second order schemes and time-step adaptivity for allen–cahn and cahn–hilliard models

Computers and Mathematics with Applications 68 (2014) 821–846

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Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

Second order schemes and time-step adaptivity forAllen–Cahn and Cahn–Hilliard modelsFrancisco Guillén-González a,∗, Giordano Tierra b,1

a Departamento de Ecuaciones Diferenciales y Análisis Numérico & IMUS, Universidad de Sevilla, 41012 Seville, Spainb Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

a r t i c l e i n f o

Article history:Received 1 July 2013Received in revised form 18 June 2014Accepted 19 July 2014Available online 2 September 2014

Keywords:Diffuse interface phase-fieldEnergy-stabilityAdaptive time steppingUnique solvabilityFinite elements

a b s t r a c t

In this paper, we focus on efficient second-order in time approximations of the Allen–Cahnand Cahn–Hilliard equations. First of all, we present the equations, generic second-orderschemes (based on a mid-point approximation of the diffusion term) and some schemesalready introduced in the literature. Then, we propose new ways of deriving second-orderin time approximations of the potential term (starting from the main schemes introducedin Guillén-González and Tierra (2013)), yielding to new second-order schemes. For theseschemes and other second-order schemes previously introduced in the literature, we studythe constraints on the physical and discrete parameters that can appear to assure theenergy-stability, unique solvability and, in the case of nonlinear schemes, the convergenceof Newton’s method to the nonlinear schemes. Moreover, in order to save computationalcost we have developed a new adaptive time-stepping algorithm based on the numericaldissipation introduced in the discrete energy law in each time step. Finally, we compare thebehaviour of the schemes and the effectiveness of the adaptive time-stepping algorithmthrough several computational experiments.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The study of problems where there is an interface evolving in time plays an increasingly role in many current scientific,engineering, and industrial applications. For instance, they arise naturally in hydrodynamics and materials science formodelling mixture of different fluids, solids or gases. The diffuse-interface model describes the dynamic of interfaces bylayers of small thickness. This idea was introduced by van der Waals [1], being the foundation for the phase-field theoryfor phase transition and critical phenomena. Thus, the structure of the interface is determined by molecular forces; thetendencies for mixing and de-mixing are balanced through a mixing energy (or free energy). LetΩ ⊂ Rd (d = 1, 2 or 3) bea bounded domain, the surface motion can be viewed as due to the physical energy dissipation of

ΩE(φ)dxwhere E(φ) is a

phase-field’s free energy functional. In the context of the surface elasticity, the value E(φ) can represent different interfacialenergies associated to the phase-field. The most basic energy functional one may introduce is

E(φ) =

Ω

12|∇φ|

2+ F(φ)

dx (1.1)

∗ Corresponding author.E-mail addresses: [email protected] (F. Guillén-González), [email protected], [email protected] (G. Tierra).

1 Current address: Mathematical Institute, Faculty of Mathematics and Physics, Charles University, 186 75 Prague 8, Czech Republic.

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822 F. Guillén-González, G. Tierra / Computers and Mathematics with Applications 68 (2014) 821–846

where F(φ) is a double-well potential. Several choices of F(φ) have been presented in the literature. In this paper, we aregoing to consider the Ginzburg–Landau double well potential

F(φ) =1

4ε2(φ2

− 1)2 and f (φ) = F ′(φ) =1ε2(φ2

− 1)φ, (1.2)

where ε > 0 is a parameter related to the interface thickness. There are other possible choices of the double well potentialthat are usually considered in the literature, like the logarithmic potential

F(φ) =θ

2[(1 + φ) log(1 + φ)+ (1 − φ) log(1 − φ)] +

θc

2(1 + φ)(1 − φ). (1.3)

The Allen–Cahn and the Cahn–Hilliard equations were introduced in [2,3] to model phase transitions in iron alloys andthe thermodynamic forces driving phase separation, respectively. Both equations are particular cases of gradient flows,and can be written as φt = −

δE(φ)δφ

, where δE(φ)δφ

stands for the variational derivative of the free energy, either takenin the L2(Ω)-norm for Allen–Cahn or in H−1(Ω) for Cahn–Hilliard. In recent times these equations have been widelystudied due to their connection with many physically motivated problems such as phase separation, liquid crystals, vesiclemembranes deformation, image processing, tumour growth, etc. For more physical background, derivation and discussionof the Cahn–Hilliard equation we refer to [4,5].

There are several challenges to obtain numerical approximations of these problems such as the existence of a nonlinearterm and the presence of the (usually small) parameter ε. An appropriate numerical resolution of the problem requires aproper relation between physical and numerical scales, that is, the (spatial) mesh size h and the (time) step size k have toproperly relate to the interaction length ε.

In the last years, many authors have been working in the numerical approximation of these problems. Implicitapproximations of the logarithmic potential (1.3) are commonly used in the literature [6–8]. Meanwhile, in the polynomialpotential case (1.2), different strategies have been considered. The implicit approximation of the potential (1.2) is consideredin [9–13], where a Newton method is usually employed in order to compute the nonlinear scheme. There is also animplicit–explicit approximation of the potential (1.2) that does not introduce any numerical dissipation (which is amodifiedmidpoint approximation) which has been widely used in phase field models [14] and in the liquid crystal context [15]. Onthe other hand, many authors follow the idea of splitting the potential F(φ) into a convex and a non-convex part [16,17]in order to assure the existence of some numerical dissipation to obtain unconditionally energy-stable schemes [18–20].Another approach that introduces some numerical dissipation to derive unconditionally energy-stable schemes has beenconsidered in [21], where Taylor’s formula is used to derive an approximation of the potential term. The resulting problemsare again nonlinear. Trying to circumvent the problem of the nonlinearity of the schemes and following Eyre’s ideas, somelinear (conditionally energy-stable) schemes are considered in [22,23]. Moreover, in the Nematic liquid crystal framework,the authors of [24] present one linear and one nonlinear first order in time unconditionally energy-stable scheme based onthe introduction of a Lagrange multiplier in the formulation of the potential term. These ideas have been extended in [25]to the Cahn–Hilliard problemwhere two new linear second order in time approximations of the potential term are derived.Moreover, the consistency errors related to each scheme are studied in [25] and the authors also compare these schemeswith other linear schemes previously introduced in the literature.

The design of adaptive time-stepping algorithms suitable to be used in the phase-field framework has attracted muchattention due to the fact of the different time scales that appear [26,27].

In this work we propose new ways of deriving second order in time approximations of the potential term and we usethese potential approximations to design new second order schemes. Then we compare the constraints on the physicaland discrete parameters appearing to assure stability, solvability and convergence (of Newton’s method to approximate thenonlinear schemes) of these schemes with other second order schemes previously introduced in the literature. Moreover,in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numericaldissipation introduced in the discrete energy laws at each time step.

The rest of this paper is organized as follows. First of all, we introduce the Allen–Cahn and Cahn–Hilliard equations andtheir corresponding generic second order schemes (based on a mid-point approximation of the diffusion) in Sections 2 and3, respectively. In Section 4we present differentways of approximating the potential term jointly to some properties of eachapproximation and in Section 5 we study Newton iterative algorithms used to approximate the schemes that are nonlinear.Section 6 is devoted to the design of the time-stepping algorithm and in Section 7 we give several numerical experimentsusing the schemes presented through the paper, comparing the effectiveness of the time-stepping algorithm. Finally, wesummarize the main achievements of the paper in Section 8.

2. Allen–Cahn model

The Allen–Cahn equation is a gradient flow from the following energy functional:

E(φ) =

Ω

12|∇φ|

2+ F(φ)

dx,

F. Guillén-González, G. Tierra / Computers and Mathematics with Applications 68 (2014) 821–846 823

where F(φ) is the Ginzburg–Landau double well potential F(φ) = (φ2− 1)2/(4ε2), and its derivative f (φ) = F ′(φ) =

(φ2− 1)φ/ε2. The Allen–Cahn equation reads,φt + γ (−1φ + f (φ)) = 0 in (0, T )×Ω,

∂nφ = 0, on (0, T )× ∂Ω, φ|t=0 = φ0 inΩ.Here, γ > 0 is a relaxation time coefficient. The weak formulation of the problem is defined as follows: Find φ ∈ L∞

(0, T ;H1(Ω))with φt ∈ L2(0, T ; L2(Ω)) such that φ(0) = φ0 inΩ and(φt , φ)+ γ (∇φ,∇φ)+ γ (f (φ), φ) = 0 ∀ φ ∈ H1(Ω), a.e. t ∈ (0, T ).

Hereafter (·, ·) denotes the L2(Ω)-scalar product. The regularity ofφ can be deduced from the following energy law obtainedby taking φ =

1γφt :

dE(φ)dt

+1γ

∥φt∥2L2 = 0. (2.4)

It is also well-known that this problem satisfies the following maximum principle:If φ0 ∈ [−1, 1] inΩ , then φ(t) ∈ [−1, 1] inΩ , for each t > 0.

To prove that, the key is to consider the following diffusion–reaction inequation for φ2:12(φ2)t + γ

−

12∆(φ2)+ f (φ)φ

≤ 0 in (0, T )×Ω,

jointly with the following property for the potential term: f (φ)φ ≥ 0 if φ ∈ [−1, 1].It is possible to deduce the existence and uniqueness of global in time weak solution [28].

2.1. A generic second order scheme

We define a generic time-discrete numerical scheme, using a mid-point finite difference approximation in time for thelinear terms. For simplicity we assume the uniform partition of the time interval: tn = nk, with k > 0 the time step. Thetask is to compute φn as an approximation of φ(tn). The scheme is defined as:

Initialization: φ0= φ0.

First step: Given φ0, find φ1 by means of an one-step scheme.Step n + 1 (n ≥ 1): Given (φn−1, φn) ∈ H1(Ω)2, find φn+1

∈ H1(Ω) such thatδtφ

n+1, φ+ γ

∇φn+ 1

2 ,∇φ

+ γ (f k(φn+1, φn, φn−1), φ) = 0 ∀ φ ∈ H1(Ω), (2.5)

where φn+ 12 := (φn+1

+ φn)/2 and δtφn+1= (φn+1

− φn)/k.In order to assure second order in time approximation, f k(φn+1, φn, φn−1) have to be defined such that

f k(φ(tn+1), φ(tn), φ(tn−1)) be a second order approximation of f (φ(tn+12 )) where tn+

12 := (tn+1

+ tn)/2. For one-stepschemes, f k(φn+1, φn, φn−1) will depend only on (φn+1, φn). Testing in (2.5) by φ =

1γδtφ

n+1 we obtain the followingdiscrete energy law

δtE(φn+1)+1γ

∥δtφn+1

∥2L2 + NDphobic(φ

n+1, φn, φn−1) = 0 (2.6)

where

δtE(φn+1) =E(φn+1)− E(φn)

kis the discrete time derivative of the energy, and

NDphobic(φn+1, φn, φn−1) =

Ω

f k(φn+1, φn, φn−1)δtφn+1

− δt

Ω

F(φn+1)

(2.7)

which could be called the phobic numerical dissipation. Therefore, depending on the approximation considered off k(φn+1, φn, φn−1)we will obtain different numerical schemes, with different discrete energy laws.

Definition 2.1. The numerical scheme (2.5) is energy-stable if it holds

δtE(φn+1)+1γ

∥δtφn+1

∥2L2 ≤ 0, ∀ n.

In particular, energy-stable schemes satisfy the energy decreasing in time property,

E(φn+1) ≤ E(φn), ∀ n.

We have to keep inmind our aim to obtain energy-stable and uniquely solvable schemes while we define the potential termf k(φn+1, φn, φn−1).


3. Cahn–Hilliard model

The Cahn–Hilliard equation is another gradient flow from the same following energy functional:

E(φ) =

Ω

12|∇φ|

2+ F(φ)

dx.

The Cahn–Hilliard equation reads,

φt − γ∆(−1φ + f (φ)) = 0 in (0, T )×Ω,

∂nφ = 0, ∂n(−1φ + f (φ)) = 0, on (0, T )× ∂Ω, φ|t=0 = φ0 inΩ.

The weak formulation is defined as follows: Find (φ,w) such that φ(0) = φ0 inΩ ,

φ ∈ L∞(0, T ;H1(Ω)) and w ∈ L2(0, T ;H1(Ω))

with φt ∈ L2(0, T ;H1(Ω)′) and satisfying the following variational formulation a.e. in (0, T ) [29]:⟨φt , w⟩ + γ (∇w,∇w) = 0 ∀ w ∈ H1(Ω),

(w, φ) = (∇φ,∇φ)+ (f (φ), φ) ∀ φ ∈ H1(Ω),

where ⟨·, ·⟩ denotes the H1(Ω)′ × H1(Ω) duality product. This problem is conservative, because the total massΩφ(t)

remains constant in time, as we can realize taking w = 1, arriving at

ddt

Ω

φ = 0, i.e.Ω

φ(t) =

Ω

φ0 ∀ t ≥ 0.

Moreover, taking w = w and φ = φt , the following energy law holds:

ddt

E(φ(t))+ γ ∥∇w(t)∥2L2 = 0. (3.8)

From these properties it is possible to deduce the existence and uniqueness of global in time weak solution [30–32,13,33].Finally, it is known that this equation does not satisfy a maximum principle property.

3.1. A generic second order scheme

Initialization: φ0= φ0.

First step: Given φ0, find φ1 by means of an one-step scheme.Step n + 1 (n ≥ 1): Given (φn−1, φn) ∈ H1(Ω)2, find (φn+1, wn+ 1

2 ) ∈ H1(Ω)2 such thatδtφ

n+1, w+ γ

∇wn+ 1

2 ,∇w

= 0 ∀ w ∈ H1(Ω),wn+ 1

2 , φ

=

∇φn+ 1

2 ,∇φ

+ (f k(φn+1, φn, φn−1), φ) ∀ φ ∈ H1(Ω).(3.9)

Unknown wn+ 12 is an approximation at midpoint tn+

12 := (tn + tn+1)/2 (directly computed), while φn+ 1

2 := (φn+1+

φn)/2.Taking w = 1, it is possible to show that these schemes are conservative because the total mass is constant in time,

Ω

φn+1=

Ω

φn=

Ω

φ0.

Again, in order to assure that previous schemes are second order in time, f k(φn+1, φn, φn−1) have to be defined as asecond order approximation of f (φ(tn+

12 )).

On the other hand, testing in (3.9) by (φ, w) = (δtφn+1, wn+ 1

2 )we obtain the following discrete energy law

δtE(φn+1)+ γ

∇wn+ 12

2

L2+ NDphobic(φ

n+1, φn, φn−1) = 0 (3.10)

where

δtE(φn+1) =E(φn+1)− E(φn)

k

and NDphobic(φn+1, φn, φn−1) is defined as in (2.7). In particular, depending on the approximation considered of

f k(φn+1, φn, φn−1)we will obtain different numerical schemes, with different discrete energy laws.


Definition 3.1. The numerical scheme (3.9) is energy-stable if it holds

δtE(φn+1)+ γ

∇wn+ 12

2

L2≤ 0, ∀ n.

In particular, energy-stable schemes satisfy the energy decreasing in time property,

E(φn+1) ≤ E(φn), ∀ n.

4. Numerical schemes

In this section, we will describe five different approximations of the potential term f k(φn+1, φn, φn−1) (where two ofthem are nonlinear and the other three are linear), studying the properties related to the energy-stability and the uniquesolvability of the schemes.

4.1. Uniquely solvable and unconditionally energy-stable scheme (MP-BDF2)

Using ideas presented in [34], we can derive a second order approximation of the potential term that allows us to defineunconditionally energy-stable schemes.We take into account that the energy density F(φ) is smooth and possesses a convex(+)–concave (−) decomposition with a quadratic concave term. In particular, we decompose F(φ) as

F(φ) = F+(φ)+ F−(φ)

such that

F+(φ) =1

4ε2(φ4

+ 1), F−(φ) = −1

2ε2φ2.

The philosophy we shall use to construct a second-order energy-stable scheme is to approximate the convex F ′+(φ) and the

quadratic concave F ′−(φ) terms using different second order approximations. Following the idea of treating the convex part

implicitly and the concave part explicitly, a second-order approximation of the potential F ′(φ) should take the followingform

f k(φn+1, φn, φn−1) = f k+(φn+1, φn)+ f k

−(φn, φn−1)

where

f k−(φn, φn−1) =

32F ′

−(φn)−

12F ′

−(φn−1) = −

12ε2

3φn

− φn−1corresponds to an explicit two-step second-order (at tn+

12 ) Backward Difference Formula (BDF2), and f k

+(φn+1, φn) should

be chosen such that let the above scheme be second-order at tn+12 and energy-stable.

Since the energy-stability is proved by taking the inner product of f k+(φn+1, φn) by the time discrete fraction (φn+1

−

φn)/k, the following inequality has to be satisfied:

(f k+(φn+1, φn), φn+1

− φn) ≥

Ω

F+(φn+1)dx −

Ω

F+(φn)dx. (4.11)

By direct calculation,Ω

F+(φn+1)dx −

Ω

F+(φn)dx =

Ω

1

0

ddτ

F+(φn+ τ(φn+1

− φn))dτdx

=

Ω

1

0F ′

+(φn

+ τ(φn+1− φn))dτ

(φn+1

− φn)dx.

Then, a natural choice for f k+(φn+1, φn) is

f k+(φn+1, φn) =

1

0F ′

+(φn

+ τ(φn+1− φn))dτ , (4.12)

and then (4.11) holds with an equality. Since a polynomial potential is considered, we can compute explicitly this integralas follows:

f k+(φn+1, φn) =

1ε2

1

0(φn

+ τ(φn+1− φn))3dτ =

14ε2

(φn

+ τ(φn+1− φn))4

φn+1 − φn

τ=1

τ=0

=1

4ε2(φn+1)4 − (φn)4

φn+1 − φn=

14ε2

(φn+1)3 + (φn+1)2φn

+ (φn)2φn+1+ (φn)3

.


Therefore,f k(φn+1, φn, φn−1) = f k

+(φn+1, φn)+ f k

−(φn, φn−1)

=1

4ε2

(φn+1)3 + (φn+1)2φn

+ (φn)2φn+1+ (φn)3

−

12ε2

3φn

− φn−1. (4.13)

Remark 4.1. In particular, the approximation of the convex part corresponds to the following mid-point approximation

f k+(φn+1, φn) =

F+(φn+1)− F+(φ

n)

φn+1 − φn.

Theorem 4.2. Using the potential approximation (4.13), schemes (2.5) (for AC) and (3.9) (for CH) are unconditionally energy-stable for a modified energyE(φn+1). In fact, the following discrete energy law holds:

δtE(φn+1)+ NDphobic(φn+1, φn, φn−1)+

1γ

∥δtφn+1

∥2L2 (for AC)

γ ∥∇wn+1/2∥2L2 (for CH)

= 0 (4.14)

where

E(φn+1) =

Ω

12|∇φn+1

|2+ F(φn+1)+

k2

4ε2|δtφ

n+1|2dx

and

NDphobic(φn+1, φn, φn−1) =

k3

4ε2∥δttφ

n+1∥2L2 ,

with δttφn+1= δt(δtφ

n+1) = (δtφn+1

− δtφn)/k and δtE(φn+1) = (E(φn+1)−E(φn))/k.

Remark 4.3. Notice thatE(φn+1) = E(φn+1)+ Epert(φn+1)where

Epert(φn+1) =k2

4ε2∥δtφ

n+1∥2L2

is a perturbation of the exact energy E(φn+1).

Proof. By the derivation of f k(φn+1, φn, φn−1) given in (4.13), the convex part of the potential does not introduce phobicnumerical dissipation when (2.5) is tested by φ =

1γδtφ

n+1 (and (3.9) by (w, φ) = (wn+ 12 , δtφ

n+1)), becauseΩ

f k+(φn+1, φn)δtφ

n+1= δt

Ω

F+(φn+1). (4.15)

With respect to the non-convex part, taking into account f k−(φn, φn−1) =

1ε2

−φn+ 1

2 +k22 δttφ

n+1we obtain

Ω

f k−(φn, φn−1)δtφ

n+1dx = −1

2ε2δt

Ω

(φn+1)2dx

+k2

2ε2

Ω

δtφn+1δttφ

n+1dx

= δt

Ω

F−(φn+1)

+

k2

4ε2δt

Ω

(δtφn+1)2dx

+

k3

4ε2

Ω

(δttφn+1)2dx.

Therefore, (4.14) holds.

Theorem 4.4 (Unconditional Unique Solvability). Schemes (2.5) (for AC) and (3.9) (for CH) with the potential approximationgiven in (4.13) are uniquely solvable.

Proof (Part 1: Allen–Cahn. Existence and Uniqueness). For each time step, scheme (2.5), (4.13) is the Euler–Lagrangeequation of the functional

J(φn+1) =12k

∥φn+1∥2L2 +

γ

4∥∇φn+1

∥2L2 + γ

Ω

F+(φn+1, φn)+

Ω

γ f k

−(φn, φn−1)−

γ

21φn

−1kφn

φn+1 (4.16)

where

F+(φn+1, φn) =

1

0

1τF+(φ

n+ τ(φn+1

− φn))dτ . (4.17)


In fact, by using (4.12) we can easily check that

f k+(φn+1, φn) =

∂F+(φn+1, φn)

∂φn+1.

The convexity of the functional F+(·, φn) follows from the convexity of the map

φn+1∈ H1(Ω) −→

1τF+(φ

n+ τ(φn+1

− φn)) ∈ R

for all τ ∈ (0, 1], thanks to the convexity of F+. Indeed,

∂2F+(φn+1, φn)

∂(φn+1)2=

1

0τF ′′

+(φn

+ τ(φn+1− φn))dτ ≥ 0 ∀φn+1, φn.

Therefore, the solution of scheme (2.5), (4.13) in each time step can be expressed as the unique minimizer of the strictlyconvex functional J , assuring the unique solvability of the scheme.

(Part 2: Cahn–Hilliard. Uniqueness of solution).

Let (φn+11 , w

n+ 12

1 ) and (φn+12 , w

n+ 12

2 ) be two possible solutions of scheme (3.9), (4.13) and we define φ = φn+11 − φn+1

2

andw = wn+ 1

21 − w

n+ 12

2 . Subtracting scheme for both solutions one has1k(φ, w)+ γ (∇w,∇w) = 0

12(∇φ,∇φ)+

f k+(φn+1

1 , φn)− f k+(φn+1

2 , φn), φ

− (w, φ) = 0.(4.18)

Then, testing by (φ, w) = ( 1kφ,w) and, since f k+(·, φn) is a monotone function,

Ω

f k+(φn+1

1 , φn)− f k+(φn+1

2 , φn)φ =

Ω

∂ f k+

∂φn+1(φξ )φ2

=

Ω

∂2F+

∂2φn+1(φξ )φ2

≥ 0

one has

γ ∥∇w∥2L2 +

12k

∥∇φ∥2L2 ≤ 0 ⇒

∇φ = 0 ⇒ φ = C1∇w = 0 ⇒ w = C2.

Then from (4.18)1 wederiveφ = 0 (i.e.φn+11 = φn+1

2 ), and combining thiswith (4.18) 2 weobtainw = 0 (i.e.wn+ 1

21 = w

n+ 12

2 ),hence uniqueness of the scheme is deduced.

(Part 3: Cahn–Hilliard. Existence of solution).Firstly, we rewrite the scheme using the change of variables

ψn+1= φn+1

− φn and zn+12 = wn+ 1

2 −

Ω

wn+ 12 = wn+ 1

2 −

Ω

f k(φn+1, φn, φn−1), (4.19)

as follows: find (ψn+1, zn+12 ) ∈ H1(Ω)2 such that, for any (ψ, z) ∈ H1(Ω)2,

ψn+1, z

+ k γ

∇zn+

12 ,∇ z

= 0,

12(∇ψn+1,∇ψ)+

f k(ψn+1

+ φn, φn, φn−1)−

Ω

f k(ψn+1+ φn, φn, φn−1), ψ

−

zn+

12 , ψ

= −(∇φn,∇ψ).

(4.20)

Notice that any solution of (4.20) satisfiesΩψn+1

=Ωzn+

12 = 0. Moreover, if (ψn+1, zn+

12 ) is a solution of (4.20) then

φn+1= ψn+1

+ φn and wn+ 12 = zn+

12 +

f k(φn+1, φn, φn−1) is a solution of (3.9), hence it suffices to prove existence of

(4.20), (4.13). For this purpose, we are going to consider a Galerkin approximation of (4.20), (4.13) (for instance with a FiniteElement approximation):


Given (φn−1h , φn

h) ∈ Xh ×Xh suitable approximations of φn−1 and φn respectively, find (ψn+1h , z

n+ 12

h ) ∈ Xh ×Wh such thatfor each (ψh, zh) ∈ Xh × Wh

ψn+1

h , zh+ kγ

∇z

n+ 12

h ,∇ zh

= 0,

12(∇ψn+1

h ,∇ψh)+

f k+(ψn+1

h + φnh , φ

nh)−

Ω

f k+(ψn+1

h + φnh , φ

nh), ψh

−

zn+ 1

2h , ψh

= −

f k−(φn

h , φn−1h )−

Ω

f k−(φn

h , φn−1h ), ψh

− (∇φn

h ,∇ψh).

(4.21)

Again, one hasΩψn+1

h =Ωzn+ 1

2h = 0 if the constant functions belong to Xh and Wh and the uniqueness of (4.21), (4.13)

can be derived using the same argument done in the continuous case. Since the constant terms (Ωf k+(ψn+1

h + φnh , φ

nh), ψ)

and (Ωf k−(φn

h , φn−1h ), ψ) vanish when

Ωψ = 0, then using the mean value spaces

Xh =

ψh ∈ Xh such that

Ω

ψh = 0, Wh =

zh ∈ Wh such that

Ω

zh = 0,

problem (4.21) can be rewritten as: find (ψn+1h , z

n+ 12

h ) ∈ Xh × Wh such that for each (ψh, zh) ∈ Xh × Whψn+1

h , zh+ kγ

∇z

n+ 12

h ,∇ zh

= 0,

12(∇ψn+1

h ,∇ψh)+

f k+(ψn+1

h + φnh , φ

nh), ψh

−

zn+ 1

2h , ψh

= −

f k−(φn

h , φn−1h ), ψh

− (∇φn

h ,∇ψh).

(4.22)

We are going to obtain energy estimates for (4.22). Taking (ψh, zh) = (ψn+1h , z

n+ 12

h ) in (4.22) and using that there is notnumerical dissipation in the approximation of f k

+(see (4.15)):

kγ∇z

n+ 12

h

2

L2+

12∥∇ψn+1

h ∥2L2 +

Ω

F+(ψn+1h + φn

h) =

Ω

F+(φnh)− (f k

−(φn

h , φn−1h ), ψn+1

h )− (∇φnh ,∇ψ

n+1h ).

In particular,

kγ∇z

n+ 12

h

2

L2+

12∥∇ψn+1

h ∥2L2 ≤

Ω

F+(φnh)+ C

∥φn

h∥H1 , ∥φn−1h ∥L2

∥ψn+1

h ∥H1 . (4.23)

SinceΩψn+1

h =Ωzn+ 1

2h = 0 and taking into account the bounds of φn

hh in H1(Ω) and φn−1h h in L2(Ω), then a priori

estimates of ψn+1h , z

n+ 12

h h in H1(Ω)2 hold. Moreover, the existence of a solution of (4.22) is consequence of (4.23) applyingfor instance Lemma 1.4, Chapter 2 of Temam’s book [35] (this problem can be seen as a nonlinear coercive problem in finitedimension):

Lemma 4.5. Let X be a finite dimensional Hilbert space with scalar product (·, ·) and norm | · | and let P be a continuousmappingfrom X into itself. Assume that there exists µ > 0 such that

(P(ξ), ξ) > 0 for |ξ | = µ > 0.

Then, there exists ξ ∈ X, |ξ | ≤ µ, such that

P(ξ) = 0.

In fact, for the appropriate definitions,

X = Xh × Wh

((ψh, zh), (ψh, zh)) = (ψh, ψh)+ (zh, zh)

and P(ψh, zh) ∈ X such thatP(ψh, zh), (ψh, zh)

= (ψh, zh)+ kγ (∇zh,∇ zh)+

12(∇ψh,∇ψh)+ (f k

+(ψh + φn

h , φnh), ψh)

− (zh, ψh)+ (f k−(φn

h , φn−1h ), ψh)+ (∇φn

h ,∇ψh) ∀ (ψh, zh) ∈ X,


one has from (4.23)

(P(ψh, zh), (ψh, zh)) ≥ kγ ∥∇zh∥2L2 +

12∥∇ψh∥

2L2 −

Ω

F+(φnh)− C

∥φn

h∥H1 , ∥φn−1h ∥L2

∥ψh∥H1 ,

hence the hypothesis of Lemma 4.5 hold for µ large enough. Then, there exists (ψh, zh) ∈ Xh × Wh such that P(ψh, zh) = 0and this pair (ψh, zh) is a solution of (4.22).

Finally, the existence of the continuous problem (4.20), (4.13) is obtained by taking limits as h → 0 in the solutions of(4.21), (4.13), using the previous energy estimates and the compactness of the sequence ψn+1

h h (and φnhh) in Lp(Ω) for

any p < 6 (owing to the estimate in H1(Ω)) which let to control the limit for the nonlinear term f k+(ψn+1

h + φnh , φ

nh).

Remark 4.6. To assure the existence and uniqueness of the finite element scheme, there are no compatibility constraintson the choice of the discrete spaces Xh andWh. Indeed, the only assumption is to consider that constant functions belong toXh and Wh.

4.2. Midpoint approximation (MP)

The following midpoint approximation of the potential term

f k(φn+1, φn) =F(φn+1)− F(φn)

φn+1 − φn(4.24)

has been widely used in the literature [12,9,15] to design one-step unconditionally energy-stable schemes preservingthe discrete energy law, i.e., without introducing phobic numerical dissipation because NDphobic(φ

n+1, φn) = 0. FromRemark 4.1,

f k(φn+1, φn) = f k+(φn+1, φn)+ f k

−(φn+1, φn) :=

F+(φn+1)− F+(φ

n)

φn+1 − φn+

F−(φn+1)− F−(φ

n)

φn+1 − φn

where f k+(φn+1, φn) is the same as in MP-BDF2 but now f k

−is defined as

f k−(φn+1, φn) = −

φn+1+ φn

2ε2= −

1ε2φn+ 1

2 .

Theorem 4.7. Schemes (2.5) (for AC) and (3.9) (for CH) with the midpoint approximation (4.24) are unconditionally energy-stable. In particular, the following discrete energy laws hold:

δtE(φn+1)+

1γ

∥δtφn+1

∥2L2 (for AC)

γ ∥∇wn+1/2∥2L2 (for CH)

= 0. (4.25)

Proof. Testing (2.5) by φ =1γδtφ

n+1 (or (3.9) by (w, φ) = (wn+ 12 , δtφ

n+1)), we arrive at the equality δtF(φn+1) =

f k(φn+1, φn)δtφn+1.

Theorem 4.8 (Conditional Unique Solvability). Under the constraints

k <2ε2

γ(for AC) and k <

4ε4

γ(for CH),

there exists a unique solution of schemes (2.5) (for AC) and (3.9) (for CH) with the midpoint approximation (4.24).

Proof (Part 1: Allen–Cahn). It suffices to follow the same arguments used in Theorem 4.4 and assuming hypothesisk < 2ε2/γ to assure the convexity of the functional

J(φn+1) =γ

4∥∇φn+1

∥2L2 +

12k

−γ

4ε2

∥φn+1

∥2L2 + γ

Ω

F+(φn+1, φn)+

Ω

−γ

21φn

+

γ

2ε2−

1k

φn

φn+1.

(Part 2: Cahn–Hilliard).The uniqueness of the scheme can be derived using again the arguments presented in Theorem 4.4 and assuming the

constraint k < 4ε4/γ to control the concave part. Furthermore, to assure the existence of the corresponding Galerkinproblem we follows Theorem 4.4 using the discrete version of the energy estimates:

12∥∇ψn+1

h ∥2L2 + k γ

∇zn+ 1

2h

2

L2≤

Ω

F(φnh)+ ∥∇φn

h∥L2∥∇ψn+1h ∥L2 .


Remark 4.9. Now, the existence and uniqueness of the finite element scheme is assured with the compatibility constraintXh ⊂ Wh on the choice of the discrete spaces Xh andWh (and the constant functions belong to Xh and Wh).

4.3. Optimal dissipation (OD2)

In [25], the authors develop the so-called optimal dissipation approach:

f k(φn+1, φn) = f (φn)+12f ′(φn)(φn+1

− φn) =1ε2

32(φn)2φn+1

−12(φn)3 −

φn+1+ φn

2

. (4.26)

It is a second-order in time linear approximation of the potential term derived by using the following Hermite quadratureformula (which is exact for P1), b

ag(x)dx = (b − a)g(a)+

12(b − a)2g ′(a)+ C (b − a)3g ′′(ξ). (4.27)

Approximation (4.26) is optimal from the phobic numerical dissipation point of view, because NDphobic(φ(tn+1), φ(tn)) =

O(k2) and it is not possible to design a linear approximation introducing a higher order in time phobic numerical dissipation.In fact [25]

NDphobic(φ(tn+1), φ(tn)) =Cε2

k2δtφ(tn+1)

3ζφ(tn+1)+ (1 − ζ )φ(tn)

= O(k2), with ζ ∈ (0, 1).

Using approximation (4.26), a second order linear scheme can be defined via (2.5) (for AC) and (3.9) (for CH). The issueof how to control the sign of NDphobic(φ

n+1, φn) to derive the energy-stability of the scheme remains as an open problem.Nevertheless, this scheme have a good behaviour in the numerical simulations. More precisely, the question that arises ishow to modify linearly the scheme, preserving the second order in time approximation, in order to introduce some extranumerical dissipation to control the negative part of NDphobic(φ

n+1, φn).

Remark 4.10. Following the ideas presented in [22], it is possible to assure the energy decreasing property E(φn+1) ≤ E(φn)in the Allen–Cahn case under a constraint of type k < ε2/γ , when a truncated approximation of the potential F(φ) isconsidered, with a quadratic increasing for large |φ|. This proof is based on controlling a bound of NDphobic(φ

n+1, φn) by thephysical dissipation 1

γ∥δtφ

n+1∥2L2. In the Cahn–Hilliard case the same argument can still hold, but the resulting problemwith

the truncate potential does not necessarily correspond with the original problem, due to the fact that there is no maximumprinciple.

On the other hand, the conditional solvability of the OD2 scheme can be deduced.

Theorem 4.11 (Conditional Unique Solvability). Schemes (2.5) (for AC) and (3.9) (for CH) with the potential approxima-tion (4.26), are uniquely solvable under the constraints

k <2ε2

γ(for AC) and k <

2ε4

γ(for CH).

Proof. Following the same arguments of Theorems 4.4 and 4.8. In particular, for the Cahn–Hilliard model it is needed toconsider the following discrete version of the energy estimates:

kγ2

∇zn+ 1

2h

2

L2+

14

−kγ8ε4

∥∇ψn+1

h ∥2L2 ≤ C

∥f (φn

h)∥2L6/5 + ∥φn

h∥2H1

. (4.28)

Remark 4.12. In this case, to assure the existence and uniqueness of a solution, the compatibility constraint Xh ⊂ Wh onthe choice of the discrete spaces Xh and Wh is needed (and the constant functions belong to Xh and Wh).

4.4. OD2–BDF2

It is possible to develop a new approach for the potential term combining some of the ideas used to derive bothMP-BDF2andOD2 approximations. The first idea consists in splitting again the potential term between convex and non-convex parts,then the convex part is approximated using OD2, while BDF2 is considered for the non-convex part as in (4.13), i.e.,

f k(φn+1, φn, φn−1) = f k+(φn+1, φn)+ f k

−(φn, φn−1)

where

f k+(φn+1, φn) = f k

+(φn)+

12f k+(φn)(φn+1

− φn) =1

2ε23(φn)2φn+1

− (φn)3,

f k−(φn, φn−1) =

32f k−(φn)−

12f k−(φn−1) = −

12ε2

3φn

− φn−1 . (4.29)


This is a second-order in time approximation and the numerical dissipation introduced is also of second-order,

NDphobic(φ(tn+1), φ(tn), φ(tn−1)) = O(k2),

but it is not clear how to control the sign of NDphobic(φn+1, φn, φn−1) (as in the OD2 case) to deduce the energy-stability.

Finally, since now the non-convex part is treated explicitly, following the same arguments presented in Theorem 4.11,no assumptions are needed to derive the existence and uniqueness of this scheme:

Theorem 4.13 (Unconditional Unique Solvability). Let us consider the schemes (2.5) (for AC) and (3.9) (for CH)with the potentialapproximation given in (4.29). Then, both schemes are uniquely solvable.

Remark 4.14. In this case, when a FE approximation is used, the only constraint on the choice of the discrete spaces Xh andWh is that these spaces contain the constant functions.

4.5. A second order two-step linear scheme (LM2)

In [24] a Lagrange multiplier is introduced to penalize the restriction of a unitary director vector in the Nematic LiquidCrystal framework and in [25] these ideas were extended to derive two unconditionally energy-stable linear schemes forthe Cahn–Hilliard model (one is a one-step and first order in time scheme while the other is a two-step and second orderscheme). Now, we present a new two-step linear second-order scheme to approximate both Allen–Cahn and Cahn–Hilliardmodels.

• Allen–Cahn (AC-LM2)Initialization: (φ0, r0) ∈ H1(Ω)× L2(Ω) approximations of (φ(0), r(0) = (|φ(0)|2 − 1)/ε2)Step n + 1 (n ≥ 1): Given (φn−1, φn, rn), find (φn+1, rn+1) ∈ H1(Ω)× L2(Ω) such that

1γ

φn+1

− φn

k, φ

+

∇φn+ 1

2 ,∇φ

+

rn+

12φ, φ

= 0,

ε2

2

rn+1− rn

k, r

−

φ φn+1− φn

k, r

= 0,

(4.30)

for each (φ, r) ∈ H1(Ω)×L2(Ω)where φn+ 12 := (φn+1

+φn)/2 and rn+12 := (rn+1

+ rn)/2. Finallyφ = (3φn−φn−1)/2,

using an explicit (BDF2) approximation.• Cahn–Hilliard (CH-LM2)

Initialization: (φ0, r0) ∈ H1(Ω)× L2(Ω) approximations of (φ(0), r(0) = (|φ(0)|2 − 1)/ε2)Step n + 1 (n ≥ 1): Given (φn−1, φn, rn), find (φn+1, wn+ 1

2 , rn+1) ∈ H1(Ω)× H1(Ω)× L2(Ω) such that

φn+1

− φn

k, w

+ γ

∇wn+ 1

2 ,∇w

= 0,∇φn+ 1

2 ,∇φ

+

rn+

12φ, φ

−

wn+ 1

2 , φ

= 0,

ε2

2

rn+1− rn

k, r

−

φ φn+1− φn

k, r

= 0,

(4.31)

for each (φ, w, r) ∈ H1(Ω)× H1(Ω)× L2(Ω).The unknowns wn+ 1

2 are a direct approximation at midpoint tn+12 := (tn + tn+1)/2, while φn+ 1

2 := (φn+1+ φn)/2 and

rn+12 := (rn+1

+ rn)/2. Finallyφ = (3φn− φn−1)/2.

Remark 4.15. If the choiceφ = φn+ 12 is considered, we obtain nonlinear second-order schemes that are equivalent to the

(MP), because from (4.30)2 and (4.31)3 one can arrive at the expression rn+1=

1ε2((φn+1)2 − 1).

Theorem 4.16. Schemes (4.30) and (4.31) are second-order in time and unconditionally energy-stable for the modified energy

E(φ, r) =

Ω

12|∇φ|

2+ε2

4|r|2

dx.

In fact, one has

δtE(φn+1, rn+1)+

1γ

∥δtφn+1

∥2L2 (for AC)

γ ∥∇wn+1/2∥2L2 (for CH)

= 0. (4.32)


Proof. Taking in (4.30), (φ, r) = (δtφn+1, rn+

12 ) for the AC case and taking in (4.31), (φ, w, r) = (δtφ

n+1, wn+ 12 , rn+

12 ) for

the CH case, energy inequality (4.32) holds.

Remark 4.17. Notice thatE(φ, r) = E(φ)+ Epert(φ, r)where

Epert(φ, r) =ε2

4

Ω

|r|2dx −

Ω

F(φ)dx.

Theorem 4.18. Schemes (4.30) and (4.31) are uniquely solvable.

Proof (Part 1: Allen–Cahn).We can rewrite problem (4.30) as:Given (φn−1, φn, rn), find (φn+1, rn+1) ∈ H1(Ω)× L2(Ω) such that

a((φn+1, rn+1), (φ, r)) = l(φ, r) ∀ (φ, r) ∈ H1(Ω)× L2(Ω)

with

a((φ, r), (φ, r)) :=1kγ(φ, φ)+

12(∇φ,∇φ)+

12(rφ, φ)+

ε2

4(r, r)−

12(φφ, r)

and

l(φ, r) :=1kγ(φn, φ)−

12(∇φn,∇φ)−

12(rnφ, φ)+

ε2

4(rn, r)−

12(φφn, r).

Since a(·, ·) is a coercive and continuous bilinear form inH1(Ω)× L2(Ω), the existence and uniqueness of a solution followsfrom the Lax–Milgram Theorem.(Part 2: Cahn–Hilliard). Using the change of variable

ψn+1= φn+1

− φn, zn+12 = wn+ 1

2 −

Ω

wn+ 12 = wn+ 1

2 −

Ω

rn+12φ and sn+

12 =

rn+1+ rn

2:

we can rewrite problem (4.31) as:Given (φn−1, φn, rn) ∈ H1(Ω)×H1(Ω)× L2(Ω), find (ψn+1, zn+

12 , sn+

12 ) ∈ H1(Ω)×H1(Ω)× L2(Ω) such that, for any

(ψ, z, s) ∈ H1(Ω)× H1(Ω)× L2(Ω),

aψn+1, zn+

12 , sn+

12

, (ψ, z, s)

= l(ψ, z, s)

with

a((ψ, z, s), (ψ, z, s)) = (ψ, z)+ k γ (∇z,∇ z)+12(∇ψ,∇ψ)

+ (sφ, ψ)− (z, ψ)−

Ω

sφ(1, ψ)+ ε2(s, s)− (φψ, s)

and

l(ψ, z, s) = −(∇φn,∇ψ)+ ε2(rn, s).

SinceΩψn+1

= 0 =Ωzn+

12 and

aψn+1, zn+

12 , sn+

12

,ψn+1, zn+

12 , sn+

12

= k γ

∇zn+12

2

L2(Ω)+

12∥∇ψn+1

∥2L2(Ω) + ε2

sn+ 12

2

L2(Ω),

the existence and uniqueness follows from the Lax–Milgram Theorem.

Remark 4.19. Theorems 4.16 and 4.18 can be easily extendedwhen a space discretization (like FE) is considered. In fact, we

have only to impose that the constant functions belong to the discrete spaces for ψn+1h and z

n+ 12

h .

5. Approximation of the nonlinear schemes by Newton’s method. Solvability and convergence

We are going to consider Newton’s method to approximate the solution φn+1 of the nonlinear schemes MP-BDF2 andMP. Iterative Newton’s method associated to a system G(φ) = 0 can be formulated as follows: given φ l, to find φ l+1 solving

G′(φ l)(φ l+1− φ l) = −G(φ l)


and we iterate until a convergence criterium is satisfied. The convergence of this iterative process reads φ l→ φn+1 as

l ↑ +∞.From this point, we are going to consider a FE space discretization based on a variational formulation in Xh ⊂ H1(Ω)

(where h denotes the mesh size) due to the fact that to prove the convergence of the algorithms we will need to use theinverse inequality ∥φh∥H1(Ω) ≤

Ch ∥φh∥L2(Ω) ∀φh ∈ Xh. The following lemmawill be also necessary to obtain the convergence

results.

Lemma 5.1. Let us consider a sequence of errors elφl≥0 with elφ = φn+1− φ l, such that

∥el+1φ ∥

2H1 ≤ C

∥elφ∥

2H1

2∀ l ≥ 0 and ∥e0φ∥

2H1 is small enough.

Then φ l+1 converges to φn+1 as l → ∞ in the H1(Ω)-norm in a quadratic way.

5.1. MP-BDF2

5.1.1. Allen–CahnIn order to approximate the solution φn+1 of the nonlinear scheme (2.5), (4.13) (previously discretized in space using the

FE space Xh ⊂ H1(Ω)), we consider Newton’s algorithm:Given φ l

∈ Xh (assuming φ0= φn at the first iteration step), to find φ l+1

∈ Xh such that ∀ φ ∈ Xh:1k(φ l+1, φ)+

γ

2(∇φ l+1,∇φ)+ γ

∂ f k

+

∂φn+1(φ l, φn) · φ l+1, φ

=

1k(φn, φ)

−γ

2(∇φn,∇φ)− γ (f k

−(φn, φn−1), φ)+ γ

∂ f k

+

∂φn+1(φ l, φn) · φ l, φ

− γ

f k+(φ l, φn), φ

(5.33)

until ∥φ l+1− φ l

∥H1 ≤ tol (tol > 0 being a tolerance parameter), where

∂ f k+

∂φn+1(φ l, φn) =

14ε2

3(φ l)2 + 2φ lφn

+ (φn)2

=1

4ε2

2(φ l)2 + |φ l

+ φn|2

≥ 0

and ∂ f k

+

∂φn+1(φ l, φn) · φ l, φ

−

f k+(φ l, φn), φ

=

14ε2

2(φ l)3 + (φ l)2φn

− (φn)3, φ.

Theorem 5.2 (Existence of Newton’s Method). There exists a unique solution of (5.33).

Proof. It suffices to follow the same arguments of Theorem 4.4 with the functional

JNewton(φ l+1) =γ

4∥∇φ l+1

∥2L2 +

Ω

12k

+γ

2∂ f k

+

∂φn+1(φ l, φn)

|φ l+1

|2

+γ

2

Ω

∇φn∇φ l+1

+

Ω

−

1kφn

+ γ f k−(φn, φn−1)− γ

∂ f k+

∂φn+1(φ l, φn) · φ l

+ γ f k+(φ l, φn)

φ l+1.

In the following, we present a convergence result.

Theorem 5.3 (Convergence of Newton’s Method). Under the constraints

kε4

≤ C and lim(k,h)→0

kh2

= 0,

the sequence φ ll≥0 of solutions of the iterative algorithm (5.33) converges in the H1(Ω)-norm to the solution φn+1 of

scheme (2.5), (4.13) in a quadratic way.

Proof. We consider J(φ) as in (4.16)

J(φ) =12k

Ω

|φ|2+γ

4

Ω

|∇φ|2+ γ

Ω

F+(φ, φn)+

γ

2

Ω

∇φn∇φ +

Ω

−

1kφn

+ γ f k−(φn, φn−1)

φ

and we can rewrite the nonlinear scheme (2.5), (4.13) as: Find φn+1∈ Xh such that

⟨J ′(φn+1), φ⟩ = 0 ∀ φ ∈ Xh.


In this case, Newton’s method is written as: Find φ l+1∈ Xh such that

⟨J ′′(φ l)(φ l+1− φ l), φ⟩ = −⟨J ′(φ l), φ⟩ ∀ φ ∈ Xh.

Since φn+1 is a root of J ′,

0 = ⟨J ′(φn+1), φ⟩ = ⟨J ′(φ l), φ⟩ + ⟨J ′′(φ l)elφ, φ⟩ +12⟨J ′′′(φn+ξ )(elφ, e

lφ), φ⟩ ∀ φ,

where elφ = φn+1− φ l

∈ Xh and φn+ξ∈ Xh is between φ l and φn+1, i.e., φn+ξ

= φn+1+ ξ(φ l

− φn+1)with ξ ∈ (0, 1). Then,manipulating both previous equalities,

⟨J ′′(φ l)(el+1φ ), φ⟩ = −

12⟨J ′′′(φn+ξ )(elφ, e

lφ), φ⟩. (5.34)

In this case, this equality is rewritten as

1k(el+1φ , φ)+

γ

2(∇el+1

φ ,∇φ)+ γ

∂ f k

+

∂φ l(φ l, φn)el+1

φ , φ

= −

γ

2

∂2f k

+

∂2φ l(φn+ξ , φn)|elφ |

2, φ

with

∂ f k+

∂φn+1(φ l, φn) =

14ε2

(3|φ l|2+ 2φ lφn

+ |φn|2),

∂2f k+

∂2φn+1(φn+ξ , φn) =

12ε2

(3φn+ξ+ φn).

Testing by φ = el+1φ /γ ,

1γ k

∥el+1φ ∥

2L2 +

12∥∇el+1

φ ∥2L2 +

Ω

∂ f k+

∂φn+1(φ l, φn)|el+1

φ |2

= −1

4ε2

Ω

(3φn+ξ+ φn)|elφ |

2el+1φ := I.

Using that ∂ f k+

∂φn+1 (φl, φn) ≥ 0 and bounding I by

I ≤1

4ε2∥3φn+ξ

+ φn∥L6∥e

lφ∥

2L6∥e

l+1φ ∥L2

≤1

2γ k∥el+1φ ∥

2L2 + C

kγε4

∥3φn+ξ+ φn

∥2H1

∥elφ∥

2H1

2,

we can arrive at

∥el+1φ ∥

2H1 ≤ C

kε4

∥3φn+ξ+ φn

∥2H1

∥elφ∥

2H1

2.

Imposing hypothesis k ≤ C ε4, one has

∥el+1φ ∥

2H1 ≤ C ∥3φn+ξ

+ φn∥2H1

∥elφ∥

2H1

2. (5.35)

In particular, since 3φn+ξ+ φn

= 3(φn+1+ ξelφ)+ φn

∥3φn+ξ+ φn

∥2H1 ≤ 2

∥3φn+1

+ φn∥2H1 + 9∥elφ∥

2H1

.

In order to use an induction strategy, we can assume the hypothesis

∥elφ∥2H1 ≤ δ0, (5.36)

where δ0 > 0 will be a small enough constant to determine below. Therefore, from (5.35) and (5.36) we obtain

∥el+1φ ∥

2H1 ≤ C(∥3φn+1

+ φn∥2H1 + 9δ0)δ20 .

Therefore, if ∥e0φ∥2H1 ≤ δ0 and choosing δ0 small enough such that

C(∥3φn+1+ φn

∥2H1 + 9δ0)δ0 ≤ 1, (5.37)

the inequality ∥el+1φ ∥

2H1 ≤ δ0 holds. Indeed, we obtain the following recurrence expression

∥el+1φ ∥

2H1 ≤ ∥elφ∥

2H1 ≤ · · · ≤ ∥e0φ∥

2H1 ≤ δ0


and from (5.35) we can derive

∥el+1φ ∥

2H1 ≤ C(δ0)

∥elφ∥

2H1

2(5.38)

where C(δ0) is bounded for small δ0. Finally, we need to find the following bound of the initial error:

∥e0φ∥2H1 = ∥φn+1

− φn∥2H1 ≤ δ0.

Indeed, by using the discrete energy law (4.14) multiplied by k and the inverse inequality,

∥e0φ∥2H1 = ∥φn+1

− φn∥2H1 ≤

Ch2

∥φn+1− φn

∥2L2 ≤ C

kh2

k∥δtφn+1∥2L2 ≤ C

γ kh2

E(φn). (5.39)

Therefore, taking δ0 = O( kh2), then constraint (5.37) holds if the following hypothesis is imposed

limkh2

= 0 as (k, h) → 0.

Hence, combining (5.38) and (5.39) we are under the hypothesis of Lemma 5.1, hence the quadratic convergence of φ l+1 toφn+1 in the H1(Ω)-norm can be assured.

Remark 5.4. Hypothesis (5.39) is becoming less restrictive in each time iteration due to the fact that the energyE(φn) isdecreasing in time.

5.1.2. Cahn–HilliardIn order to approximate the nonlinear scheme (4.21), (4.13), we consider Newton’s algorithm:Given (ψ l, z l) ∈ Xh × Wh (assuming (ψ0, z0) = (0, zn−

12 ) at the first step), to find (ψ l+1, z l+1) ∈ Xh × Wh solving

(ψ l+1, z)+ k γ∇z l+1,∇ z

= 0,

12(∇ψ l+1,∇ψ)+

∂ f k

+

∂φn+1(ψ l

+ φn, φn) · ψ l+1, ψ

−

Ω

∂ f k+

∂φn+1(ψ l

+ φn, φn) · ψ l+1, ψ

− (z l+1, ψ)

= −(∇φn,∇φ)− (f k−(φn, φn−1), ψ)+

Ω

f k−(φn, φn−1), ψ

+

∂ f k

+

∂φn+1(ψ l

+ φn, φn) · φ l, ψ

−

Ω

∂ f k+

∂φn+1(ψ l

+ φn, φn) · φ l, ψ

−

f k+(ψ l

+ φn, φn), ψ+

Ω

f k+(ψ l

+ φn, φn), ψ

,

(5.40)

for each (ψ, z) ∈ Xh × Wh until ∥ψ l+1− ψ l

∥H1 ≤ tol.

Theorem 5.5 (Existence of Newton’s Method). There exists a unique solution of the iterative algorithm (5.40).

Proof. Following the same arguments presented in Theorem 4.4.

Remark 5.6. The only constraint on the FE space Xh to assure the existence and uniqueness of a solution of system (5.40),is that the constant functions belong to Xh.

Theorem 5.7 (Convergence of Newton’s Method). Under hypotheses

k1/2

ε4< C and lim

(k,h)→0

kh2

= 0,

the sequence of solutions (ψ l, z l)l≥0 of the iterative algorithm (5.40) converges to the solution (ψn+1, zn+12 ) of scheme

(4.21), (4.13) in the H1(Ω)-norm.

Proof. We can define the problem (4.21), (4.13) in a vectorial way,

Gψn+1, zn+

12

=

G1

ψn+1, zn+

12

,G2

ψn+1, zn+

12

= (0, 0)

where Gi corresponds with Eqs. (4.21) i–(4.13) (i = 1, 2). Therefore, Newton’s method (5.40) reads

G′(ψ l, z l)(ψ l+1− ψ l, z l+1

− z l) = −G(ψ l, z l).


Equivalently to the expression (5.34) obtained in AC case, but now using a vectorial Taylor’s formula of G′(ψn+1, zn+12 )with

centre at (ψ l, z l) and the previous equality, we arrive at∂G1

∂ψ l(ψ l, z l)(el+1

ψ ), z+

∂G1

∂z l(ψ l, z l)(el+1

z ), z

= −12

(elψ , e

lz)

tG′′

1(ψn+ξ , zn+ξ )(elψ , e

lz), z

and

∂G2

∂ψ l(ψ l, z l)(el+1

ψ ), ψ

+

∂G2

∂z l(ψ l, z l)(el+1

z ), ψ

= −

12

(elψ , e

lz)

tG′′

2(ψn+ξ , zn+ξ )(elψ , e

lz), φ

where elψ = ψn+1

− ψ l, elz = zn+12 − z l, ψn+ξ

= ξψn+1+ (1 − ξ)ψ l, zn+ξ = ξzn+

12 + (1 − ξ)z l and G′

i and G′′

i denote theJacobian and the Hessian of Gi (i = 1, 2) respectively. In particular,

G′′

1(ψ, z) = (0, 0) and G′′

2(ψ, z) =

∂2f k

+

∂2φn+1(ψ + φn, φn), 0

,

hence, we arrive at

(el+1ψ , z)+ k γ

∇el+1

z ,∇ z

= 0,

12(∇en+1

ψ ,∇ψ)+

∂ f k

+

∂φn+1(ψ l

+ φn, φn) · el+1ψ , ψ

−

Ω

∂ f k+

∂φn+1(ψ l

+ φn, φn) · el+1ψ , ψ

− (el+1

z , ψ)

= −12

∂2f k

+

∂2φn+1(ψn+ξ , φn)|elφ |

2, ψ

+

12

Ω

∂2f k+

∂2φn+1(ψn+ξ , φn)|elφ |

2, ψ

.

(5.41)

Testing by (ψ, z) =

el+1ψ , el+1

z

in (5.41)

k γ ∥∇el+1z ∥

2L2 +

12∥∇el+1

ψ ∥2L2 +

Ω

∂ f k+

∂φn+1(ψ l

+ φn, φn)|el+1ψ |

2= −

14ε2

Ω

(3ψn+ξ+ 4φn)|elψ |

2el+1ψ := I, (5.42)

and testing in (5.41) by z = el+1ψ /k1/2 ∈ Xh (if Xh ⊂ Wh):

1k1/2

∥el+1ψ ∥

2L2 ≤

k γ2

∥∇el+1z ∥

2L2 +

γ

2∥∇el+1

ψ ∥2L2 . (5.43)

Using (5.43) we can bound I by

I ≤1

4ε2∥3ψn+ξ

+ 4φn∥L6∥e

lψ∥

2L6∥e

l+1ψ ∥L2

≤β

2k1/2∥el+1ψ ∥

2L2 + C

k1/2

β ε4∥3ψn+ξ

+ 4φn∥2H1

∥elψ∥

2H1

2

≤ βk γ4

∥∇el+1z ∥

2L2 + β

γ

4∥∇el+1

ψ ∥2L2 + Cβ

k1/2

ε4∥3ψn+ξ

+ 4φn∥2H1

∥elψ∥

2H1

2.

Therefore, taking into account that ∂ f k+

∂φn+1 (ψl+ φn, φn) ≥ 0 and considering β small enough, from (5.42) we obtain

k γ ∥∇el+1z ∥

2L2 + ∥∇el+1

ψ ∥2L2 ≤

C(γ ) k1/2

ε4∥3ψn+ξ

+ 4φn∥2H1

∥elψ∥

2H1

2

where C(γ ) > 0 is a constant independent of k, ε and h. Using thatΩel+1ψ =

Ω(ψn+1

−ψ l+1) = 0 (taking z = 1 in (5.41))and a Poincaré inequality with ψ l+1

−1

|Ω|

Ωψn+1 we can bound the complete H1-norm of el+1

ψ by:

∥el+1ψ ∥

2H1 ≤ C

k1/2

ε4∥3ψn+ξ

+ 4φn∥2H1

∥elψ∥

2H1

2.

Assuming hypothesis k1/2 ≤ C ε4, one has

∥el+1ψ ∥

2H1 ≤ C∥3ψn+ξ

+ 4φn∥2H1

∥elψ∥

2H1

2. (5.44)

Using the same arguments of Theorem 5.3 (using (5.44) instead of (5.35)), we derive

∥el+1ψ ∥

2H1 ≤ C(δ0)

∥elψ∥

2H1

2.


Finally, we need to find the bound of the initial error

∥e0ψ∥2H1 = ∥ψn+1

− ψ l=0∥H1 = ∥φn+1

− φn∥2H1 ≤ δ0.

Testing by w = δtφn+1

∈ Xh in (3.9) and using an inverse inequality,

∥δtφn+1

∥2L2 ≤ γ

∇wn+ 12

L2

∥∇δtφn+1

∥L2 ≤Ch

∇wn+ 12

L2

∥δtφn+1

∥L2 ,

hence

∥δtφn+1

∥2L2 ≤

Ch2

∇wn+ 12

2

L2.

Then by using the discrete energy law (4.14),

h2

k2∥φn+1

− φn∥2L2 ≤ C

∇wn+ 12

2

L2≤ CE(φn).

Using again the inverse inequality from the H1-norm to L2-norm, we obtain the bound

∥e0ψ∥2H1 = ∥φn+1

− φn∥2H1 ≤

Ch2

∥φn+1− φn

∥2L2 ≤ C

k2

h4E(φn). (5.45)

Take δ0 = O( k2

h4) then δ0 is small enough from the constraint

limkh2

= 0 as (k, h) → 0.

Hence, combining (5.44) and (5.45) we are under the hypothesis of Lemma 5.1 and the quadratic convergence of ψ l+1 toψn+1 in the H1(Ω)-norm can be assured.

The convergence of z l+1 to zn+12 can be derived using similar arguments. First of all, taking z = el+1

z in (5.41)1

k γ ∥∇el+1z ∥

2L2 ≤ ∥el+1

ψ ∥L2∥el+1z ∥L2 . (5.46)

SinceΩel+1z = 0, one has the Poincaré inequality ∥el+1

z ∥L2 ≤ P ∥∇el+1z ∥L2 . Then, from (5.46) we can deduce

k γ ∥∇el+1z ∥

2L2 ≤ ∥el+1

ψ ∥L2P ∥∇el+1z ∥L2

and we arrive at

k γ ∥∇el+1z ∥

2L2 ≤ P ∥∇el+1

z ∥L2 .

Finally, using again the Poincaré inequality, we derive ∥el+1w ∥H1 → 0.

5.2. MP

5.2.1. Allen–CahnIn order to approximate the MP nonlinear scheme (2.5), (4.24), we consider the Newton’s algorithm: Given φ l

∈ Xh(assuming φ0

= φn at the first iteration step), to find φ l+1∈ Xh such that ∀ φ ∈ Xh:

1k(φ l+1, φ)+

γ

2(∇φ l+1,∇φ)+ γ

∂ f k(φ l, φn)

∂φ lφ l+1, φ

=

1k(φn, φ)

−γ

2(∇φn,∇φ)+ γ

∂ f k(φ l, φn)

∂φ lφ l, φ

− γ

f k(φ l, φn), φ

(5.47)

until ∥φ l+1− φ l

∥H1 ≤ tol, where

∂ f k(φ l, φn)

∂φ l=

14ε2

3(φ l)2 + 2φ lφn

+ (φn)2 − 2

=∂ f k

+(φ l, φn)

∂φ l−

12ε2

and ∂ f k(φ l, φn)

∂φ lφ l, φ

−

f k(φ l, φn), φ

=

14ε2

2(φ l)3 + (φ l)2φn

− (φn)3 + 2φn, φ.


Theorem 5.8 (Existence of Solution of Newton’s Method). Under the constraint k < 2ε2, there exists a unique solution of theiterative algorithm (5.47).

Proof. Following the same arguments of Theorem 4.4 and assuming the constraint k < 2ε2 to assure the convexity of thefunctional

JNewton(φ l+1) =γ

4∥∇φ l+1

∥2L2 +

Ω

12k

+γ

2∂ f k

+(φ l, φn)

∂φ l−

γ

4ε2

|φ l+1

|2

+γ

2

Ω

∇φn∇φ l+1

+

Ω

−

1kφn

+ γ f k(φn, φn−1)− γ∂ f k(φ l, φn)

∂φ lφ l

φ l+1.

Theorem 5.9 (Convergence of Newton’s Method). Under the constraints

kε4

≤ C and lim(k,h)→0

kh2

= 0,

the sequence of solutions φ ll≥0 of the iterative algorithm (5.47) converges to the solution φn+1 of scheme (2.5), (4.24) in the

H1(Ω)-norm in a quadratic way.

Proof. The proof follows from the same arguments of Theorem 5.3. In particular, the constraint k < 2ε2/γ derived inTheorem 5.8 must be imposed to be exactly under the same hypothesis considered in Theorem 5.3.

5.2.2. Cahn–HilliardIn order to approximate the nonlinear scheme (4.21), (4.24), we consider Newton’s algorithm:Given (ψ l, z l) ∈ Xh × Wh (assuming (ψ0, z0) = (0, zn−

12 ) at the first step), to find (ψ l+1, z l+1) ∈ Xh × Wh solving

(ψ l+1, z)+ k∇z l+1,∇ z

= 0,

12(∇ψ l+1,∇ψ)+

∂ f k

∂φn+1(ψ l

+ φn, φn)ψ l+1, ψ

−

Ω

∂ f k

∂φn+1(ψ l

+ φn, φn)ψ l+1, ψ

− (z l+1, ψ)

= −(∇φn,∇ψ)+

∂ f k

∂φn+1(ψ l

+ φn, φn)ψ l, ψ

−

Ω

∂ f k

∂φn+1(ψ l

+ φn, φn)ψ l, ψ

−

f k(ψ l

+ φn, φn), ψ+

Ω

f k(ψ l+ φn, φn), ψ

,

(5.48)

for each (ψ, z) ∈ Xh × Wh, until ∥ψ l+1− ψ l

∥H1 ≤ tol.

Theorem 5.10 (Existence of Solution of Newton’s Method). There exists a unique solution of the iterative algorithm (5.48) underthe constraint k < 4ε4

γ.

Proof. In this case, (5.48) is a linear square system and the existence of the solution follows from the uniqueness. Using thesame arguments of Theorem 5.5, the uniqueness can be derived imposing the constraint k < 4ε4/γ .

Theorem 5.11 (Convergence of Newton’s Method). The sequence of solutions (ψ l, z l)l≥0 of the iterative algorithm (5.48) con-verges to the solution (ψn+1, zn+

12 ) of scheme (4.21), (4.24) in the H1(Ω)-norm in a quadratic way under the hypothesis

k1/2

ε4< C and lim

(k,h)→0

kh2

= 0.

Proof. The proof follows as in Theorem 5.7. In particular, the constraint k < 4ε4/γ derived in Theorem 5.10 must beimposed to be exactly under the same hypothesis of Theorem 5.7.

6. Time-step adaptivity

In many phase field models, adaptive time-stepping is of primordial importance due to the different time scales thatarise in the dynamic of the models. In the following, we propose an adaptive time-stepping method where the time step isselected by using a criterion related to a ‘‘residual’’ of the discrete energy law. The key point is to assure that the numericaldissipation introduced in the discrete energy law is always small enough to maintain the accuracy of the scheme. This ideaallows us to reduce the computational time of our simulations by factors of hundreds while ensuring that sufficient timeaccuracy is achieved.


Table 1Parameters.

ε γ dt0 resmax resmin θ tol

10−2 10−4 10−5 10 1 1.1 10−3

The definition of the residual REn+1 is based on evaluating the continuous energy laws (2.4) and (3.8) at t = tn+12 , doing

a second order approximation of this equalities by using φ(tn+1) and φ(tn), and then, we define the residual of the energyreplacing φ(tn+i) by φn+i (for i = 0, 1). Therefore, we compute the residual as (see [25] for the CH case):

REn+1:=

E(φn+1)− E(φn)

dtn+

1γ

Ω

φn+1

− φn

dtn

2

dx (for AC)

γ

Ω

|∇wn+1/2|2dx (for CH).

(6.49)

Although it is possible to consider other choices of the residual of the energy, wemake this selection because a second orderapproximation of the continuous energy laws (2.4) and (3.8) is the optimal approximation considering only φ(tn+1) andφ(tn).

Then, the time-adapting algorithm in the (n + 1) time-step, with n ≥ 1, reads:Given φn, φn−1, dtn−1, dtn and a parameter θ > 1 (θ = 1.1 in our numerical simulations):

1. Compute φn+1 and obtain REn+1.2. If |REn+1

| > resmax, take dtn = dtn/θ and go to 1.3. If |REn+1

| < resmin, take dtn+1= θdtn.

4. Take tn+1= tn + dtn and go to next time step.

We have to take into account that in all the two-step schemes previously introduced, we need to change the way ofdefiningφ because it was designed as a linear second-order approximation of φ with a constant time-step.

In the case of non-constant time-step, let dtn−1 and dtn with tn = tn−1+ dtn−1 and tn+1

= tn + dtn, and we consider thefollowing expressions:

φ(tn−1) = φtn+ 1

2

−

dtn−1

+dtn

2

φt

tn+ 1

2

+ O((dtn−1)2 + (dtn)2),

φ(tn) = φtn+ 1

2

−

dtn

2φt

tn+ 1

2

+ O((dtn)2).

Then, 2dtn−1

dtn+ 1

φ(tn)− φ(tn−1) =

2dtn−1

dtn

φ

tn+ 1

2

+ O((dtn−1)2 + (dtn)2).

Therefore, we can define

φ =

dtn

2dtn−1+ 1

φn

−dtn

2dtn−1φn−1. (6.50)

Remark 6.1. The choice of resmax and resmin for each scheme is not a trivial task, because its values depend on the physicaland discrete parameters. In fact, if resmin is large, the accuracy of the solutions could be deteriorated andwrong equilibriumsolutions could be achieved. By the contrary, if resmax is small, the time step will keep decreasing and no improvement inthe computational cost is obtained. Consequently, a trial and error choice of resmax and resmin must be done, dependingon the results obtained in each numerical experiment.

7. Numerical comparison of the adaptive time-stepping criterion

In this sectionwe compare the behaviour of the numerical schemes presented through the paperwhen the adaptive time-stepping algorithm is considered. For this purpose we have carried out several simulations considering a FE approximationin space using FreeFem++ software [36].

We show the results related to the Cahn–Hilliard model in the domainΩ = [0, 1]2 with a 90 × 90 triangular mesh andall the unknowns approximated by P1 FE.

We take random initial data φ0 with values between −0.01 and 0.01 (Fig. 1) to simulate a spinodal decomposition andwe consider for all the simulations the fixed parameters (see Table 1):


Fig. 1. Random initial data.

Fig. 2. Dynamic of the schemes.

Here tol denotes a tolerance parameter for Newton’s algorithms approximating the nonlinear schemes MP-BDF2 andMP. In particular, we consider the criterium ∥∇(φ l+1

− φ l)∥L2 ≤ tol (sinceΩ(φ l+1

− φ l)dx = 0, then ∥∇(φ l+1− φ l)∥L2 is

a norm equivalent to ∥φ l+1− φ l

∥H1 ).When Newton’s method does not converge in 10 iterations, we decrease the time step.Moreover, the stop criterium for all the schemes, i.e., the criterium to consider that the schemes have reached a steady

equilibrium solution is set to ∥∇wn+ 12 ∥L2 < 10−2 because it is well known that ∥∇wn+ 1

2 ∥L2 tends to zero in the equilibriumstates.

Fig. 2 shows the dynamic of the system with the initial data considered in Fig. 1, which is exactly with the same initialcondition considered in the simulations presented in [25] where a fixed time step is used.

As we expected, the time-adapt algorithm does not work with schemesMP and LM2 due to the fact that they are derivedas schemes where there is no phobic numerical dissipation in the discrete energy laws (4.25) and (4.32), respectively. Inthe MP case, it is possible to use the algorithm related with the convergence of Newton’s method to adapt the time step,arriving almost at the correct equilibrium solution, being the same solution obtained with the other schemes but with someperturbations (see Fig. 3). In the LM2 case the time-adapt algorithm leads to very big time-steps obtaining a meaninglessbehaviour of the scheme.

The evolution in time of themixing energy for each scheme is shown in Figs. 4–7where the schemesMP-BDF2+Newton,OD2 and OD2–BDF2 arrives to the same equilibrium solutions andMP+Newton to a slightly different one.


Fig. 3. Equilibrium solution of MP+Newton.

Fig. 4. Free energy in [0, 0.5].

Fig. 5. Free energy in [0.5, 1].

In Figs. 8–11 the evolution of the time steps considered for each scheme is shown. It is clearly observed thatMP+Newtontends to take very big time steps. This fact is due to theMP scheme satisfies exactly the residual REn+1 (without consideringrun-off errors), therefore this residual criterium is not good for the MP scheme. On the other hand, it can also be observedhow in the rest of schemes the time-step is reduced while coarsening phenomena are occurring in the system.


Fig. 6. Free energy in [1, 5].


Fig. 8. Time steps in [0, 0.5].

In Fig. 12 we show an x–y–φ plot of each one of the schemes to show how the smoothness of φ is not affected by thechoice of the numerical schemes. In the case of the MP we can observe how there are some numerical instabilities in thepure phases (φ = ±1) due to the fact that the time steps considered are not small enough.

Finally, the computational cost of each scheme is presented in Table 2. If we compare these results with fixed timestep simulations, the improvement is spectacular. In [25], a numerical study of the behaviour of linear schemes for theCahn–Hilliard equation considering a fixed time step is presented, arriving at the conclusion that the optimal choice is toconsider scheme OD2 with k = 10−4, because it is the largest time step that can be considered without losing accuracy ofthe solution. In other words, it is shown that if a larger time step is considered, the system will arrive at wrong equilibriumstates. Using this optimal case (scheme OD2 with k = 10−4) the equilibrium solution is achieved when t = 8, i.e. in thiscase it was needed to solve 80000 time steps (i.e. 80 000 (φ,w) coupled linear systemswere solved), while only 3533 linearsystems have to be solved to arrive at the correct equilibrium state when the adaptive-time algorithm is imposed.


Fig. 9. Time steps in [0.5, 1].

Fig. 10. Time steps in [1, 5].


Table 2Computational cost.

MP+Newton OD2 OD2–BDF2 MP-BDF2+Newton

# Time steps 339 2642 4340 3691# Linear systems solved 3896 3533 5687 12812

The better behaviour from the efficiency point of view of OD2 with respect to OD2–BDF2 is related to the fact thatOD2–BDF2 introduces more numerical dissipation, depending on the approximation of the non-convex potential.

One can think that the results using MP+Newton scheme could be improved taking a lower tolerance parametertol in the stop criterium for Newton’s algorithm. In Figs. 14–15 it can be observed how the time steps obtained are


Fig. 12. x–y–φ for each scheme.


still higher than the time steps in the other schemes when the tolerance parameter is decreased (concretely for tol =

10−3, 10−8, 10−12 and 10−13) and in Fig. 13 it is shown how theMP+Newton scheme never arrives at the same equilibriumsolution obtained with the other schemes, even considering a very low value of tol.

The results of the numerical experiments show that the most efficient scheme is OD2. In fact, this scheme requires theminimum number of iterations (linear systems to be solved) to achieve the correct equilibrium solution (Table 2) and itscharacteristic time of arriving at the equilibrium energy correspond with the correct one (around t = 6.4) presented in [25](using a small enough and constant time step) while the other schemes arrive at the equilibrium solution later (see Figs. 7and 13), around t = 6.8 forMP-BDF2, between t = 7 and t = 7.5 forMP and around t = 6.9 for OD2–BDF2.


Fig. 14. Time steps in [0, 6].


8. Conclusions

In this paper we have derived and compared second-order in time numerical schemes to approximate Allen–Cahn andCahn–Hilliard problems, where three of them are linear (OD2, OD2–BDF2 and LM2) and two of them nonlinear (MP andMP-BDF2).

Wehave obtained the unconditional unique solvability ofOD2–BDF2,MP-BDF2 and LM2. To assure the unique solvabilityof MP we have to impose the constraints k < 2ε2/γ (for AC) and k < 4ε4/γ (for CH) meanwhile for OD2 is imposed theconstraints k < 2ε2/γ (for AC) and k < 2ε4/γ (for CH).

The unconditional energy-stability ofMP,MP-BDF2 and LM2 is derived through the paper. AlthoughOD2 andOD2–BDF2seem to be conditionally energy-stable we have not been able to obtain some constraints to assure their stability. Our guessis that these constraints should not be very restrictive because in the numerical simulations we have not observed anyviolation of the correspondent discrete energy law.

Therefore, at this moment the unique linear unconditionally energy-stable scheme (with respect to a modified energy)and uniquely solvable is LM2 (although using this scheme the adapt-time technique designed in this paper does not work).

In order to compare all the constraints related to each scheme, we have also focused on the conditions that arise tryingto assure the convergence of Newton’s algorithms to the nonlinear schemesMP andMP-BDF2. In both cases, we obtain thatthe algorithms are convergent under the constraints

kε4

≤ C (for AC),k1/2

ε4< C (for CH), and lim

(k,h)→0

kh2

= 0.

Wehave also presented a new time-stepping algorithm based on controlling the residual term of the discrete energy law.In the numerical simulations this time-stepping algorithm does not work with the schemes that do not introduce numericaldissipation asMP and LM2.

The main features of each scheme in this paper can be summarized in Table 3.


Table 3Features of schemes.

MP OD2 OD2–BDF2 MP-BDF2 LM2

Linear × ×

Unconditionally unique solvable × ×

Conditionally unique solvable

Unconditionally energy-stable E(φ) × × × ×

Uncond. (modified-energy)-stableE(φ) × ×

One-step algorithm × × ×

Time-step adaptivity × ×

Acknowledgements

The work of the first author has been partially supported by MTM2009-12927 and MTM2012-32325 (Ministerio deEconomía y Competitividad, Spain). The work of the second author has been partially supported by MTM2009-12927 andMTM2012-32325 (Ministerio de Economía y Competitividad, Spain), NIH 1 R01GM095959-01A1 (United States) and ERC-CZproject LL1202 (Ministry of Education, Youth and Sports of the Czech Republic).

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second order schemes and time-step adaptivity for allen–cahn and cahn–hilliard models

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