Nonoverlapping Schwarz Waveform Relaxation Algorithm ... - IOS Press

Fundamenta Informaticae 151 (2017) 231–240

231

DOI 10.3233/FI-2017-1489 IOS Press

Nonoverlapping Schwarz Waveform Relaxation Algorithm for a Class of Time-Fractional Heat Equations Shu-Lin Wu∗† School of Science Sichuan University of Science and Technology Zigong 643000, Sichuan, P. R. China wushulin [email protected]

Guo-Cheng Wu‡ Data Recovery Key Laboratory of Sichuan Province College of Mathematics and Information Science Neijiang Normal University Neijiang 641100, Sichuan, P. R. China [email protected]

Abstract. In this paper, we analyze the convergence properties of the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions (TCs) for a class of heat equations with Riemann-Liouville fractional derivative. The Robin TCs contain a free parameter, which has a significant effect on the convergence rate of the SWR algorithm, and optimizing this parameter is an important step for the convergence analysis of the SWR algorithm. By studying the monotonic properties of the convergence factor obtained by applying the Fourier transform to the error functions, we provide a realiable choice of the Robin parameter in the nonoverlapping case. Numerical results are provided, which show that the analyzed Robin parameter results in satisfactory convergence rate. Keywords: Schwarz waveform relaxation, fractional heat equations, parameter optimization ∗

Address for correspondence: Sichuan University of Science and Technology, Zigong 643000, Sichuan, P. R. China The first author is supported by the NSF of China (No. 11301362), the China Postdoctoral Foundation (No. 2015M580777, No. 2016T90841), the NSF of Sichuan Province (No. 2014JQ0035, 15ZA0220) and the NSF of SUSE (No. 2015LX01). ‡ The second author is supported by the NSF of China (No. 11301257) and the Seed Funds for Major Science and Technology Innovation Projects of Sichuan Provincial Education Department (14CZ0026) †

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S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

Introduction

In the last few decades, fractional calculus has attracted great interest by many researchers. Fractional integrals and derivatives are used more and more by scientists and engineers to simulate many real world phenomena, such as the transport of fluid in porous media [1], diffusion on liquid surfaces [9], turbulent flow [12], anomalous diffusion in spiny neuronal dendrites [3], etc. Unlike the integer order differential equations, a simple and explicit analytical solution to a fractional problem is usually unavailable and thus numerical methods play an important role in investigating the properties of these problems. Moreover, compared to the integer order problems, numerical computation for fractional problems is much more time consuming, because we need to treat convolutions with singular kernels at each time grid. In the past few years, many numerical methods are studied for time-fractional differential equations and for this issue we refer the interested reader to [7, 11, 14, 15, 17, 18]. Here, our interest is to apply the Schwarz waveform relaxation(SWR) algorithms to solve a class of heat equations with Riemann-Liouville temporal fractional derivative. These algorithms belong to the widely used domain decomposition methods, but with completely new implementation strategy for time-dependent problems: the classical strategy employs the alternating or parallel Schwarz methods to the elliptic PDEs which result from semi-discretizing the time-dependent PDEs in time by some implicit time-integrator, but in the SWR framework one directly decomposes the space domain into many subdomains and then solve each subproblem simultaneously or alternately. This provides flexibility for treating different subproblems numerically differently with an adapted procedure for each subdomain; the interested reader can refer to [4] and [5] for the original idea of the SWR algorithms. The main feature of the SWR algorithm is that it decouples the whole space domain into several overlapping (or nonoverlapping) subdomains and then by imposing some artificial boundary conditions for each subdomain the time-dependent PDEs can be solved simultaneously on these subdomains. The solutions along the artificial boundaries are updated through iterations. Upon convergence, we get solutions of the underlying PDEs on all these subdomains. The artificial boundary conditions are often called transmission conditions (TCs) in this field and the Robin TCs received lots of attention in the past few years. This TCs contain a free parameter, which has a significant effect on the convergence rate of the SWR algorithm, and optimizing this parameter is one of the top-priority matters. For integer order PDEs, optimizing the Robin parameter for the SWR algorithms has been investigated widely and deeply in the literature [2, 6, 13]. However, much less results are known about the algorithms applied to fractional order PDEs. In this paper, we analyze the algorithms with Robin transmission conditions (TCs) for time-fractional heat equations, with special attention for the optimization of the Robin parameter. We introduce the time-fractional heat equations in Section 2, where we show that the study of this paper can be straightforwardly generalized to other time-fractional problems. Section 3 presents the main results of this paper, a detailed optimization of the Robin parameter for the SWR algorithm. We provide numerical results to validate the efficiency of the proposed Robin parameter by numerical results in Section 4. We conclude this paper in Section 5.

S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

2.

233

The model problem and the SWR algorithm

Our model problem is the following 1-D time-fractional heat equation ( Dtα u(x, t) = ∂xx u(x, t) + f (x, t), u(x, 0) = u0 (x),

(x, t) ∈ R × (0, T ), x ∈ R, t = 0,

(2.1)

where α ∈ (0, 1) and Dtα u denotes the Riemann-Liouville fractional derivative, defined by Dtα u(x, t)

d 1 = Γ(1 − α) dt

Z 0

t

1 u(x, τ )dτ. (t − τ )α

(2.2)

From [10, Chapter 2], for any v ∈ C1 (0, T ) it holds that limα→1− Dtα v = v 0 (t)(∀t ∈ (0, T )). Hence, the Riemann-Liouville fractional derivative builds a ‘bridge’ from v(t) to v 0 (t). For numerical solution of (2.1), the interested reader can refer to [8, 16] and references therein. In the field of fractional calculous, the Caputo derivative is another popular fractional derivative, α c Dt v(t)

1 = Γ(1 − α)

Z 0

t

1 v 0 (τ )dτ. (t − τ )α

(2.3)

The two definitions are linked by the following relation [10, pp. 62-77]: Dtα v(t)

Z t v 0 (τ ) 1 v(0)t−α dτ , ∀α ∈ (0, 1). = + Γ(1 − α) 0 (t − τ )α Γ(1 − α) | {z }

(2.4)

Caputo derivative c Dtα v(t)

Thanks to this connection, we shall only consider the Riemann-Liouville derivatives in this paper, since generalizing the results to problems with Caputo derivatives is straightforward. We now introduce the Schwarz waveform relaxation algorithms for (2.1). For simplicity, we restrict ourselves to the case of two nonoverlapping subdomain case; such a study will be generalized to multi-subdomain case in the future. We divide the spatial domain into two subdomains, Ω = Ω1 ∪ Ω2 , with Ω1 = (−∞, 0] and Ω2 = [0, +∞). Then, the SWR algorithms for (2.1) is  α k  ukj (x, t) + f (x, t), x ∈ Ωj , Dt uj (x, t) = ∂xx

∂x + (−1)j+1 p ukj (0, t) = ∂x + (−1)j+1 p uk−1 3−j (0, t),   k uj (x, 0) = u0 (x),

(2.5)

where p ∈ R is a free parameter, k ≥ 1 is iteration index and j = 1, 2. For k = 0, the functions u01,2 (x, t)(x, t) satisfying u0 (x, t)1,2 (x, 0) = u0 (x) are arbitrarily chosen.

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S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

Convergence analysis

In this section, we perform a convergence analysis for the SWR algorithm (2.5). Our analysis is based on the Fourier transform for Riemann-Liouville fractional derivative Dtα v with zero initial value v(0) = 0: Z +∞ Z +∞ F(Dtα v) = e−iωt Dtα v(t)dt = (iω)α vˆ(ω), with vˆ(ω) := F(v) := e−iωt v(t)dt. (3.1) 0

0

(Note that, in (3.1) we have used an extension technique, that is: we extend v(t) = 0 for t ≤ 0 and we denote the extension by v(t), too.) If v(0) 6= 0, then the Fourier transform of the RiemannLiouville fractional derivative is slightly different from (3.1). The Laplace transform of the Caputo derivative c Dtα is also (iω)α vˆ(ω), if v(0) = 0. More details about the Fourier transform of the fractional derivatives/integrals can be found in [10, Chapter 2]. Let ekj (x, t) = ukj (x, t) − u(x, t)|x∈Ωj . Then, it is easy to know that ekj (x, t) satisfies  k α k  Dt ej (x, t) − ∂xx
ej (x, t) = 0,
∂x + (−1)j+1 p ekj (0, t) = ∂x + (−1)j+1 p ek−1 3−j (0, t),   k ej (x, 0) = 0.

(3.2)

To analyze the convergence of (2.5), it is sufficient to study the error functions ekj (x, t) for j = 1, 2. Theorem 3.1. Let p ∈ R, t ∈ (0, T ) and ∆t be the temporal step-size. Then, it holds that



2k

max |ˆ ρ(p, ω|k e0j (x, ·) 2 ,

ej (x, ·) ≤ 2

|ω|∈[ωmin ,ωmax ]

(3.3)

where for any real-valued function v(t) with t ∈ (0, t) the 2-norm kv(·)k2 is defined by kv(·)k2 := q RT 2 ˆ(p, ω), which we call the conver0 v (t)dt. In (3.3), the quantities ωmin , ωmax and the function ρ gence factor in the frequency domain of the SWR algorithm, are defined by !2 p p − (iω)α π π (3.4) p ωmin = , ωmax = , ρˆ(p, ω) = . T ∆t p + (iω)α Here, for any s ∈ C the real part of the square root of s is a non-negative. Proof: Based on (3.1), applying Fourier transform to the error equation (3.2) gives ( p ∂x2 eˆkj = λ2 (ω)ˆ ekj , λ(ω) = (iω)α (with non-negative real part),

∂x + (−1)j+1 p eˆkj (0, t) = ∂x + (−1)j+1 p eˆk−1 3−j (0, t).

(3.5)

The solutions eˆkj can be expressed in the general form, as eˆkj = Akj eλ(ω)x + Bjk e−λ(ω)x (∀x ∈ Ωj ). Then, using the fact that eˆk1,2 do not increase exponentially at infinity, we get eˆk1 = Ak1 eλ(ω)x and eˆk2 = B2k e−λ(ω)x .

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Substituting this into the transmission conditions in (3.5), we get (p + λ(ω))Ak1 = (p − λ(ω))B2k−1 , (p + λ(ω))B2k = (p − λ(ω))Ak−1 1 , and this gives Ak+2 = ρˆ(p, ω)Ak1 and B2k+2 = ρˆ(p, ω)B2k , where ρˆ(p, ω) is given in (3.4). Hence, 1 eˆkj (x, ω) = ρˆ(p, ω)ˆ ek−2 (x, ω), ∀x ∈ Ωj . Now, by using Plancherel’s theorem, namely ’the L2 -norm j of the Fourier transform of a function is equal to the L2 -norm of the function itself’, it is easy to get





k



2k

0 = ρ ˆ (p, ω)ˆ e (x, ·) (x, ·) (3.6)

.

ej (x, ·) = eˆ2k j j 2

2

2

In a numerical computation, a numerical grid in time with spacing ∆t cannot carry arbitrarily high and low angular frequencies; the minimal (resp. maximal) frequency supported by the time π discretization, independent of the input waveforms, is Tπ (resp. ∆t ) (see [6]). This implies that





k

max |ˆ ρ(p, ω)|k e0j (x, ·) 2 . e0j (x, ·) ≤ max |ˆ ρ(p, ω)|k eˆ0j (x, ·) 2 =

ρˆ (p, ω)ˆ 2

|ω|∈[ωmin ,maxmax ]

|ω|∈[ωmin ,maxmax ]

t u

Substituting this into (3.6) gives (3.3). Mathematically, we want |ˆ ρ(p, ω)|  1, which leads to the following min-max problem min

max

p∈R |ω|∈[ωmin ,ωmax ]

|ˆ ρ(p, ω)|.

 π α  απ 
α α α  α 2 For any ω > 0, we have (±iω) 2 = ω 2 e± 2 i = ω 2 e± 4 i = ω 2 cos απ 4 ± i sin Let  απ   απ  α α α 2 2 c = cos , ηmax = ωmax , s = sin , η = ω 2 , ηmin = ωmin 4 4 Then, we have |ˆ ρ(p, ±ω| = min

(p−cη)2 +s2 η 2 . (p+cη)2 +s2 η 2

max

p>0 η∈[ηmin ,ηmax ]

(3.7) απ 4


 .

(3.8)

Hence, the above min-max problem (3.7) is equivalent to

R(p, η), with R(p, η) =

(p − cη)2 + s2 η 2 . (p + cη)2 + s2 η 2

(3.9)

Here, we used the fact that R(p, η) ≥ 1 for p ≤ 0 and therefore we only need to consider p > 0.
απ Theorem 3.2. Let T > 0 be the length of time-interval, ∆t > 0 be the step-size, c = cos , 4
απ s = sin 4 and ηmin,max be the quantities defined by (3.8). Then, for α ∈ (0, 1) the solution of the min-max problem (3.9), denoted by popt , is given by √ (3.10) popt = ηmin ηmax . For any Robin parameter p, we denoted by ρ(p) the convergence factor of the SWR algorithm in the temporal domain, i.e., ρ(p) = maxω∈[ωmin , ωmax ] |ˆ ρ(p, ω)| = maxη∈[ηmin , ηmax ] R(p, η). Then, with optimal Robin parameter popt it holds that √ ηmin − 2c ηmin ηmax + ηmax ρ(popt ) = R(popt , ηmin ) = R(popt , ηmax ) = . (3.11) √ ηmin + 2c ηmin ηmax + ηmax

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S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

Proof: We first claim that for any p > 0 the function R(p, η) does not have local maximums for η ∈ [ηmin , ηmax ]. To this end, a derivative of R(p, η) with respect to η gives ∂η R(p, η) =

4cp(η 2 − p2 )

[(p + cη)2 + s2 η 2 ]2

,

(3.12)

where we have used c2 +s2 = 1. Hence, R(p, η) has at most one extremum located at η = p. We have 2 ∂η2 R(p, p) = [(p+cη)8cp > 0, and this implies that η = p must correspond to a local minimum. 2 +s2 η 2 ]2 From (3.11) it is easy to know popt ∈ [ηmin , ηmax ], since otherwise we can further uniformly reduce R(popt , η) by increasing or decreasing popt . Hence, the min-max problem (3.10) is equivalent to min

max

p∈[ηmin ,ηmax ] η∈[ηmin ,ηmax ]

R(p, η).

(3.13)

This analysis implies that max

η∈[ηmin ,ηmax ]

R(p, η) = max {R(p, ηmin ), R(p, ηmax )} .

(3.14)

T =50, ∆t =0.01 0.7 0.6

ρ(popt)

0.5 0.4 0.3 0.2 0.1

0.1

0.2

0.3

0.4

0.5

0.6

α

0.7

T = 50

0.8

0.9

∆t =0.01

0.9

0.6

α = 0.8

α = 0.8

0.8

0.5

0.6 0.5

ρ(popt)

ρ(popt)

0.7

α = 0.5

0.4 0.3

0.4 0.3

α = 0.5

0.2

α = 0.3

0.2

α = 0.3

0.1

0.1 −6

10

−4

10

∆t

−2

10

0

10

1

10

2

10

T

Figure 3.1. The dependence of the optimal convergence rate ρ(popt ) given by (3.11) on the fractional order α (top), the temporal step-size ∆t (bottom left) and the length of time-interval T (bottom right).

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237

For p ∈ [ηmin , ηmax ], from (3.12) it is easy to know that R(p, ηmin ) and R(p, ηmax ) are respectively increasing and decreasing functions of p. Hence, the parameter p that minimizes R(p, η) for √ η ∈ [ηmin , ηmax ] is the solution of R(p, ηmin ) = R(p, ηmax ), which is ηmin ηmax . t u η

√ −2c η

η

+η

√ min max max depends on the fractional order The optimal convergence factor ρ(popt ) = ηmin min +2c ηmin ηmax +ηmax α, the step-size ∆t and the length of time-interval T , because the quantities c, ηmin and ηmax are connected with these parameters (see (3.8)). In Figure 3.1, we show the effect of these parameters on ρ(popt ). For fixed ∆t and T , the convergence factor ρ(popt ) is an increasing function of α ∈ (0, 1); for fixed α and T , ρ(popt ) increases as ∆t decreases; for fixed α and ∆t the convergence factor increases as T increases. We will provide numerical results in the next section to validate these predictions.

4.

Numerical experiments

In this section, we provide numerical results to validate the theoretical analysis. We consider the following 
α α 3 2 α  Dt u(x, t) = ∂x u(x, t) + t sin 5t x , (x, t) ∈ (0, 1) × (0, T ), (4.1) u(x, 0) = 0, x ∈ (0, 1),   u(0, t) = u(1, t) = 0, t ∈ (0, T ). Pn

ω u

(x)

r n−r We use the Gr¨unwald-Letnikov formula [10] to discretize Dtα u(x, t), as Dtα u(x, t) ≈ r=0 ∆t , b
1+α where ω0 = 1, ωr = 1 − r ωr−1 for r ≥ 1 and un (x) denotes the numerical approximation to the exact solution at time point t = tn and at space point x. At each time point t = tn , the second-order spatial derivative ∂x2 un (x) are discretized by the centered finite difference scheme.

The spatial domain [0, 1] is divided into two subdomains Ω1 = [0, 0.5] and Ω2 = [0.5, 1]. The SWR iterations stop when the error between the iterate and the reference solution satisfies max |ukm,n − um,n | ≤ 10−12 , m,n

(4.2)

where ukm,n denotes the discrete SWR solution at the k-th iteration. The reference solution {um,n } is obtained by using the same discretization on the whole space-time domain. All the computations were performed using the software MATLAB 2010a on a computer with a 3.7 GHz CPU and 4 GB RAM. We first intuitively illustrate the space-time approximation of the iterates generated by the SWR algorithm to the solution u(x, t) of (4.1) along the a curve in the space-time domain: √ t = 2 sin(π(1 − x)x)T. (4.3) We choose T = 10, α = 0.6, ∆x = 0.01, ∆t = 0.01. In Figure 4.1 on the left we show the curve (4.3). On the right, we show the approximation of the k-th iterate (dotted line) to the solution u(x, t) (solid line) of (4.1) along the curve (4.3): from top to bottom, k = 0, k = 2, k = 4 and k = 6. In Figure 4.2 on the left, we show the measured convergence rate, together with the error predicted k by the linear bound (i.e., ρ 2 (popt ) max(x,t)∈[0,1]×[0,T ] |e0 (x, t)|). The results shown in this subfigure

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S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

10

0.6 0.4

8

t = 10

0.2

2 sin(π(1 − x)x)

0

t

6

√

−0.2 10

4

2

5

t 0 0

0.2

0.4

0.6

x

0.8

Ω2

Ω1 0

1

0.1

0

0.2

0.3

0.4

0.6

0.5

0.7

0.8

0.9

1

x

√ Figure 4.1. Left: the curve t = 2 sin(π(1 − x)x)T with T = 10. Right: the approximation of the k-th iterate (dotted line) generated by the nonoverlapping SWR algorithm to the solution u(x, t) (solid line) of (4.1) along the curve shown on the left subfigure: from top to bottom, k = 0, k = 2, k = 4 and k = 6.

imply that the convergence factor analyzed by using the Fourier transform at the continuous level can precisely predict the measured error and therefore the linear bound is sharp. It would be interesting to verify if the optimal choice for the parameter p = popt derived using the continuous Fourier analysis in Theorem 3.2 really corresponds to the best choice one can make in the fully discretized algorithm. In Figure 4.2 on the right, we show the error obtained after running the Schwarz waveform relaxation algorithm with Robin transmission conditions for five iterations using various values for the Robin parameter p in the transmission conditions. The optimal choice popt from Theorem 3.2 is indicated by a circle. Clearly the Fourier analysis predicts the optimal choice of the parameter p very well.

0

Measured Error Linear Bound

10

−2

Error After 5 Iterations

10

−4

Error

10

−6

10

−8

10

0

10

−1

10

−10

10

−12

10

5

10

15

Iteration Number: k

20

25

0

2

4

6

8

10

Robin Parameter: p

Figure 4.2. Left: the measured error, together with the error predicted by the linear bound (i.e., k ρ 2 (popt ) max(x,t)∈[0,1]×[0,T ] |e0 (x, t)|). Right: the error obtained running the SWR algorithm for 5 iterations and various choices of the Robin parameter p, and indicated by a circle the choice popt predicted by Theorem 3.2.

We next verify the dependence of the convergence rate of the SWR algorithm with the optimal Robin parameter popt on the fractional order α, the temporal step-size ∆t and the length of the time-

S.L. Wu and G.C. Wu / SWR for Time-Fractional Heat Equations

239

interval. For three choices of α, we show in Figure 4.3 the measured number of iterations needed to satisfy (4.2) for several values of the temporal step-size ∆t (left) and the quantity T —the length of time-interval (right). In each subfigure, the iteration number predicted by the optimal convergence factor ρ(popt ), i.e., log(10−12 )/ log(ρ(popt )), is also presented (the lines without markers). We see that the number of iterations increases as α becomes larger and that for fixed α the number of iterations increases as ∆t becomes smaller (or as T becomes larger). This confirms what we have observed in Figure 3.1 very well. Moreover, we can see in Figure 4.3 that the iteration number predicted by the convergence factor is very close to the measured one. Again, this implies that the convergence factor analyzed by using the Fourier transform at the continuous level is sharp. T =20

50

∆t =0.01

60

α = 0.8

α = 0.8

45 50

Number of Iterations

Number of Iterations

40 35 30 25

α = 0.5

20 15

α = 0.3

40

30

α = 0.5 20

α = 0.3 10

10 5

−2

−1

10

10

∆t

0

0

10

1

10

2

10

T

Figure 4.3. For fixed T = 20 (resp. ∆t = 0.01), the measured number of iterations needed to satisfy (4.2) for several values of the temporal step-size ∆t (resp. T , the length of time-interval). In each subfigure, we consider three fractional order α and the iteration number predicted by the optimal convergence factor ρ(popt ), i.e., log(10−12 )/ log(ρ(popt )), is also presented (the lines without markers).

5.

Conclusions

We have studied a nonoverlapping Schwarz waveform relaxation algorithm with Robin transmission conditions (TCs) for a class of time-fractional heat equations. The free parameter contained in the Robin TCs was fixed by using the Fourier transform at the continuous level. Numerical results indicated that the results analyzed by using the Fourier transform predict the practical computation very well.

Acknowledgment The authors are very grateful to the anonymous referees for the careful reading of a preliminary version of the manuscript and their valuable suggestions and comments, which greatly improved the quality of this paper.

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