Provably Unconditionally Stable, Second-order Time-accurate, Mixed Variational Methods for

Phase-field Models

Hector Gomez and Thomas J.R. Hughes

The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustin, Texas 78712

by

ICES REPORT 10-38

September 2010

Reference: Hector Gomez and Thomas J.R. Hughes, "Provably Unconditionally Stable, Second-order Time-accurate, Mixed Variational Methods for Phase-field Models", ICES REPORT 10-38, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, September 2010.

Provably Unconditionally Stable,

Second-order Time-accurate, Mixed

Variational Methods for Phase-field Models

Hector Gomez1 ∗ , Thomas J.R. Hughes2

1: Group of Numerical Methods in EngineeringUniversity of A Coruna

Department of Mathematical MethodsCampus de Elvina, s/n

15192, A Coruna

2: Institute for Computational Engineering and SciencesThe University of Texas at Austin

1 University Station, C0200201 E. 24th Street, Austin, TX 78712

Abstract

We introduce provably unconditionally stable mixed variational methods for phase-field models. Our formulation is based on a mixed finite element method for spacediscretization and a new second-order accurate time integration algorithm. Thefully-discrete formulation inherits the main characteristics of conserved phase dy-namics, namely, mass conservation and nonlinear stability with respect to the freeenergy. We illustrate the theory with the Cahn-Hilliard equation, but our methodmay be applied to other phase-field models. We also propose an adaptive time-stepping version of the new time integration method. We present some numericalexamples that show the accuracy, stability and robustness of the new method.

Key words: Nonlinear stability, Phase-field, Mixed finite element, Timeintegration, Cahn-Hilliard

∗ Correspondence to: University of A Coruna, Deparment of Mathematical MethodsEmail address: hgomez@udc.es (Hector Gomez1).

Preprint submitted to Elsevier Science 26 September 2010

1 Introduction

The phase-field method is a recently introduced mathematical theory for modeling interfacialphenomena (see [2] for a brief introduction in the context of fluid mechanics). The keyidea of the phase-field methodology is to replace sharp interfaces by thin transition regionswhere interfacial forces are smoothly distributed. These transition layers (also called diffuseinterfaces) are part of the solution of the phase-field equations and, thus, front-tracking isnot necessary. Another fundamental feature of phase-field models is that diffuse interfacesare described by higher-order partial-differential operators.

The phase-field method was originally devised for microstructure evolution [14,27] and phasetransition [36,52,62,76], but it has been recently used to model foams [31], planet formation[71], ferroelectric ceramics [42,54], dendritic growth [48,51], phase separation of block copoly-mers [15], growth of cancerous tumors [20,32,61], solid-solid transitions [33,57,58], infiltrationof water into a porous medium [21,22], dewetting and rupture of thin liquid films [9] andmany other phenomena.

One of the main reasons for the success of the phase-field methodology is that it is basedon rigorous mathematics and thermodynamics. Most phase-field models satisfy a nonlinearstability relationship, usually expressed as a time-decreasing free-energy functional. The goalof this paper is to devise space and time discretization schemes that inherit the nonlinearstability relationship of the continuous model.

The idea of designing numerical techniques that satisfy thermodynamic relations at thediscrete level has been extensively studied in the context of solid [3,60,66,67] and fluidmechanics [39,45,69,70], but it is significantly less developed in the realm of phase-fieldmodels. The first study we know of was carried out by Du and Nicolaides [24] who derived asecond-order accurate unconditionally stable time-stepping scheme for a simplified versionof the Cahn-Hilliard equation. This paper was followed up by several works of an analyticalcharacter, but to our knowledge, it has not been applied to large scale calculations. In fact,we applied this method to the general version of the Cahn-Hilliard equation and obtainedpoor numerical results. Later on, Eyre [28] derived a first-order accurate unconditionallystable time-stepping scheme for a simplified version of the Cahn-Hilliard equation. AlthoughEyre’s work is unpublished, it has had significant impact. Eyre’s method has been extensivelyused by the computational community and it has served as inspiration for many other timeintegration schemes. Other significant works, which deal with simplified versions of the Cahn-Hilliard equation, are due to de Mello et al. [59], Furihata [34], He et al. [40], Kim [50] andVollmayr-Lee et al. [73]. Other noteworthy contributions for the phase-field crystal equationand a bistable epitaxial thin film equation are due to Hu et al. [41] and Xu and Tang [75],respectively.

In this paper we introduce provably unconditionally stable, second-order time accurateschemes for phase-field models. We illustrate the theory using the Cahn-Hilliard equation,but the method may also be applied to other phase-field models. We present the method

2

for a spatial discretization based on a mixed finite element formulation, but we feel thatour time integration scheme may also be applied to finite difference or spectral collocationspatial discretizations, which are ubiquitous in computational phase-field modeling.

The rest of the paper is organized as follows: In section 2 we briefly review the derivation ofthe Cahn-Hilliard equation to illustrate its conservation and stability properties, which willbe inherited by our numerical scheme. Section 3 presents our space and time discretizations.In section 4 we present some numerical examples that illustrate the accuracy, stability androbustness of the proposed method. We draw conclusions in Section 5.

2 The Cahn-Hilliard equation

The Cahn-Hilliard equation describes the separation of mixtures with a miscibility gap intheir phase diagrams and it is probably the most widely known phase-field model. The Cahn-Hilliard equation is derived from the Ginzburg-Landau free energy [53], which we describein the following section.

2.1 The Ginzburg-Landau free energy

We focus on isothermal binary mixtures. In this setting, the thermodynamic state of themixture is defined by a single scalar field c representing the concentration of one of thespecies. The concentration of the other component is 1 − c. Assuming that the mixture iscontained within an open domain Ω ⊂ R3, the Ginzburg-Landau free energy is defined bythe functional,

E(c) =∫

Ω(Ψc + Ψs) dx (1)

where Ψc is the chemical free energy and Ψs is the surface free energy. According to theoriginal model of Cahn and Hilliard [12,13], the chemical free energy is given by

Ψc = NkT (c log c+ (1− c) log(1− c)) +Nωc(1− c) (2)

while the surface free energy takes on the form

Ψs =1

2λ|∇c|2 (3)

In equations (2) and (3), N is the number of molecules per unit volume, k is the Boltzmann’sconstant, T is the absolute temperature and ω is an interaction energy defined by

ω = 2kTc (4)

3

In equation (4) Tc is the so-called critical temperature, that is, the maximum temperatureat which phase separation is possible. Finally, λ is a positive constant defined as

λ = Nωε2 (5)

where ε is a length scale of the problem.

For θm = Tc/T ≤ 1 the chemical free energy has a single well and admits only one phase. Incontrast, for θm > 1, it is non-convex, with two local minima to which the concentration isdriven. These local minima are called binodal points. The binodal points are not the purephases, although they are close to them for physically relevant values of the parameters. Inthis paper we will focus on the case θm > 1.

2.2 The Cahn-Hilliard equation

Our numerical scheme inherits the conservation and dissipation properties of the Cahn-Hilliard equation. As a consequence, we feel that it is important to start off with the classicalderivation of the Cahn-Hilliard equation, which clearly illustrates those properties. Thestarting point is the mass balance law,

∂c

∂t+∇ · J = 0 (6)

where J is the mass flux defined as

J = −M(c)∇(δEδc

)(7)

In equation (7) M is a nonlinear and positive function, the mobility, and

δEδc

= µ(c)− λ∆c (8)

denotes the variational derivative of the functional E . Finally, µ is the chemical potential,that is, the derivative of the chemical free energy with respect to c. Gathering equations (6),(7) and (8) we get the Cahn-Hilliard equation

∂c

∂t= ∇ · (M(c)∇(µ(c)− λ∆c)) (9)

This is the standard derivation of the Cahn-Hilliard phase-field model. For an alternativederivation based on a microforce balance, the reader is referred to [37]. The previous deriva-tion clearly shows (see equation (6)) that the Cahn-Hilliard equation conserves the mass ofthe mixture as time evolves. It can also be proven that the Ginzburg-Landau free energydoes not increase with time. The latter result can be obtained multiplying (9) with the vari-ational derivative of the free energy. If we define the real-valued function E(t) = E(c(·, t)),

4

it follows thatdE

dt= −

∫Ω∇(µ(c)− λ∆c)M(c)∇(µ(c)− λ∆c)dx ≤ 0 (10)

We consider (10) the fundamental stability property of the Cahn-Hilliard equation.

Remarks:

(1) In most analytic studies and numerical simulations the mobility is assumed to be apositive constant. This assumption significantly simplifies the mathematical analysisand the numerical simulation of the Cahn-Hilliard equation. However, to accuratelydescribe the physics of phase separation, pure phases must have vanishing mobility.A positive function of c satisfying this requirement is called degenerate mobility. Themost common choice in the literature is

M(c) = Dc(1− c) (11)

where D is a positive constant. We will use the mobility defined in (11) throughout thispaper.

(2) Due to the complexity of the function Ψc, some simpler approximations are normallyemployed. In particular, a polynomial of degree four has been used to approximate thechemical free energy in most analytic studies and numerical simulations. The quarticpolynomial may be an acceptable approximation of the chemical free energy for shallowquenches, but it is a poor approximation for deep quenches [17]. Moreover, the solutionto the Cahn-Hilliard equation with the quartic polynomial and constant mobility maytake values outside the physical range c ∈ [0, 1] (see the work of Elliot and Garcke [26]).This demonstrates the importance of developing numerical algorithms that can handlethe logarithmic free energy. In what follows we will focus on the logarithmic chemicalfree energy defined in (2). For further reference, we note that µ′′′ = Ψiv

c > 0, which maybe easily verified. This property is important in the sequel.

(3) In order to define a well posed initial/boundary-value problem, equation (9) needs tobe complemented with suitable initial and boundary conditions. Let us assume thatequation (9) is posed on a space/time domain Ω× (0, T ), where Ω is an open subset ofR3 and (0, T ) is the time interval. We assume that Ω has a smooth boundary Γ with unitoutward normal n. A suitable initial condition consists of setting the concentration fieldat the initial time. Regarding boundary conditions, there are several options. Probablythe most common choice in the literature is to assume periodic boundary conditions(the reason for this may be that the literature on numerical methods for the Cahn-Hilliard equation is dominated by spectral methods). Another option, considered abetter alternative by some authors [59], is setting on the boundary

∇c · n = 0 (12)

M(c)∇(µ(c)− λ∆c) = 0 (13)

Equations (12)–(13) may be considered as natural boundary conditions for the Cahn-Hilliard equation in a variational formulation.

5

3 Numerical Formulation

3.1 Continuous problem in the weak form

Our theory is based on a mixed finite element [43] formulation of the Cahn-Hilliard equation.Thus, we split equation (9) as

∂c

∂t= ∇ · (M(c)∇v) in Ω× (0, T ) (14)

v = µ(c)− λ∆c in Ω× (0, T ) (15)

Let V denote the trial solution and the weighting functions spaces which are assumed to bethe same. Equations (14)–(15) can be recast in the weak form as: find c, v ∈ V such thatfor all w, q ∈ V

(w,∂c

∂t

)Ω

+ (∇w,M(c)∇v)Ω = 0 (16)

(q, v)Ω − (q, µ(c))Ω − (∇q, λ∇c)Ω = 0 (17)

where (·, ·)Ω denotes the L2 inner product over the domain Ω.

The integration by parts of the previous equations under the assumption of sufficient regu-larity leads to the Euler-Lagrange form of (16)–(17),

(w,∂c

∂t−∇ · (M(c)∇v)

)Ω

+ (w,M(c)∇v · n)Γ = 0 (18)

(q, v − µ(c) + λ∆c)Ω − (q, λ∇c · n)Γ = 0 (19)

Equations (18)–(19) enforce weak satisfaction of the partial-differential equation (9) andboundary conditions (12)–(13).

3.2 Semidiscrete formulation

To perform the space discretization of (16)–(17) we make use of the Galerkin method. Weapproximate (16)–(17) by the following finite-dimensional problem over the finite elementspace V h ⊂ V . The problem can be stated as: find ch, vh ∈ V h such that for all wh, qh ∈ V h

6

(wh,

∂ch

∂t

)Ω

+(∇wh,M(ch)∇vh

)Ω

= 0 (20)

(qh, vh − µ(ch)

)Ω− (∇qh, λ∇ch)Ω = 0 (21)

In the previous equations ch takes on the form

ch(x, t) =nb∑A=1

cA(t)NA(x) (22)

where nb is the dimension of the discrete space, the NA’s are the basis functions of the discretespace, and the cA’s are the coordinates of ch on V h. The rest of the variables utilized in(20)–(21) are defined analogously to ch.

Remark:

(1) We use the same discrete space for ch and vh. This type of mixed finite element formu-lation was shown to lead to stable space discretizations in [17] and [30] (a somewhatdifferent mixed finite element formulation was employed in [74]).

(2) We will use Non-Uniform Rational B-Splines (NURBS) basis functions to define ourdiscrete spaces. For an introduction to NURBS see [63] or [65]. The use of the NURBStechnology in the context of analysis has led to the concept of Isogeometric Analy-sis, which is a generalization of finite element analysis with several advantages [4,6–8,11,19,25,29,35,44,56]. For a detailed description of Isogeometric Analysis, the readeris referred to [18].

3.3 Time integration

Our time integration scheme is based on the following splitting of the chemical free-energy,

Ψc = Ψ1 + Ψ2 (23)

where Ψiv1 ≥ 0, Ψiv

2 ≤ 0 and Ψiv denotes the fourth derivative of Ψ. This decomposition

always exists, but it is not unique. For the Cahn-Hilliard equation, the splitting is just

Ψ1 = Ψc; Ψ2 = 0 (24)

but for other phase-field models Ψ1 and Ψ2 will be both nonzero, so we will assume thatΨ1 6= 0 and Ψ2 6= 0 to present our new method in its full generality.

In the sequel we will make use of the notation µ1 = Ψ′1, µ2 = Ψ′2 and, thus

µ′′′1 ≥ 0, µ′′′2 ≤ 0 (25)

7

Our time integration algorithm may be described as follows: Let us assume that the timeinterval I = (0, T ) is divided into N subintervals In = (tn, tn+1), n = 0, . . . , N − 1. We usethe notation chn and vhn for the fully discrete solutions. Our time integration scheme may bedefined as follows: Given chn and vhn, find chn+1, vhn+1 such that for all wh, qh ∈ V h

(wh,

JchnK∆tn

)Ω

+(∇wh,M(chn+α)∇vhn+1

)Ω

= 0 (26)

(qh, vhn+1

)Ω−(qh,

1

2

(µ(chn) + µ(chn+1)

)− JchnK2

12

(µ′′1(chn) + µ′′2(chn+1)

))Ω

−(∇qh, λ∇chn+α

)Ω

= 0 (27)

where

∆tn = tn+1 − tn, (28)

JchnK = chn+1 − chn, (29)

chn+α = chn + αJchnK (30)

In equation (30), α is a real-valued parameter that controls the accuracy and stability of themethod. We define α as

α = 1/2 + η (31)

In equation (31), η takes on the form

η =1

2tanh

(∆tnδ

)(32)

where δ is an intrinsic time scale of the problem. We define δ making use of a natural timescale of the Cahn-Hilliard equation, namely, λ/D. Thus, we assume that

δ = Cλ

D(33)

where C is a nondimensional constant that takes the value C = 103. We will use this valueof C for all the numerical examples in this paper. However, we remark that this parametermay be used to increase the robustness or the accuracy of the method when necessary. Ingeneral, larger values of C render a more accurate method and smaller values of C lead toa more robust algorithm.

The following theorem summarizes the main features of our numerical scheme.

Theorem 1 The fully-discrete variational formulation (26)–(32):

8

(1) Verifies mass conservation, that is,∫Ωchndx =

∫Ωch0dx ∀n = 1, . . . , N

(2) Verifies the nonlinear stability condition

E(chn) ≤ E(chn−1) ∀n = 1, . . . , N

(3) Gives rise to a local truncation error τ that may be bounded as |τ(tn)| ≤ K∆t2n for alltn ∈ [0, T ], where K is a constant independent of ∆tn.

Proof:

(1) Taking wh = 1 in equation (26), it follows by induction that∫Ωchndx =

∫Ωch0dx ∀n = 1, . . . , N

(2) Let f : [a, b] 7→ R be a sufficiently smooth function. We will make use of the followingquadrature formulas:

∫ b

af(x)dx =

b− a2

(f(a) + f(b))− (b− a)3

12f ′′(a)− (b− a)4

24f ′′′(ξ); ξ ∈ (a, b) (34)

∫ b

af(x)dx =

b− a2

(f(a) + f(b))− (b− a)3

12f ′′(b) +

(b− a)4

24f ′′′(ζ); ζ ∈ (a, b) (35)

Since these quadrature formulas are new as far as we are aware, we provide a completederivation of one of them in the Appendix (the derivation of the other is analogous).Let us apply the quadrature formula (34) to the right-hand side of the identity

∫ chn+1

chn

Ψ′k(t)dt =∫ chn+1

chn

µk(t)dt; k = 1, 2 (36)

with k = 1. Basic algebraic manipulation leads to the expression

JΨ1(chn)KJchnK

+JchnK3

24µ′′′1 (chn+ξ) =

1

2

(µ1(chn) + µ1(chn+1)

)− JchnK2

12µ′′1(chn); ξ ∈ (0, 1) (37)

Applying the quadrature formula (35) to the right-hand side of (36) with k = 2, itfollows that

JΨ2(chn)KJchnK

− JchnK3

24µ′′′2 (chn+ζ) =

1

2

(µ2(chn) + µ2(chn+1)

)− JchnK2

12µ′′2(chn+1); ζ ∈ (0, 1) (38)

Taking wh = vhn+1 in (26) and qh = JchnK/∆tn in (27) and applying equations (37) and(38), we get

9

−(∇vhn+1,M(chn+α)∇vhn+1

)Ω−(

JchnK∆tn

,JΨc(c

hn)K

JchnK+

JchnK3

24

(µ′′′1 (chn+ξ)− µ′′′2 (chn+ζ)

))Ω

−(∇(

JchnK∆tn

), λ∇chn+α

)Ω

= 0 (39)

Using the relation chn+α = chn+1/2 + ηJchnK, it follows that

−(∇vhn+1,M(chn+α)∇vhn+1

)Ω− 1

∆tn

∫ΩJΨc(c

hn)KdΩ−

(JchnK4

24∆tn, µ′′′1 (chn+ξ)− µ′′′2 (chn+ζ)

)Ω

−(∇(

JchnK∆tn

), λ∇chn+1/2

)Ω

−(∇JchnK, λ

η

∆tn∇JchnK

)Ω

= 0. (40)

Making use of the identity

(∇JchnK, λ∇chn+1/2

)Ω

=∫

Ω

λ

2J|∇ch|2Kdx (41)

we conclude that

JE(ch)K∆tn

=−(∇vhn+1,M(chn+α)∇vhn+1

)Ω

−(

JchnK4

24∆tn, µ′′′1 (chn+ξ)− µ′′′2 (chn+ζ)

)Ω

−(∇JchnK, λ

η

∆tn∇JchnK

)Ω

(42)

Since µ′′′1 (c) ≥ 0 and µ′′′2 (c) ≤ 0, it follows that

JE(ch)K ≤ 0 (43)

which completes the proof.(3) We derive a bound on the local truncation error by comparing our method with a known

second-order accurate method, the midpoint rule, given as follows:

(wh,

JchnK∆tn

)Ω

+(∇wh,M(chn+1/2)∇vhmid

)Ω

= 0 (44)

(qh, vhmid

)Ω

=(qh, µ(chn+1/2)

)Ω

+(∇qh, λ∇chn+1/2

)Ω

(45)

The expression for the local truncation error is obtained by replacing the time dis-crete solution chn with the time continuous solution ch(tn) in the above equations. Thetime continuous solution does not satisfy equations (44)–(45), giving rise to the localtruncation error,

10

(wh,

Jch(tn)K∆tn

)Ω

+(∇wh,M(ch(tn+1/2))∇vhmid

)Ω

= (wh, τmid)Ω (46)

(qh, vhmid

)Ω

=(qh, µ(ch(tn+1/2))

)Ω

+(∇qh, λ∇ch(tn+1/2)

)Ω

(47)

where τmid is the local truncation error. Assuming sufficient smoothness, Taylor seriescan be utilized to show that τmid = O(∆t2n). Proceeding in similar fashion with ouralgorithm, we obtain

(wh, τ

)Ω

=

(wh,

Jch(tn)K∆tn

)Ω

+(∇wh,M(ch(tn+α))∇vh

)Ω

(48)

(qh, vh

)Ω

=(qh,

1

2

(µ(ch(tn)) + µ(ch(tn+1))

))Ω

−(qh,

Jch(tn)K2

12

(µ′′1(ch(tn)) + µ′′2(ch(tn+1))

))Ω

+(∇qh, λ∇ch(tn+α)

)Ω

(49)

Assuming smoothness, Taylor series can be utilized to prove

1

2

(µ(ch(tn)) + µ(ch(tn+1))

)= µ(ch(tn+1/2)) +O(∆t2n) (50)

Jch(tn)K2

12

(µ′′1(ch(tn)) + µ′′2(ch(tn+1))

)= O(∆t2n) (51)

Let us use the identity

ch(tn+α) = ch(tn+1/2) +O(η∆tn) (52)

Considering that

η =1

2tanh

(∆tnδ

)≤ ∆tn

2δ(53)

we conclude thatch(tn+α) = ch(tn+1/2) +O(∆t2n) (54)

Combining all these results, it follows that(qh, vh

)Ω

=(qh, vhmid

)Ω

+O(∆t2n) (55)

(wh, τ

)Ω

=(wh, τmid

)Ω

+O(∆t2n) (56)

which completes the proof.

Remarks:

11

(1) Our new algorithm may be viewed as a second-order perturbation of the midpoint rulewhich achieves unconditional stability, in contrast with the midpoint rule.

(2) It is worth further analyzing the stability relationship (42). The first term on the righthand side is what we may call physical dissipation. That term mimics the right handside of equation (10). The last two terms of equation (42) may be considered numericaldissipation. Note that these terms vanish as the time step tends to zero. When the timestep is large, those terms enhance the stability properties of the scheme, rendering avery robust method.

3.4 Numerical implementation

Let Cn and V n be the vector of global degrees of freedom of chn and vhn, respectively. Ourtime stepping scheme may be implemented as follows: given Cn and V n, find Cn+1, V n+1,such that,

RM(Cn+1,V n+1) = 0 (57)

RE(Cn+1,V n+1) = 0 (58)

where the above residual vectors are defined as

RM = RMA (59)

RMA =

(NA,

JchnK∆tn

)Ω

+(∇NA,M(chn+α)∇vhn+1

)Ω

(60)

RE = REA (61)

REA =

(NA, v

hn+1

)Ω−(NA,

1

2

(µ(chn) + µ(chn+1)

)− JchnK2

12

(µ′′1(chn) + µ′′2(chn+1)

))Ω

−(∇NA, λ∇chn+α

)Ω

(62)

Equations (57)–(58) constitute a nonlinear system of algebraic equations that needs to besolved at each time step. We linearize system (57)–(58) using Newton’s method which leadsto a two-stage predictor multicorrector algorithm, that may be described as follows,

Predictor stage: Set

Cn+1,(0) = Cn (63)

V n+1,(0) = V n (64)

where the subscript 0 on the left-hand-side quantities denotes the iteration index of thenonlinear solver.

Multicorrector stage: Repeat the following steps for i = 1, 2, . . . , imax

12

(1) Evaluate the iterates at the α level

Cn+α = Cn + α(Cn+1,(i−1) −Cn

)(65)

(2) Use Cn, Cn+α and V n+1,(i−1) to assemble the residual and the tangent matrix of thelinear system K11,(i) K12,(i)

K21,(i) K22,(i)

∆Cn+1,(i)

∆V n+1,(i)

=

RM(i)

RE(i)

(66)

Solve this linear system using a preconditioned GMRES algorithm to a specified toler-ance (see Saad and Schultz [68]).

(3) Use ∆Cn+1,(i) and ∆V n+1,(i) to update the iterates as

Cn+1,(i) = Cn+1,(i−1) + ∆Cn+1,(i) (67)

V n+1,(i) = V n+1,(i−1) + ∆V n+1,(i) (68)

This completes one nonlinear iteration. The nonlinear iterative algorithm should berepeated until both residuals RM and RE have been reduced to a given tolerance.

Remarks:

(1) The concentration control variables at the initial time C0 are straightforwardly obtainedfrom the initial condition. We obtain the control variables V 0 by solving equation (21).

(2) We use a consistent tangent matrix in our computations. Two to four nonlinear itera-tions are normally required to reduce the nonlinear residual to 10−3 of its initial valuein a time step.

3.5 Time-step adaptivity

Time-step adaptivity is of prime importance to simulate the entire dynamics of the Cahn-Hilliard equation [23,35,64] accurately and efficiently. We propose an adaptive time-steppingstrategy for our provably stable scheme. The method is presented in Algorithm 1.

We update the time step using the equation

F (e,∆t) = ρ

(tol

e

)1/2

∆t (69)

which is frequently used in adaptive time-stepping algorithms based on embedded Runge-Kutta methods [10,38,72]. Our default values for the safety coefficient ρ and the tolerancetol are those suggested in [55], that is, ρ = 0.9 and tol = 10−3.

Remark:

13

Algorithm 1 Time step adaptive process

Given: Cn, V n and ∆tn1: Compute CBE

n+1 using the Backward Euler method and ∆tn2: Compute Cn+1 using equations (26)–(27) and ∆tn3: Calculate en+1 = ||CBE

n+1 −Cn+1||/||Cn+1||4: if en+1 > tol then5: Recalculate the time step ∆tn ←− F (en+1,∆tn)6: goto 17: else8: Update the time step ∆tn+1 = F (en+1,∆tn)9: continue

10: end if

Note that when the accuracy criterion in Algorithm 1 (Step 4) is not satisfied, the computedsolution is rejected and recalculated using a smaller time step. Typically, fewer than 10% ofthe time steps are rejected using the safety coefficient ρ = 0.9.

4 Numerical examples

In this section we present some numerical examples that illustrate the robustness, stability,and accuracy of our numerical formulation.

For the space discretization we employ C1 quadratic Non-Uniform Rational B-Splines (NURBS)for all variables. Although, our variational formulation is well defined when C0 basis functionsare employed, the C1 quadratic elements have been shown to exhibit superior approximabilityproperties [1,19,29,46].

4.1 Dimensionless form of the Cahn-Hilliard equation

To minimize the number of independent parameters in our numerical examples we will usea dimensionless form [5] of the Cahn-Hilliard equation. Introducing the arbitrary length andtime scales L0 and T0 = L4

0/(NωDε2), we obtain the dimensionless equation,

∂c

∂t= ∇ ·

(M(c)∇

(χµ(c)− ∆c

))(70)

where the “hats” indicate that the corresponding variable/operator has been nondimension-alized using the scales L0 and T0. The rest of the notation is as follows:

µ(c) = c log(c) + (1− c) log(1− c) + 2θmc(1− c) (71)

M(c) = c(1− c) (72)

14

and

χ =L2

0

2θmε2(73)

is a dimensionless group of the problem. As shown in [35], the thickness of the interfacelayers scales as χ−1/2.

The dimensionless Ginzburg-Landau free-energy is given by

E(c) =E(c)

NkTL30

∫Ω

(c log(c) + (1− c) log(1− c) + 2θmc(1− c) +

1

2χ|∇c|2

)dx (74)

We will take L0 = 1 and θm = 3/2 for all the numerical examples. As a consequence, thevalue of χ completely characterizes the solution.

Henceforth, we will use the dimensionless form of the Cahn-Hilliard equation. Thus, let usdrop the hats for the sake of notational convenience.

4.2 Phase separation on a periodic square: constant time step

In this example we analyze the performance of our time integration scheme for a wide rangeof time step sizes. We feel that this example demonstrates the robustness, accuracy andstability of our method.

The computational domain is the square Ω = [0, 1]2. The problem is defined by χ = 600. Weassume periodic boundary conditions. As initial condition, we take a randomly perturbedhomogeneous concentration state c. We set c = 0.7 in this example. The random perturbationis uniformly distributed on [−0.05, 0.05] and it is directly applied to the global vector ofdegrees of freedom C0.

It is known that the Cahn-Hilliard equation is unstable under random perturbations withinthe spinodal region and, thus, two separate phases will develop. The dynamics of the Cahn-Hilliard equation are driven by the minimization of the Ginzburg-Landau free energy, whichis composed of two terms, namely, the chemical free energy and the surface free energy. Theminimization of these two components of the Ginzburg-Landau free energy occurs almostindependently, and at significantly different time scales. The first part of the dynamics isphase separation. This process is driven by the minimization of the chemical free energyand takes place at very short time scales. After phase separation, the chemical free energyis essentially minimized, because the mass of the two phases is conserved during the entireprocess. The phenomenon that follows phase separation is known as coarsening and it isdriven by the minimization of the surface free energy. When the dimensionless number χis very large, the minimization of the surface free energy may also be thought of as theminimization of the interface length. The process finishes when there are only two regionsoccupied by the two components of the mixture and they achieve the equilibrium topology.We will observe this behavior in our simulations.

15

Fig. 1. Phase separation on a periodic square: constant time step. Numerical solution using theGeneralized-α method for time integration. Each row of the figure shows snapshots of the solutionat the times indicated at the bottom and has been calculated using the time step indicated on theleft hand side of the figure.

For the space discretization, we employ a uniform mesh of 642 quadratic elements in thecalculations. We integrate in time using our new method and analyze its behavior for differentvalues of the time step ∆t. In an effort to illustrate the properties of our time integrationscheme we will use constant time steps in this example, although we are aware that this isa very inefficient strategy for the Cahn-Hilliard equation.

We will compare our algorithm with the Generalized-α method [16,47] and Eyre’s scheme.In our experience, the Generalized-α method is a very accurate and robust scheme, but onethat is not provably stable for the Cahn-Hilliard equation. Eyre’s scheme is a first-orderaccurate provably unconditionally stable algorithm for the Cahn-Hilliard equation.

Figure 1 shows the solution of the Cahn-Hilliard equation using the space discretizationdefined in (20)–(21) and the Generalized-α method with ρ∞ = 0.5 (see [47] for a descriptionof the Generalized-α method). We have used different time steps, as shown on the left-hand side of the figure. The top row shows snapshots of the solution at different timesfor ∆t = 10−5, while the lower rows show the time-history of the numerical solution for∆t = 10−6 and ∆t = 10−7. There are no discernible differences between the solutionscomputed using different time steps. Thus, we take any of these solutions as our referencesolution.

Figure 2 shows the solution to the Cahn-Hilliard equation using the space discretizationdefined in (20)–(21) and Eyre’s method for time integration. The implementation of the

16

Fig. 2. Phase separation on a periodic square: constant time step. Numerical solution using Eyre’smethod for time integration. Each row of the figure shows snapshots of the solution at the timesindicated at the bottom and has been calculated using the time step indicated on the left handside of the figure.

method in a mixed finite element framework follows [49]. Eyre’s method is based on asplitting of the chemical free energy on a convex (ΨD) and a concave (ΨP ) function. AlthoughEyre’s method was originally proposed for the quartic chemical free energy, we extend ithere to the logarithmic free energy to make the comparison meaningful. For the logarithmicchemical free energy there is a very natural splitting, which we adopt here. It is defined as,

Ψc = ΨD + ΨP (75)

where

ΨD(c) =NkT (c log c+ (1− c) log(1− c)) (76)

ΨP (c) =Nωc(1− c) (77)

Figure 2 presents the numerical results using Eyre’s method for time integration. It showsthat Eyre’s method is very robust, but also significantly inaccurate. The first row shows thatfor ∆t = 10−5 Eyre’s scheme is clearly less accurate than the Generalized-α method. Theearly dynamics of the equation are completely missed by Eyre’s method. The last row of thefigure shows that, even for very small time steps, Eyre’s method is significantly inaccurate,leading to incorrect morphologies in the phase-separation process.

In Figure 3 we plot the transient and stationary solutions using our new method. We again

17

Fig. 3. Phase separation on a periodic square: constant time step. Numerical solution using ourprovably stable method. Each row of the figure shows snapshots of the solution at the timesindicated at the bottom and has been calculated using the time step indicated on the left handside of the figure.

use different time step sizes, as shown on the left-hand side of the figure. The top row of thefigure shows the solution for ∆t = 10−5. The lower rows show snapshots of the solution for∆t = 10−6 and ∆t = 10−7, as indicated in the figure. From Figures 1 and 3 we conclude thatthe accuracy of our provably stable algorithm is similar to that of the Generalized-α method.The solutions in Figure 3 are indistinguishable from those obtained using the Generalized-αmethod. From Figures 2 and 3 we conclude that our method is significantly more accuratethan the most widely used provably stable method for the Cahn-Hilliard equation, namely,Eyre’s algorithm.

Finally, Figure 4 shows the evolution of the Ginzburg-Landau free energy using our timeintegration algorithm. We plot three energy curves that correspond to different time steps,as indicated in the figure. We observe that the free energy does not increase with time, evenfor very large time steps. This supports our theoretical results.

Remark:

We have also performed this calculation using standard second-order accurate time inte-gration schemes, such as the trapezoidal rule and the midpoint rule. These methods ledto unstable results for ∆t = 10−5 and produced similar results to those of our method for∆t = 10−6 and ∆t = 10−7.

18

0 0.005 0.01 0.015 0.02 0.025

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

t

E∆t = 10−5

∆t = 10−6

∆t = 10−7

Fig. 4. Phase separation on a periodic square: constant time step. Evolution of the Ginzburg-Lan-dau free energy calculated using our provably stable algorithm. We represent three curves, eachcorresponding to the time step labeled. We observe that the free energy does not increase withtime irrespectively of the time step.

4.3 Phase separation on a periodic square: adaptive time step

In this section we recompute the previous example using our adaptive time-stepping algo-rithm. Our goal is to show that the proposed time integration scheme may be easily combinedwith an adaptive technique, rendering an accurate, efficient and provably stable algorithm.

Figure 5 shows snapshots of the solution at different times and the stationary configuration.The images are all indistinguishable from those calculated with a constant time step ∆t =10−7 (see Figure 3). We note that there are significantly fewer time steps taken in theadaptive case

The evolution of the time step may be observed in Figure 6. As we had anticipated, weobserve two main processes that take place at very different time scales, namely, phaseseparation and coarsening. Phase separation takes place at the beginning of the simulationand corresponds to the initial part of the dynamics marked with a circle in Figure 6. Duringthe coarsening process that follows phase separation, bubbles merge with each other toreduce the surface energy.

19

(a) t = 2.00 · 10−4 (b) t = 4.03 · 10−4 (c) t = 8.09 · 10−4 (d) t = 32.1 · 10−4 (e) Stationary

Fig. 5. Phase separation on a periodic square: adaptive time step. Numerical solution using ourprovably stable method. The solution is indistinguishable from those calculated using a constanttime step ∆t = 10−7.

Fig. 6. Phase separation on a periodic square: adaptive time step. Evolution of the time stepusing our adaptive provably stable algorithm. We depict snapshots of the solution appended to thetime-step curve. These show that local minima in the time step correspond to significant dynamicalevents in the evolution of the mixture.

Figure 6 shows a complex pattern in the evolution of the time step. This pattern is actuallyreflecting the physics of the problem. Every local minimum in the time step in Figure 6corresponds to a significant dynamical event in the evolution of the mixture. We observethat the three main local minima in Figure 6 (these are marked with arrows in the figurewith accompanying snapshots of the solution) correspond to either the disappearance of abubble or to the coalescence of two bubbles. These are the two main mechanisms by which

20

Fig. 7. Phase separation on a periodic square: adaptive time step. Detail of the evolution of thetime step using the adaptive provably stable algorithm. Snapshots of the solution accompany thetime-step curve. These show that local minima in the time step correspond to significant dynamicalevents in the evolution of the mixture.

coarsening takes place.

Let us now look at the small area marked with a circle in Figure 6. We have zoomed inthis area and plotted it in Figure 7. We observe again that every local minimum of thetime step corresponds to a significant dynamical event (see the snapshots of the solutionthat accompany the time-step curve). In particular, the first local minimum corresponds tophase separation. The subsequent local minima correspond to either the disappearance of abubble or the coalescence of two bubbles.

Remarks:

(1) In this example the ratio ∆t/δ varies from approximately 0.142 · 10−5 to 0.537, demon-strating the good performance of our method for a wide range of values of ∆t/δ.

(2) The evolution of the time step depicted in Figure 6 is fairly complex. However, itcorresponds to a simple example of the Cahn-Hilliard equation. The dynamics of theCahn-Hilliard equation become much more complex as χ increases and, thus, time-stepadaptivity becomes even more important.

21

4.4 Phase separation on an annular surface

In this example we solve the Cahn-Hilliard equation on an annular surface. The objectiveof this example is to show that our numerical formulation may be applied to non-trivialgeometries while mantaining its accuracy, stability and robustness.

The interior radius of the annular surface is ri = 0.5, while the exterior is re = 2. We employa uniform mesh composed of 256 elements in the circumferencial direction and 64 in theradial direction. We use quadratic NURBS, which permit exact geometrical modeling ofthis problem. On the boundary we weakly enforce natural boundary conditions by way of(18)–(19).

The initial condition is a randomly perturbed homogeneous concentration state c. We takec = 0.7 and a random perturbation uniformly distributed on [−0.05, 0.05]. In this examplewe take the value χ = 200.

We use the discrete formulation (26)–(27) and adaptive time stepping. Figure 8 shows severalsnapshots of the time-history of the numerical solution. The physical process is the same asin the previous example and we observe again phase separation followed by coarsening.

In Figure 9 we plot the evolution of the time step on a doubly logarithmic scale. We observea richer behavior than in the previous example and larger variations of the time step. In thisexample the ratio ∆t/δ varies from approximately 1.693 · 10−7 to 2.941 · 101, showing thegood performance of our algorithm for a wide range of values of ∆t/δ.

Figure 10 shows the evolution of the Ginzburg-Landau free energy. We have zoomed in onthe part of the evolution between t = 1 and t = 16 because it looked almost flat at the scaleof the plot. The figure clearly shows that the free energy does not increase at any time.

5 Conclusions

We have introduced provably unconditionally stable, second-order time accurate, mixedvariational methods for phase-field models. Our numerical formulation is based on a mixedfinite element formulation for the space discretization and a new second-order accuratetime integration scheme. We prove that our formulation inherits the main characteristics ofconserved phase dynamics, namely, mass conservation and nonlinear stability with respectto the free energy. We also propose an adaptive time-stepping algorithm that may be easilycombined with our method. We present numerical examples that illustrate the robustness,accuracy and stability of our formulation.

22

(a) t = 2.0816 · 10−3 (b) t = 1.2015 · 10−2 (c) t = 1.6251 · 10−1

(d) t = 4.1066 · 10−1 (e) t = 1.4920 (f) Steady state

Fig. 8. Phase separation on an annular surface: Solution at different times and steady-state config-uration using our new provably stable algorithm.

6 Acknowledgements

H. Gomez was partially supported by the J. Tinsley Oden Faculty Fellowship ResearchProgram at the Institute for Computational Engineering and Sciences. H. Gomez gratefullyacknowledges the funding provided by Xunta de Galicia (grants # 09REM005118PR and#09MDS00718PR), Ministerio de Ciencia a Innovacin (grants #DPI2009-14546-C02-01 and#DPI2010-16496) cofinanced with FEDER funds, and Universidad de A Coruna. T.J.R.Hughes was partially supported by the Office of Naval Research under Contract NumberN00014-08-1-0992.

Appendix

In this appendix we derive the following quadrature formula:

∫ b

af(x)dx =

b− a2

(f(a) + f(b))− (b− a)3

12f ′′(a)− (b− a)4

24f ′′′(ξ); ξ ∈ (a, b) (78)

23

10−4

10−2

100

102

10−4

10−3

10−2

10−1

100

101

t

∆t

Fig. 9. Phase separation on an annular surface. Evolution of time step on a doubly logarithmicscale. We use our adaptive provably stable algorithm.

where f : [a, b] 7→ R is a sufficiently smooth function.

Let P2 be the quadratic polynomial that satisfies

P2(a) = f(a); P2(b) = f(b); P ′′2 (a) = f ′′(a) (79)

We define the function R2 as,R2(x) = f(x)− P2(x) (80)

From the definition of R2 and P2 it follows that

R2(a) = R2(b) = 0 (81)

Let us rewrite R2 asR2(x) = w2(x)S2(x) (82)

wherew2(x) = (x− a)(x− b)(x+ b− 2a) (83)

and S2 is an unknown function such that (82) is verified. For further reference, we note thatw2 verifies the conditions w2(a) = w2(b) = w′′2(a) = 0.

Let us define the function

F (z) = f(z)− P2(z)− w2(z)S2(x) (84)

24

Fig. 10. Phase separation on an annular surface. Evolution of the Ginzburg-Landau free energyusing our new adaptive provably stable algorithm. We have zoomed in the part of the evolutionbetween t = 1 and t = 16 because it appears almost flat at the scale of the plot. The figure clearlyshows that the free energy does not increase at any time.

where x is a fixed parameter. This function clearly satisfies the conditions F (a) = F (b) =F (x) = 0. Applying Rolle’s theorem twice we conclude the there exists at least one point inthe open interval (a, b) at which F ′′ vanishes. In addition, we know that F ′′(a) = 0, whichallows us to apply Rolle’s theorem again to conclude that there exists θ ∈ (a, b) such thatF ′′′(θ) = 0. Taking derivatives in equation (84) we can obtain a explicit expression for F ′′′,namely

F ′′′(z) = f ′′′(z)− w′′′2 (z)︸ ︷︷ ︸=6

S2(x) (85)

Evaluating this expression at z = θ, it follows that

S2(x) =f ′′′(θ)

6(86)

Taking all of this into account, we may integrate (80) to conclude

∫ b

af(x)dx =

∫ b

aP2(x)dx+

∫ b

aw2(x)

f ′′′(θ(x))

6dx (87)

Since w2 does not change sign on the open interval (a, b), we can apply the mean value

25

theorem to the last integral in (87) to obtain

∫ b

aw2(x)

f ′′′(θ(x))

6dx =

f ′′′(ξ)

6

∫ b

aw2(x)dx; ξ ∈ (a, b) (88)

This analysis leads to the quadrature formula∫ b

af(x)dx =

b− a2

(f(a) + f(b))− (b− a)3

12f ′′(a)− (b− a)4

24f ′′′(ξ); ξ ∈ (a, b) (89)

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