with mesh steps n } for m = 0, 1, . . . , 2M and h/2 and τ /2, compute the numerical solution {zm n = 0, 1, . . . , 2N with the scheme (5). Then, calculate the two-mesh differences dδM,N :=
max 0 ≤ m ≤ M, 0≤n≤N
2n |unm − z2m |;
(9)
and from these values one computes the estimated orders of convergence by ! dδM,N δ qM,N = log2 {orders} . (10) dδ2M,2N ¯ ∗ to estimate the Analogous quantities will be computed on the subdomain Q maximum errors and rates of convergence on that subdomain.
{BoundsTruncCrude}
Numerical results for the time-fractional heat equation
5
1 1
0.9 0.9
0.8 0.8 Computed solution
Computed solution
0.7 0.6 0.5 0.4 0.3 0.2
0.7 0.6 0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
1
0.5
3 0
0 0
0.5
1
1.5
2
2.5
0.5
1
Space
Space
{fig:ex1}
3 0
Time
Time
Fig. 1. Solution for δ = 0.2 (left) and δ = 0.8 (right).
{ex1}
Example 1. In (1) let the initial condition be φ(x) = sin x. One can easily verify that the solution of (1) is then u(x, t) = Eδ (−tδ ) sin x, where Eδ (·) is the MittagLeffler function [1], which is defined by Eδ (z) :=
∞ X k=0
zk . Γ (δk + 1)
Figure 1 shows the solutions for δ = 0.2 and δ = 0.8. The layer at t = 0 is clearly visible; it is sharper for δ = 0.2. The rest of Section 3 presents numerical results for Example 1. We first show that the temporal truncation error estimate of Lemma 1 is sharp. Tables 1 and 2 display for Example 1 the maximum value of the truncation δ error |DN u(xm , tn ) − Dtδ u(xm , tn )| and their two-mesh orders of convergence in ¯ and Q ¯ ∗ . These tables indicate that the temporal truncation error the domains Q ¯ as N increases (this agrees with (7)) of the L1 scheme does not converge in Q ¯ ∗ , in agreement with (8). but it is first-order convergent in the subdomain Q ¯ Table 1. Example 1: Temporal truncation errors in Q {Table:TE} N=M=64 δ = 0.2 1.283E-001 -0.047 δ = 0.4 2.459E-001 -0.023 δ = 0.6 2.678E-001 0.010 δ = 0.8 1.801E-001 0.036
N=M=128 1.325E-001 -0.042 2.498E-001 -0.017 2.660E-001 0.008 1.757E-001 0.022
N=M=256 1.364E-001 -0.037 2.527E-001 -0.012 2.645E-001 0.006 1.730E-001 0.013
N=M=512 1.400E-001 -0.033 2.549E-001 -0.009 2.635E-001 0.004 1.715E-001 0.008
N=M=1024 1.433E-001 2.565E-001 2.628E-001 1.706E-001
Next, the convergence of the solution of scheme (5) is investigated. Although the exact solution of Example 1 is known, the errors and rates of convergence
6
J.L. Gracia, E. O’Riordan and M. Stynes ¯∗ Table 2. Example 1: Temporal truncation errors in the subdomain Q {Table:TESubdomain} N=M=64 δ = 0.2 5.327E-003 1.083 δ = 0.4 6.492E-003 1.303 δ = 0.6 7.837E-003 1.328 δ = 0.8 1.052E-002 1.143
N=M=128 2.514E-003 1.197 2.632E-003 1.447 3.122E-003 1.469 4.760E-003 1.236
N=M=256 1.097E-003 1.185 9.651E-004 1.437 1.128E-003 1.457 2.021E-003 1.222
N=M=512 4.824E-004 1.163 3.564E-004 1.411 4.108E-004 1.428 8.664E-004 1.199
N=M=1024 2.154E-004 1.340E-004 1.526E-004 3.773E-004
are computed using the two-mesh principle in accordance with the examples of Section 4 whose exact solutions are unknown. Table 3 displays the maximum δ two-mesh differences dδM,N and orders of convergence qM,N for Example 1. They δ indicate that the method converges at the rate O(τ ) in agreement with the error estimate (6) (recall that M = N , so temporal error dominates spatial error). Table 3. Example 1: Maximum two-mesh differences and orders of convergence. {tb:ex1Differences} N=M=64 δ = 0.2 2.235E-002 0.109 δ = 0.4 1.775E-002 0.340 δ = 0.6 7.774E-003 0.588 δ = 0.8 2.863E-003 0.817
N=M=128 2.072E-002 0.117 1.402E-002 0.354 5.172E-003 0.592 1.624E-003 0.810
N=M=256 1.910E-002 0.126 1.097E-002 0.365 3.431E-003 0.595 9.263E-004 0.808
N=M=512 1.751E-002 0.134 8.520E-003 0.373 2.271E-003 0.597 5.293E-004 0.805
N=M=1024 1.595E-002 6.578E-003 1.501E-003 3.030E-004
Figure 2 gives the pointwise two-mesh differences at x = π/2 for δ = 0.2 and the discretisation parameters M = N = 64 and M = N = 128. This plot reveals that the maximum two-mesh difference for both solutions occurs at the first interior mesh point in this example. If one computes the two-mesh differences in the subdomain [0, π] × [p, 1] with p a positive constant independent of N , the computed orders of convergence indicate that the method is first¯ ∗ for order convergent. In particular, the numerical results in the subdomain Q Example 1 are displayed in Table 4. The observed order of convergence for the solution of (5) is of the same order as the order of the truncation error stated in (8) (see also Table 2). Table 4. Example 1: Maximum two-mesh differences and orders of convergence in the ¯∗. subdomain Q
{tb:ex1SubdomainDifferences} N=M=64 δ = 0.2 2.023E-003 0.984 δ = 0.4 3.872E-003 0.989 δ = 0.6 4.371E-003 0.944 δ = 0.8 2.826E-003 0.878
N=M=128 1.023E-003 1.051 1.952E-003 1.053 2.271E-003 1.008 1.537E-003 0.917
N=M=256 4.938E-004 1.027 9.406E-004 1.033 1.129E-003 1.008 8.142E-004 0.937
N=M=512 2.423E-004 1.001 4.596E-004 1.009 5.618E-004 0.998 4.254E-004 0.946
N=M=1024 1.210E-004 2.284E-004 2.814E-004 2.208E-004
Numerical results for the time-fractional heat equation
7
0.025 N=M=64 N=M=128
Two−mesh differences
0.02
0.015
0.01
0.005
0
{fig:ex1_differences}
0
0.02
0.04
0.06
0.08 Time
0.1
0.12
0.14
0.16
Fig. 2. Example 1: Pointwise two-mesh differences at x = π/2 with δ = 0.2, N = M = 64 () and N = M = 128 (◦).
4
Numerical experiments for non-smooth data {sec:NumerExp}
In this section it will be shown that the improvement in the order of convergence ¯ ∗ that was observed numerically in Section 3 also occurs of the scheme (5) in Q in problems where the initial condition is not smooth or corner compatibility conditions between the data of (1) are not satisfied. {ex:C0IC}
Example 2. Consider now an example whose initial condition φ is continuous in [0, π] but fails to be differentiable at one interior point of (0, π). Let the initial condition be ( 2x/π if 0 ≤ x ≤ π/2, φ(x) = (11) 2(π − x)/π if π/2 < x ≤ π, which is not differentiable at x = π/2. The computed solutions for δ = 0.2 and δ = 0.8 are shown in Figure 3. The maximum two-mesh differences and the ¯ and the subdomain Q ¯ ∗ appear orders of convergence for the scheme (5) in Q in Tables 5 and 6, respectively. The numerical results in Table 5 shows that the order of global convergence is smaller than δ (compare with the numerical results in Table 3) — perhaps it is δ/2. On the other hand, Table 6 shows again ¯∗. first-order convergence on the subdomain Q Table 5. Example 2: Maximum two-mesh differences and orders of convergence.
{tb:singulardifferences N=M=64 δ = 0.2 2.096E-002 0.090 δ = 0.4 2.169E-002 0.217 δ = 0.6 1.582E-002 0.332 δ = 0.8 1.014E-002 0.469
N=M=128 1.969E-002 0.087 1.866E-002 0.206 1.257E-002 0.312 7.326E-003 0.431
N=M=256 1.854E-002 0.089 1.618E-002 0.202 1.012E-002 0.305 5.433E-003 0.414
N=M=512 1.743E-002 0.092 1.407E-002 0.201 8.194E-003 0.302 4.077E-003 0.406
N=M=1024 1.636E-002 1.224E-002 6.648E-003 3.077E-003
J.L. Gracia, E. O’Riordan and M. Stynes 1
1
0.9
0.9
0.8
0.8
0.7
0.7
Computed solution
Computed solution
8
0.6 0.5 0.4 0.3
0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
3 0
0.5
1
0 0
0.5
1
1.5
2
2.5
3 0
1
0.5 Time
Time Space
Space
Fig. 3. Computed solution un m of Example 2 when δ = 0.2 (left) and δ = 0.8 (right).
{fig:C0IC}
Table 6. Example 2: Maximum two-mesh differences and orders of convergence in the ¯∗. subdomain Q
{tb:SingularSubdomainDi N=M=64 δ = 0.2 1.881E-003 1.037 δ = 0.4 3.581E-003 1.022 δ = 0.6 4.378E-003 0.987 δ = 0.8 3.861E-003 0.952
N=M=128 9.167E-004 1.077 1.764E-003 1.071 2.209E-003 1.039 1.995E-003 0.995
N=M=256 4.345E-004 1.041 8.396E-004 1.042 1.075E-003 1.024 1.001E-003 0.988
N=M=512 2.111E-004 1.008 4.077E-004 1.013 5.285E-004 1.006 5.047E-004 0.978
N=M=1024 1.049E-004 2.020E-004 2.632E-004 2.562E-004
{exa:discontinuous}
Example 3. Consider now an example whose initial condition φ is discontinuous at one interior point of (0, π). Suppose the initial condition is ( 2x/π, if 0 ≤ x ≤ π/2, u(x, 0) = (12) 0, if π/2 < x ≤ π, which is discontinuous at x = π/2. The computed solutions for δ = 0.2 and δ = 0.8 are shown in Figure 4. Observe that the solution is now more complicated: it exhibits layer regions caused by the time-fractional derivative and the discontinuity of the initial condition. Table 7 gives the maximum two-mesh ¯ from these numerical results, the differences and the orders of convergence in Q; scheme does not appear to converge. Nevertheless, if we compute the two-mesh ¯ ∗ (see Table 8), then the method appears to be differences in the subdomain Q first-order convergent. {exa:IncompCornerOrder1}
{CompCondOrder0}
Example 4. In Example 1 the initial condition satisfies the zero-order compatibility condition φ(0) = 0, φ(π) = 0, (13) and first-order compatibility condition
{CompCondOrder1}
φ00 (0) = φ00 (π) = 0,
(14)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
Computed solution
Computed solution
Numerical results for the time-fractional heat equation
0.6 0.5 0.4 0.3 0.2
0.6 0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
3 0
1
0.5
0 0
0.5
1
1.5
2
2.5
Time
3 0
1
0.5 Time
Space
{fig:discontinuous}
9
Space
Fig. 4. Computed solution un m of Example 3 when δ = 0.2 (left) and δ = 0.8 (right). Table 7. Example 3: Maximum two-mesh differences and orders of convergence.
lardifferencesDiscontinuous} N=M=64 δ = 0.2 2.420E-002 0.205 δ = 0.4 3.610E-002 0.222 δ = 0.6 4.445E-002 0.251 δ = 0.8 5.426E-002 0.280
N=M=128 2.099E-002 0.122 3.096E-002 0.151 3.734E-002 0.168 4.470E-002 0.194
N=M=256 1.929E-002 0.078 2.789E-002 0.105 3.325E-002 0.118 3.908E-002 0.141
N=M=512 1.828E-002 0.055 2.593E-002 0.076 3.063E-002 0.082 3.545E-002 0.097
N=M=1024 1.759E-002 2.459E-002 2.895E-002 3.315E-002
(see [5]). We now consider an example that satisfies the zero-order compatibility condition (13) but not the first-order compatibility condition (14). Define the initial condition by 4 φ(x) = 2 x(π − x). (15) π Observe that φ00 (0) 6= 0 and φ00 (π) 6= 0. The computed solutions for δ = 0.2 and δ = 0.8 are shown in Figure 5. Although this example does not satisfy the compatibility condition (14), we reach the same conclusions as for Example 1: The method converges with order δ in the whole domain and with first order in ¯ ∗ (see Tables 9 and 10, respectively). the subdomain Q
{exa:IncompCornerOrder0
Example 5. Finally, we consider an example that fails to satisfy the zero-order compatibility condition (13). Choose φ(x) ≡ 1.
(16)
The computed solutions for δ = 0.2 and δ = 0.8 are shown in Figure 6. The structure of the solution is very complicated; it has layer regions caused by the time-fractional derivative and the incompatibilities at the corners (0, 0) and (π, 0). The numerical results in Table 11 shows that the method is not conver¯ but Table 12 shows that the method is first-order gent in the whole domain Q ¯∗. convergent in the subdomain Q
10
J.L. Gracia, E. O’Riordan and M. Stynes
Table 8. Example 3: Maximum two-mesh differences and orders of convergence in the ¯∗. subdomain Q
ainDifferencesDiscontinuous} N=M=128 3.925E-003 1.013 5.023E-003 1.020 5.876E-003 1.019 6.612E-003 1.024
N=M=256 1.944E-003 1.007 2.476E-003 1.011 2.899E-003 1.010 3.253E-003 1.012
1
1
0.9
0.9
0.8
0.8
0.7
0.7 Computed solution
Computed solution
N=M=64 δ = 0.2 7.855E-003 1.001 δ = 0.4 1.004E-002 0.999 δ = 0.6 1.169E-002 0.993 δ = 0.8 1.309E-002 0.985
0.6 0.5 0.4 0.3 0.2
N=M=512 9.677E-004 1.001 1.229E-003 1.002 1.439E-003 1.000 1.612E-003 0.999
N=M=1024 4.837E-004 6.136E-004 7.195E-004 8.066E-004
0.6 0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
3 0
0.5
1
0 0
0.5
1
1.5
2
Time Space
2.5
3 0
0.5
1
Time Space
Fig. 5. Computed solution un m of Example 4 when δ = 0.2 (left) and δ = 0.8 (right).
Conclusions Using a standard finite difference scheme for the time-fractional heat equation, we see that the convergence rate of the numerical method is affected by the smoothness of the initial condition. However, we observe that the reduction in the convergence rate is confined to an initial region and first-order convergence is seen away from t = 0. Acknowledgments. The research of Jos´e Luis Gracia was partly supported by the Institute of Mathematics and Applications (IUMA), the project MTM201340842-P and the Diputaci´ on General de Arag´on. The research of Martin Stynes was supported in part by the National Natural Science Foundation of China under grant 91430216.
References 1. Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010, An application-oriented exposition using differential operators of Caputo type. 2. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), vol. 16, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1750671 (2001c:65003)
{fig:IncompCornerOrder1
Numerical results for the time-fractional heat equation
11
Table 9. Example 4: Maximum two-mesh differences and orders of convergence.
{tb:singulardifferences N=M=64 δ = 0.2 2.227E-002 0.116 δ = 0.4 1.658E-002 0.372 δ = 0.6 6.601E-003 0.603 δ = 0.8 2.520E-003 0.826
N=M=128 2.054E-002 0.126 1.281E-002 0.386 4.347E-003 0.600 1.422E-003 0.802
N=M=256 1.883E-002 0.136 9.807E-003 0.394 2.867E-003 0.600 8.154E-004 0.801
N=M=512 1.714E-002 0.145 7.462E-003 0.399 1.891E-003 0.600 4.680E-004 0.800
N=M=1024 1.551E-002 5.660E-003 1.248E-003 2.687E-004
Table 10. Example 4: Maximum two-mesh differences and orders of convergence in ¯∗ the subdomain Q
{tb:SingularSubdomainDi N=M=64 δ = 0.2 2.050E-003 0.982 δ = 0.4 3.902E-003 0.988 δ = 0.6 4.289E-003 0.940 δ = 0.8 2.520E-003 0.884
N=M=128 1.038E-003 1.050 1.968E-003 1.052 2.236E-003 1.002 1.365E-003 0.898
N=M=256 5.013E-004 1.027 9.490E-004 1.033 1.116E-003 1.004 7.325E-004 0.920
N=M=512 2.460E-004 1.001 4.638E-004 1.009 5.564E-004 0.996 3.872E-004 0.935
N=M=1024 1.229E-004 2.305E-004 2.790E-004 2.026E-004
3. Jos´e Luis Gracia, Eugene O’Riordan, and Martin Stynes, Faster convergence outside the initial layer of a finite difference method for a time-fractional diffusion equation, In preparation. 4. Bangti Jin, Raytcho Lazarov, and Zhi Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36 (2016), no. 1, 197–221. MR 3463438 5. Yuri Luchko, Initial-boundary-value problems for the one-dimensional timefractional diffusion equation, Fract. Calc. Appl. Anal. 15 (2012), no. 1, 141–160. MR 2872116 6. Keith B. Oldham and Jerome Spanier, The fractional calculus, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Vol. 111. MR 0361633 (50 #14078) 7. Martin Stynes, Too much regularity may force too much uniqueness, ArXiv 1607.01955. 8. Martin Stynes, O’Riordan Eugene, and Jos´e Luis Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, Submitted for publication.
J.L. Gracia, E. O’Riordan and M. Stynes
1
1
0.9
0.9
0.8
0.8
0.7
0.7 Computed solution
Computed solution
12
0.6 0.5 0.4 0.3 0.2
0.6 0.5 0.4 0.3 0.2
0.1
0.1
0 0
0.5
1
1.5
2
2.5
3 0
0.5
1
0 0
0.5
1
1.5
2
Time
2.5
3 0
0.5
1
Time
Space
Space
Fig. 6. Computed solution un m of Example 5 when δ = 0.2 (left) and δ = 0.8 (right).
{fig:IncompCornerOrder0
Table 11. Example 5: Maximum two-mesh differences and orders of convergence.
{tb:singulardifferences N=M=64 δ = 0.2 2.466E-002 0.039 δ = 0.4 3.823E-002 -0.003 δ = 0.6 4.923E-002 -0.006 δ = 0.8 5.546E-002 -0.014
N=M=128 2.400E-002 0.033 3.831E-002 -0.001 4.944E-002 -0.005 5.600E-002 -0.018
N=M=256 2.346E-002 0.026 3.834E-002 -0.001 4.959E-002 -0.002 5.670E-002 -0.005
N=M=512 2.305E-002 0.019 3.836E-002 -0.000 4.965E-002 -0.001 5.691E-002 -0.001
N=M=1024 2.274E-002 3.837E-002 4.967E-002 5.695E-002
Table 12. Example 5: Maximum two-mesh differences and orders of convergence in ¯∗. the subdomain Q
{tb:SingularSubdomainDi N=M=64 δ = 0.2 2.157E-003 0.954 δ = 0.4 4.009E-003 0.970 δ = 0.6 5.413E-003 0.958 δ = 0.8 7.613E-003 0.935
N=M=128 1.113E-003 1.036 2.046E-003 1.043 2.786E-003 1.043 3.981E-003 1.041
N=M=256 5.431E-004 1.020 9.929E-004 1.028 1.352E-003 1.029 1.935E-003 1.032
N=M=512 2.679E-004 0.998 4.868E-004 1.007 6.623E-004 1.008 9.462E-004 1.011
N=M=1024 1.341E-004 2.422E-004 3.293E-004 4.694E-004
