Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model Hong-lin LiaoNan LiuXuan Zhao

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Asymptotically compatible energy of variable-step fractional BDF2

formula for time-fractional Cahn-Hilliard model

Hong-lin Liao∗Nan Liu†Xuan Zhao‡

October 25, 2022

Abstract

A new discrete energy dissipation law of the variable-step fractional BDF2 (second-order

backward diﬀerentiation formula) scheme is established for time-fractional Cahn-Hilliard model

with the Caputo’s fractional derivative of order α∈(0,1), under a weak step-ratio constraint

0.4753 ≤τk/τk−1< r∗(α), where τkis the k-th time-step size and r∗(α)≥4.660 for α∈(0,1).

We propose a novel discrete gradient structure by a local-nonlocal splitting technique, that

is, the fractional BDF2 formula is split into a local part analogue to the two-step backward

diﬀerentiation formula of the ﬁrst derivative and a nonlocal part analogue to the L1-type

formula of the Caputo’s derivative. More interestingly, in the sense of the limit α→1−, the

discrete energy and the corresponding energy dissipation law are asymptotically compatible

with the associated discrete energy and the energy dissipation law of the variable-step BDF2

method for the classical Cahn-Hilliard equation, respectively. Numerical examples with an

adaptive stepping procedure are provided to demonstrate the accuracy and the eﬀectiveness of

our proposed method.

Keywords: time-fractional Cahn-Hilliard model, fractional BDF2 formula, discrete gradient

structure, asymptotically compatible energy, energy dissipation law

AMS subject classiﬃcations. 35Q99, 65M06, 65M12, 74A50

1 Introduction

Linear and nonlinear diﬀusion equations with fractional time derivatives have become widely mod-

els describing anomalous diﬀusion processes [8,12, 33]. These models always exhibit multi-scaling

time behaviour, which makes them suitable for the description of diﬀerent diﬀusive regimes and

characteristic crossover dynamics in complex systems. To capture the multi-scale behaviors in

time-fractional diﬀerential equations, adaptive time-stepping strategies, namely, small time steps

are utilized when the solution varies rapidly and large time steps are employed otherwise, would be

practically useful especially in the simulations of long-time coarsening dynamics [9,10,17,22–24]. It

is natural to require practically and theoretically reliable time-stepping methods on general setting

of time step-size variations, or on a wider class of nonuniform time meshes.

This work develops a new discrete energy dissipation law of the second-order fractional back-

ward diﬀerentiation formula (FBDF2) for the time-fractional Cahn-Hilliard (TFCH) ﬂow in mod-

elling the coarsening process with a general coarsening rate [2, 6, 10, 25, 30, 31, 33, 34],

∂α

tΦ = κ∆µwith µ:= δE

δΦ=f(Φ) −2∆Φ,(1.1)

∗ORCID 0000-0003-0777-6832. College of Mathematics, Nanjing University of Aeronautics and Astronautics,

Nanjing 211106, China; Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles

(NUAA), MIIT, Nanjing 211106, China. Emails: liaohl@nuaa.edu.cn and liaohl@csrc.ac.cn. This author’s work is

supported by NSF of China under grant number 12071216.

†College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China. Email:

liunan@nuaa.edu.cn.

‡School of Mathematics, Southeast University, Nanjing 210096, China. Email: xuanzhao11@seu.edu.cn.

arXiv:2210.12514v1 [math.NA] 22 Oct 2022

where f(Φ) := F0(Φ) and E[Φ] is the Ginzburg-Landau energy functional

E[Φ] := ZΩ2

2|∇Φ|2+F(Φ)dxwith the potential F(Φ) := 1

4Φ2−12.(1.2)

The notation ∂α

t:= C

0Dα

trepresents the Caputo’s derivative of order α∈(0,1), deﬁned by

(∂α

tv)(t) := Zt

ω1−α(t−s)v0(s) dswith ωβ(t) := tβ−1/Γ(β) for β > 0. (1.3)

The real valued function Φ represents the concentration diﬀerence in a binary system on the domain

Ω⊆R2, > 0 is an interface width parameter, and κ > 0 is the mobility coeﬃcient.

Let h·,·i and k·k be the L2(Ω) inner product and the associated norm, respectively. Always

we use the standard norms of the Sobolev space Hm(Ω) and the Lp(Ω) space. For vand w

belonging to the zero-mean space ˚

V:= v∈L2(Ω) | hv, 1i= 0, we use the H−1-like inner

product hv, wi−1:= (−∆)−1v, wand the induced norm kvk−1:= qhv, vi−1. The well-possness

of the TFCH model (1.1) has been established in [2,6] and the analysis indicated that the solution

lacks the smoothness near the initial time while it would be smooth away from t= 0, also see [5,6].

The motions of interfaces and coarsening rates governed by the TFCH equation were studied by

Tang, Wang and Yang [34]. For the constant mobility, the sharp interface limit model is a fractional

Stefan problem at the timescale t=O(1); while on the timescale t=O(−1/α), the sharp interface

limit model is a fractional Mullins–Sekerka model. The TFCH model with constant mobility was

proven to preserve an O(t−α/3) coarsening rate, which is asymptotically compatible (as α→1−)

with the coarsening rate O(t−1/3) of the following classical Cahn–Hilliard (CH) equation,

∂tΦ = κ∆µfor t > 0.(1.4)

As is well known, the CH model (1.4) conserves the initial volume hΦ(t),1i=hΦ(0),1iand has

the following energy dissipation law

dt+κk∇µk2= 0 for t > 0.(1.5)

Some eﬀorts have been made to seek the reasonable versions of the energy E[Φ] and the energy

dissipation law (1.5) for the TFCH equation (1.1). Tang, Yu and Zhou [33] established a global

energy dissipation law, E[Φ(t)] ≤E[Φ(0)] for t > 0, see also [9, 31]. In [29], Quan, Tang and Yang

derived a nonlocal energy decaying law, ∂α

tE≤0, and a weighted energy dissipation law, ∂tEη≤0,

for a nonlocal energy Eη(t) := R1

0η(θ)E(θt) dθ, also cf. the dissipation-preserving augmented

energy in [6]. They are quite diﬀerent from the classical energy law (1.5). By reformulating the

Caputo form (1.1) into the Riemann-Liouville form and applying the equality in [1, Lemma 1], one

can obtain the following variational energy dissipation law, cf. [22, Section 1],

dEα

dt+κ

2ωα(t)k∇µk2−κ

2Zt

ωα−1(t−s)k∇µ(t)− ∇µ(s)k2ds= 0,(1.6)

where the variational (modiﬁed) energy

Eα[Φ] := E[Φ] + κ

2Iα

tk∇µk2for t > 0.(1.7)

We remark that the variational energy law (1.6) is naturally consistent with the energy dissipation

law (1.5) of the CH model. Actually, the discrete energy dissipation laws of the variable-step L1 or

L1Rtime-stepping schemes in [10,22,24] were proven to be asymptotically compatible (in the limit

α→1−) with the associated discrete energy laws of their integer-order counterparts. However, the

variational energy Eα[Φ] is not asymptotically compatible with the Ginzburg-Landau energy E[Φ].

Very recently, Quan et al. [30] applied an equivalent deﬁnition (obtained from the integration by

parts) of the Caputo derivative to ﬁnd a new energy dissipation law (in our notations)

Eα

dt−ω−α(t)

2κkΦ(t)−Φ(0)k2

−1+1

2κZt

ω−α−1(t−s)kΦ(t)−Φ(s)k2

−1ds= 0,(1.8)

where the modiﬁed energy e

Eα[Φ] is deﬁned by

Eα[Φ] := E[Φ] + ω1−α(t)

2κkΦ(t)−Φ(0)k2

−1−1

2κZt

ω−α(t−s)kΦ(t)−Φ(s)k2

−1ds. (1.9)

An interesting property is that the new modiﬁed energy e

Eα[Φ] is asymptotically compatible with

the Ginzburg-Landau energy [30, Proposition 5.1], that is, e

Eα[Φ] →E[Φ] as α→1−. They also

derived the modiﬁed energy dissipation laws at discrete time levels for the L1 and L2-type implicit-

explicit stabilized schemes on the uniform time mesh. Nonetheless, it is noticed that the modiﬁed

discrete energies of these time-stepping schemes are not asymptotically compatible (except the

steady state case) with the associated discrete energies of their integer-order counterparts, see

more details in [30, Theorem 5.2] or [30, Theorems 4.1 and 4.2].

The above mentioned variable-step schemes are built by the piecewise linear or piecewise con-

stant interpolating polynomial and only have a low order accuracy of O(τ2−α) or O(τ1+α) in time,

see the related error analysis in [14, 18, 19, 27, 28]. In this paper, we shall build a discrete gradi-

ent structure (DGS) of fractional BDF2 formula on general nonuniform meshes and establish an

asymptotically compatible discrete energy dissipation law at each time level with an asymptoti-

cally compatible energy. As far as we know, there are few studies on the high-order energy stable

schemes with unequal time-step sizes for time-fractional phase ﬁeld models, see [9, 31, 32].

Application of quadratic interpolating polynomial to obtain the high-order approximations of

Caputo derivative (1.3) was suggested and investigated in recent years, see [7,20,26]. The so-called

L2-type (fractional BDF2) methods in [7,26] were shown to be (3−α)-order accurate on the uniform

mesh for suﬃciently smooth solutions. By using a nonuniform mesh such as the well-known graded

meshes [14,15,18–20,32], we will restore the optimal convergence when the solution is not smooth

near the initial time. Recently, Kopteva [15] applied the inverse-monotonicity of discrete fractional-

derivative operator to obtain sharp pointwise-in-time error bounds on quasi-graded meshes for the

linear subdiﬀusion equation and derived a mild constraint of grading parameter to recover the

optimal order convergence at any positive time.

We study the variable-step fractional BDF2 scheme on a general class of time meshes. Consider

the nonuniform time levels 0 = t0< t1<··· < tk−1< tk<··· < tN=Twith the time-step

sizes τk:= tk−tk−1for 1 ≤k≤Nand the maximum time-step size τ:= max1≤k≤Nτk. Also, let

tk−1/2:= (tk+tk−1)/2 for k≥1, and the adjacent time-step ratio r1:= 0 and rk:= τk/τk−1for

k≥2. For any time sequence vk=v(tk), deﬁne the backward diﬀerence Oτvk:= vk−vk−1and

the diﬀerence quotient ∂τvk:= Oτvk/τk. Let Π2,kv,k≥1, denote the quadratic interpolant with

respect to three nodes tk−1,tkand tk+1 for k≥1. It is easy to ﬁnd (for instance, by using the

Newton forms of the interpolating polynomials) that

(Π2,kv)0=∂τvk+2(t−tk−1/2)

τk+τk+1 ∂τvk+1 −∂τvk=∂τvk+1 +2(t−tk+1/2)

τk+τk+1 ∂τvk+1 −∂τvk.

For any time-level tnwith n≥2, applying the quadratic interpolating polynomial Π2,kv, we have

the following (3 −α)-order BDF2 type formula [7,26] of Caputo derivative (1.3),

(∂α

τv)n:= Ztn

tn−1

ω1−α(tn−s) (Π2,n−1v)0(s) ds+

n−1

k=1 Ztk

tk−1

ω1−α(tn−s) (Π2,kv)0(s) ds

k=1

a(n)

n−kOτvk+rnη(n)

1 + rnOτvn−rnOτvn−1+

n−1

k=1

η(n)

n−kOτvk+1 −rk+1Oτvk

rk+1(1 + rk+1)(1.10)

for n≥2, where the positive coeﬃcients a(n)

n−kand η(n)

n−kare deﬁned by

a(n)

n−k:= 1

τkZtk

tk−1

ω1−α(tn−s) dsfor 1 ≤k≤n, (1.11)

η(n)

n−k:= 2

τkZtk

tk−1

s−tk−1/2

τk

ω1−α(tn−s) dsfor 1 ≤k≤n. (1.12)

To start with the stepping formula (1.10), we use the standard L1 formula at the ﬁrst time level,

(∂α

τv)1,a(1)

0Oτv1with a(1)

0:= 1

τ1Zt1

ω1−α(t1−s) ds. (1.13)

Notice that if the fractional order α→1−, then ω3−α(t)→t,ω2−α(t)→1 and ω1−α(t)→0,

uniformly for t > 0. Thus, a(n)

0=ω2−α(τn)/τn→1/τnand η(n)

0=ατ−α

n/Γ(3−α)→1/τn, whereas

a(n)

n−k→0 and η(n)

n−k→0 for 1 ≤k≤n−1. We shall call (1.10) together with (1.13) as the

fractional BDF2 (FBDF2) formula for brevity, since the approximation (1.10) degrades into the

standard BDF2 formula [16, 17, 21, 23] of the derivative ∂tv, that is,

(∂α

τv)n→D2vn:= 1+2rn

(1 + rn)τn

Oτvn−r2

(1 + rn)τn

Oτvn−1as α→1−. (1.14)

We can reformulate (1.10) and (1.13) into a compact form,

(∂α

τv)n,

k=1

B(n)

n−kOτvkfor n≥1, (1.15)

where the kernels B(n)

n−kare determined via (1.10) and (1.13). The above asymptotic property

(1.14) arises the main diﬃculty in the numerical analysis of the FBDF2 formula (1.10). That is,

the second kernel B(n)

1would be negative when αis close to 1, and the discrete kernels B(n)

n−klose

the monotonicity. The established stability and convergence theory [14, 18–20] for the variable-

step L1 and L2-1σformulas can not be applied here directly. Recently, the uniform FBDF2

formula was showed in [31] to be positive semideﬁnite with the BDF2 multiplier D2vn, that is,

k=1(D2vk)(∂α

τv)k≥0. By combining with the recent scalar auxiliary variable technique, the

resulting time-stepping scheme was proven to preserve a global energy law [33] of time-fractional

phase-ﬁeld equations. Quan and Wu [32] investigated the H1stability of the FBDF2 formula (1.10)

on general nonuniform time meshes for a linear subdiﬀusion equation and established the positive

semideﬁniteness with the diﬀerence multiplier Oτvn, that is, Pn

k=1(Oτvk)(∂α

τv)k≥0, under the

following step-ratio condition

0.4573328 ≤rk≤3.5615528 for k≥2. (1.16)

Nonetheless, these results in [31, 32] would be inadequate to build an asymptotically compatible

discrete energy law for the TFCH equation (1.1).

The current work is inspired by the following DGS [17] of the BDF2 formula,

(Oτvn)D2vn≥r3/2

n+1

2(1 + rn+1)τn

(Oτvn)2−r3/2

2(1 + rn)τn−1

(Oτvn−1)2for n≥2. (1.17)

This DGS holds if the adjacent step-ratio restriction 0 < rk< r?≈4.864 for k≥2, where r?is

the positive root of 1 + 2r?−(r?)3/2= 0. This DGS has been proven to be useful to establish the

discrete energy dissipation laws at each time level of the BDF2 scheme for the classical gradient

ﬂows [17]. It is natural to ask whether the FBDF2 formula (1.10) has a similar gradient structure

under some constraint on the step-ratio rk, and whether the resulting FBDF2 time-stepping for

time-fractional gradient ﬂows also has an energy dissipation law at discrete time levels.

In the next section, we update the step-ratio stability constraint (1.16) into

0.4753 ≤rk< r∗(α) for k≥2 with r∗(α)≥4.660,

and establish a novel DGS for the FBDF2 formula (1.10). Our main tools include a novel local-

nonlocal splitting technique and the step-scaling technique so that the present analysis is quite

diﬀerent and would be more concise than those in [30–32]. As an application, we consider an

implicit FBDF2 stepping for the TFCH model (1.1) in Section 3 and prove that it is uniquely

solvable and has an asymptotically compatible energy law at each time level. More importantly,

our discrete energy is also asymptotically compatible with the associated discrete energy of the

BDF2 scheme for the CH equation. Numerical tests show that the proposed method is accurate

with an order of O(τ3−α) in time. An adaptive time-stepping procedure is also provided to speedup

the simulations of long-time coarsening dynamics.

2 Discrete gradient structure of FBDF2 formula

To derive the DGS of the nonuniform FBDF2 formula (1.10), we propose a local-nonlocal splitting

technique by splitting it into two parts

(∂α

τv)n,1

2−αa(n)

0+rnη(n)

1 + rnOτvn−r2

nη(n)

1 + rn

Oτvn−1

| {z }

k=1

ˆa(n)

n−kOτvk

| {z }

for n≥2,(2.1)

B2Jn

where the discrete kernels ˆa(n)

n−kare deﬁned by

ˆa(1)

0:= a(1)

0,ˆa(n)

0:= 1−α

2−αa(n)

0+η(n)

rn(1 + rn)for n≥2, (2.2)

ˆa(n)

n−k:= a(n)

n−k−η(n)

n−k

1 + rk+1

+η(n)

n−k+1

rk(1 + rk)for 2 ≤k≤n−1 (n≥3), (2.3)

ˆa(n)

n−1:= a(n)

n−1−η(n)

n−1

1 + r2

for n≥2.(2.4)

The ﬁrst part Jn

B2represents a variable-step BDF2-type formula [16, 17, 21, 23]. The second part

L1represents the nonuniform L1-type formula [19, 24] of Caputo derivative (1.3). As seen, the

discrete kernel a(n)

0is split into two parts with diﬀerent weights 1

2−αand 1−α

2−α. The technical

reason of the above local-nonlocal splitting is that the local term Jn

B2→D2vnas α→1−, while

the convolution kernels ˆa(n)

n−k(n≥2) of the nonlocal term Jn

L1will vanish, ˆa(n)

n−k→0 as α→1−.

So we can build a discrete energy that is asymptotically compatible with the discrete energy of

the BDF2 scheme for the CH model, see Remark 4. It signiﬁcantly updates the previous energy

laws in [10, 22,24], in which the discrete energies are always not asymptotically compatible in the

fractional order limit α→1−although the associated energy laws are asymptotically compatible.

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Asymptoticallycompatibleenergyofvariable-stepfractionalBDF2formulafortime-fractionalCahn-HilliardmodelHong-linLiao*NanLiuXuanZhaoOctober25,2022AbstractAnewdiscreteenergydissipationlawofthevariable-stepfractionalBDF2(second-orderbackwarddierentiationformula)schemeisestablishedfortime-fractionalCah...

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Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model Hong-lin LiaoNan LiuXuan Zhao

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