AsecondorderoperatorsplittingmethodforAllen–Cahntypeequationswithnonlinearsourceterms PhysicaA

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Physica A

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

A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms

Hyun Geun Lee

^a

, June-Yub Lee

^b,∗

aInstitute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, Republic of Korea

bDepartment of Mathematics, Ewha Womans University, Seoul 120-750, Republic of Korea

h i g h l i g h t s

• The proposed operator splitting method is second order accurate and stable.

• The method can be easily applicable to wide class of Allen–Cahn type equations with non-linear source terms.

• Simulations for multi-component phase separation and dendrite growth show the feasibility of the method.

a r t i c l e i n f o

Article history:

Received 17 April 2014

Received in revised form 11 February 2015 Available online 17 March 2015

Keywords:

Vector-valued Allen–Cahn equation Phase-field equation for dendritic crystal

growth

Operator splitting method Second order convergence Fourier spectral method

a b s t r a c t

Allen–Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decom- pose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to com- bine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half- time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

The Allen–Cahn (AC) equation was originally introduced as a phenomenological model for antiphase domain coarsening in a binary alloy [1]. The AC equation arises from minimization of the Ginzburg–Landau free energy

E

(φ) := 

Ω



F

(φ)

ϵ

²

+ |∇ φ|

² 2



dx

,

∗Corresponding author.

E-mail address:jyllee@ewha.ac.kr(J.-Y. Lee).

http://dx.doi.org/10.1016/j.physa.2015.03.012 0378-4371/©2015 Elsevier B.V. All rights reserved.

whereΩis a domain in R^d

(

^d

=

1

,

²

,

³

)

. The quantity

φ(

^x

,

^t

)

is defined as the difference between the concentrations of the two components in a mixture (e.g.,

φ(

^x

,

^t

) = (

^mα

−

m_β

)/(

^mα

+

m_β

)

^{, where m}αand m_βare the masses of phases

α

^and

β

^).

The function F

(φ) =

⁰

.

²⁵

(φ

²

−

1

)

²is the Helmholtz free-energy density for

φ

, which has a double-well form, and

ϵ >

^{0 is} the gradient energy coefficient. The AC equation is the L²-gradient flow of the free energyE

(φ)

^:

∂φ

∂

^t

= −

gradE

(φ),

⁽¹⁾

where the symbol ‘‘grad’’ denotes the gradient in the sense of the Gâteaux derivative. Let the domain of the functionalEbe D

= { φ ∈

^H2

(

Ω

) |

^∂φ_∂n

=

0 on

∂

Ω

}

. For any

φ, ψ ∈

^{D, we have}

(

^gradE

(φ), ψ)

L²

=

^d

d

θ

^E

(φ + θψ)









_θ=

0

=

lim θ→0

1

θ



E

(φ + θψ) −

^E

(φ)

=



Ω



F^′

(φ) ϵ

²

−

₁

φ

 ψ

^dx

=



F^′

(φ)

ϵ

²

−

₁

φ, ψ



L²

.

See Ref. [2] for details of the notation and the derivation of the formula. We identify gradE

(φ) ≡

^F′

(φ)/ϵ

²

−

₁

φ

^{, then} Eq.(1)becomes the AC equation

∂φ(

^x

,

^t

)

∂

^t

= −

¹

ϵ

^2F

′

(φ(

^x

,

^t

))

  

Free-energy evolution

+

₁

φ(

^x

,

^t

)

  

Heat evolution

,

^x

∈

_Ω

,

⁰

<

^t

≤

T

.

⁽²⁾

The main difficulty when developing a numerical method for solving Eq.(2)is that the free-energy evolution term F^′

(φ)

yields a severe stability restriction on the time step. In order to deal with this restriction, Eyre [3,4] proposed a semi-implicit method, which is first order accurate in time and unconditionally gradient stable, and Eyre’s method was used to solve Eq. (2) in Ref. [5]. Many researchers employed first and second order stabilized semi-implicit methods for solving Eq. (2) [6–9], in which the heat evolution term1

φ

is treated implicitly and the free-energy evolution term F^′

(φ)

is treated explicitly to avoid the expensive process of solving nonlinear equations at each time step with an extra stabilizing term added to alleviate the stability constraint while maintaining accuracy and simplicity.

Alternative methods implemented for the AC equation are first and second order operator splitting methods [8,10,11], which become the base of our proposed method for more general form of AC type equations. Operator splitting schemes have been and continue to be used for many types of evolution equations [12–14]. It is easy to construct a first order solution A

(

^tn+1

)

of time evolution equation

∂

^A

∂

^t

=

f₁

(

^A

) +

^f2

(

^A

)

by computing A

(

^tn

+

₁t

) ≈ 

^S11^tS₂^1t



A

(

^tn

)

^whereS1^1tandS₂^1tare the solution operators for^∂_∂^A_t

=

f₁

(

^A

)

^{and ∂}_∂^A_t

=

f₂

(

^A

)

^, respectively. Then a second order scheme can be derived simply by symmetrizing the first order scheme:

A

(

^tn

+

_1t

) ≈ 

S₁^1t/2S₂^1tS₁^1t/2



A

(

^tn

).

The basic idea of the operator splitting methods for AC equation is to decompose the original equation into heat and free-energy evolution subequations at each time step, in which the free-energy evolution subequation has a closed-form solution. The first and the second order operator splitting methods in Refs. [8,10,11] are unconditionally stable thanks to the unconditional stability of each substep. The derivation and numerical properties of the higher order operator splitting method for AC equation will appear in Ref. [15].

Our main concern in this paper is the AC type equations with a nonlinear source term in the form of

∂φ(

^x

,

^t

)

∂

^t

= −

¹

ϵ

^2F

′

(φ(

^x

,

^t

))

  

Free-energy evolution

+

₁

φ(

^x

,

^t

)

  

Heat evolution

+

S

(φ(

^x

,

^t

))

  

Nonlinear source

.

⁽³⁾

This type of equations have been applied to a wide range of problems such as phase transitions [1], crystal growth [16–19], grain growth [20–24], image analysis [25–28], motion by mean curvature [29–33], two-phase fluid flows [34], and vesicle membranes [35,36]. Since available analytical solutions for these equations are limited, numerical methods are important tools for understanding dynamics of the equations. In contrast to the well developed first and second order methods for the AC equation, only first order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically [17,19,21,23,25–28,35,37,38].

In this paper, we propose a second order operator splitting method for solving the AC type equations with nonlinear source terms. Our method is a generalization of the second order operator schemes for three subequations with the free- energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these

three subequations in proper order to achieve the second order accuracy and stability. It is also worth noting that the free- energy and heat evolution subequations can be solved in semi-analytic form as in the method for the AC equation [11].

Our proposed scheme combines with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver.

This paper is organized as follows. In Section2, we review the second order operator splitting methods for the AC equa- tion. In Section3, we briefly review a derivation of the vector-valued AC equation and propose a second order operator splitting method with the numerical experiment for phase separation. In Section4, we also propose a second order operator splitting method for the phase-field equation for dendritic crystal growth and present the numerical experiment. Finally, conclusions are drawn in Section5.

2. Review of the second order operator splitting methods for the Allen–Cahn equation

We consider the AC equation(2)in two-dimensional spaceΩ

= (

⁰

,

^L1

) × (

⁰

,

^L2

)

. For simplicity of notation, we define the ‘‘free-energy evolution operator’’F^1tas follows:

F^1t

(φ(

^tn

)) := φ(

^tn

+

₁t

),

where

φ(

^tn

+

₁t

)

is a solution of the first order differential equation

∂φ

∂

^t

= −

^F

′

(φ) ϵ

²

with an initial condition

φ(

^tn

)

. For given F^′

(φ) = φ

³

− φ

, we have an analytical formula (see Refs. [8,10,11]) for the evolution operatorF^1tin the physical space

F^1t

(φ) = φ



φ

²

+ (

¹

− φ

²

)

^e−2ϵ12t

for

ϵ =

^constant

.

⁽⁴⁾

Here it is worth to note that

|

_F^1t

(φ)|

is bounded by max

(

¹

, |φ|)

^since

|

_F^1t

(φ)| ≤ | φ|

 φ

²

=

1

,

^for

| φ| ≤

¹

|

_F^1t

(φ)| ≤ | φ|

 φ

²

+ (

¹

− φ

²

) = | φ|,

^for

| φ| ≥

¹

.

(5)

On the other hand, for any fixed1^t,

|

_F^1t

(φ)|

is also bounded by some constant B

(

1^t

)

regardless of size of

| φ|

^,

|

_F^1t

(φ)| ≤ | φ|

 φ

²



1

−

e⁻

21^t ϵ²



=

¹



1

−

e⁻

21^t ϵ²

=

_B

(

1^t

).

⁽⁶⁾

We also define the ‘‘heat evolution operator’’H^1tas follows:

H^1t

(φ(

^tn

)) := φ(

^tn

+

₁t

),

where

φ(

^tn

+

₁t

)

is a solution of the first order differential equation

∂φ

∂

^t

=

₁

φ

with an initial condition

φ(

^tn

)

. In this paper, we employ the discrete cosine transform [39] to solve the AC equation with the zero Neumann boundary condition: for k₁

=

0

, . . . ,

^N1

−

1 and k₂

=

0

, . . . ,

^N2

−

1,

 φ

k1k2

= α

k1

β

k2 N1−1



l1=0 N2−1



l2=0

φ

l1l2cos

 π

N₁k₁



l₁

+

¹

2



cos

 π

N₂k₂



l₂

+

¹

2



,

where

φ

l1l2

= φ 

L1 N1



l₁

+

¹

2

 ,

_N^L2₂



l₂

+

¹

2

 

and

α

0

= √

1

/

^N1,

β

0

= √

1

/

^N2,

α

k1

= √

2

/

^N1,

β

k2

= √

2

/

^N2for k₁

,

^k2

≥

1.

(Here we slightly abuse the notation

φ

which itself depends continuously on time and space. The subscripts l₁

,

^l2are used for spatial discretization, thus

φ

l1l2denotes a time dependent function. We later use superscript n to denote time discretization of

φ

.) Then, we have a semi-analytical formula for the evolution operatorH^1tin the discrete cosine space

H^1t

(φ) =

^C−1



e^Ak1k21tC[

φ

^]



,

⁽⁷⁾

where A_k1k2

= − π

²

 

k1 L1



2

+



k2 L2



2



andCdenotes the discrete cosine transform.

In Ref. [11], Lee and Lee proposed a second order semi-analytical method based on operator splitting scheme for the AC equation:

φ

ⁿ⁺¹

= 

H^1t/2F^1tH^1t/2



φ

ⁿ

,

⁽⁸⁾

where

φ

^nand

φ

ⁿ⁺¹are numerical approximations of

φ(

^tn

)

^and

φ(

^tn

+

₁t

)

, respectively. It is easy to show that the Lee and Lee’s method is unconditionally stable (in the sense that the solution remains to be bounded by 1 for all time). Let

φ

^∗

=

_H^1t/2

(φ

ⁿ

)

^and

φ

^∗∗

=

_F^1t

(φ

^∗

)

. A function defined as a cosine series

ψ

^1t/2

(

^x1

,

^x2

) :=

N1−1



k1=0 N2−1



k2=0

e^Ak1k21t/2

ψ 

⁰k1k2cos

 π

N₁k₁x₁



cos

 π

N₂k₂x₂



is a solution of the heat equation with an initial condition

ψ

⁰

(

^x1

,

^x2

)

thus satisfies the maximum principle, max_x1,x2

| ψ

^1t/2

(

^x1

,

^x2

)| ≤

^maxx1,x2

| ψ

⁰

(

^x1

,

^x2

)|

. Therefore,

max

l1,l2

| φ

l^∗1l2

| =

max

x1,x2

| φ

^∗

(

^x1

,

^x2

)| ≤

^max

x1,x2

| φ

ⁿ

(

^x1

,

^x2

)| =

^max

l1,l2

| φ

ⁿl1l2

| .

Next, if we assume that

| φ

l^∗1l2

| ≤

1 for each l₁

,

^l2(which is a natural consequence of the assumption

| φ

ⁿl1l2

| ≤

1), we have

| φ

l^∗∗1l2

| = |

_F^1t

(φ

l^∗1l2

)| = | φ

l^∗1l2

|



(φ

l^∗1l2

)

²

+



1

− (φ

^∗l1l2

)

²



e

−21^t ϵ²

≤

1

.

Finally, it can also be shown that max_l1,l2

| φ

lⁿ1⁺l2¹

| ≤

_maxl

1,l2

| φ

l^∗∗1l2

|

. Therefore, the Lee and Lee’s method is unconditionally stable regardless of time step size in the sense that max_l1,l2

| φ

lⁿ1⁺l2¹

| ≤

1 for given max_l1,l2

| φ

lⁿ1l2

| ≤

1.

When solving the AC equation using the operator splitting method in the type of(8), the numerical results are almost not affected by the order of operator evaluation. The desired order of accuracy is achieved regardless of operator evaluation order since bothF^1tandH^1thave semi-analytic formulas satisfying semigroup properties and the stability is guaranteed since both

|

_F^1t

(φ)|

^and

|

_H^1t

(φ)|

are bounded by 1 for

| φ| ≤

1. However, in the case of AC type equations with nonlinear source terms, the accuracy and stability of numerical results are affected by the order of operator evaluations. Therefore, when one wants to develop a high order time discretization method for solving such type equations, it needs to consider the proper order of operator evaluations.

Before we close this section, we just want to mention that there exists another second order unconditionally stable operator splitting method by Yang [8] for the AC equation:

φ

ⁿ⁺¹

= 

F^1t/2H_CN^1tF^1t/2

 φ

ⁿ

,

where the heat evolution operatorH_CN^1tusing standard second order Crank–Nicolson is defined as

H_CN^1t

(φ) =

^C−1

 

1

+

₁tA_k1k2

/

²



C[

φ

^] 1

−

₁tA_k1k2

/

²

 .

It is worth noting that the method consists of two symmetrized half steps of free-energy evolution operatorF^1t/2and a second order full step of heat evolution operatorH_CN^1tin between.

3. A vector-valued Allen–Cahn equation

Our first example for the AC type equations with nonlinear source terms is the vector-valued AC equation. The equation was introduced in Garcke et al. [40] and has been extensively used [27,37,38,41], which allows one to consider an arbitrary number of components to the order parameter. In this section, we give some basic information about the vector-valued AC equation and propose a second order operator splitting method for the equation. Numerical experiments finally illustrate the accuracy and stability of the proposed method.

We consider the evolution of multi-component systems on a polygonal (polyhedral) domainΩ

⊂

_R^d

,

^d

=

1

,

²

,

^{3. Let} c

= (

^c1

, . . . ,

^cN

)

be a vector-valued phase-field. The components

{

c_i

}

^N_i=1represent mole fractions of different components in the system. Clearly the total mole fractions must sum to 1,

c₁

+ · · · +

_cN

=

₁

,

⁽⁹⁾

hence admissible states will belong to the Gibbs N-simplex

G

:=



c

∈

_R^N











N



i=1

c_i

=

1

,

⁰

≤

c_i

≤

1



.

Without loss of generality, we postulate that the free energy can be written as follows:

E

(

^c

) := 

Ω



F

(

^c

) ϵ

²

+

¹

2

N



i=1

|∇

c_i

|

²



dx

,

where F

(

^c

) =

⁰

.

²⁵



N

i=1c_i²

(

¹

−

c_i

)

². The natural boundary condition for the vector-valued AC equation is the zero Neumann boundary condition:

∇

c_i

·

n

=

0 on

∂

Ω^{for i}

=

1

, . . . ,

^N

,

⁽¹⁰⁾

where n is the unit normal vector to

∂

Ω^.

Now we review a derivation of the vector-valued AC equation as a gradient flow under the additional constraint(9), which has to hold everywhere at any time. It is natural to seek a law of evolution in the form

∂

^c

∂

^t

= −

gradE

(

^c

).

⁽¹¹⁾

In order to ensure the constraint(9), we use (see Ref. [40]) a variable Lagrange multiplier

β(

^c

) := (−

¹

/

^N

) 

^Ni=1f

(

^ci

)

^where f

(

^c

) =

^c

(

^c

−

0

.

⁵

)(

^c

−

1

)

for the previously given F

(

^c

)

^{and f}

(

^c

) = (

^f

(

^c1

), . . . ,

^f

(

^cN

)) :=

^∂_∂^F_c

=



∂F

∂^c1

, . . . ,

_∂^∂_c^F_N



. Choosing a general smooth vector-valued function

ξ ∈

R^Nand setting d

:= (

^d1

, . . . ,

^dN

) = ξ −

_N¹



N

i=1

ξ

i1 where 1

= (

¹

, . . . ,

¹

) ∈

R^N, then



N

i=1d_i

=

0 and we have the following equation for the evaluation of gradE

(

^c

)

^:

(

^gradE

(

^c

),

^d

)

L²

=

^d

d

η

^E

(

^c

+ η

^d

)









_η=

0

=

^d d

η



Ω N



i=1



1

4

ϵ

²

(

^ci

+ η

^di

)

²

(

¹

− (

^ci

+ η

^di

))

²

+

¹

2

|∇ (

^ci

+ η

^di

)|

²



dx











_η=

0

=



Ω



1

ϵ

²



f

(

^c

) · ξ −

^f

(

^c

) ·

¹ N

N



i=1

ξ

i1

 +

N



i=1

∇

 ξ

i

−

¹

N

N



j=1

ξ

j



· ∇

c_i



dx

=



Ω



f

(

^c

) + β(

^c

)

¹

ϵ

²

−

₁c



· ξ

^dx

=



Ω



f

(

^c

) + β(

^c

)

¹

ϵ

²

−

₁c



·

d dx

=



f

(

^c

) + β(

^c

)

¹

ϵ

²

−

₁c

,

^d



L²

,

where we have used the boundary condition(10). We identify gradE

(

^c

) = (

^f

(

^c

) + β(

^c

)

¹

) /ϵ

²

−

₁c, then Eq.(11)becomes the vector-valued AC equation

∂

^c

∂

^t

= −

^f

(

^c

)

ϵ

²

+

₁c

− β(

^c

)

ϵ

^{2 1}

.

⁽¹²⁾

Before we present our method, let us define the following notations:

F^1t

(

^c

) = (

^F1t

(

^c1

), . . . ,

^F1t

(

^cN

))

^{and H1t}

(

^c

) = (

^H1t

(

^c1

), . . . ,

^H1t

(

^cN

)),

whereF^1t

(

^ci

)

is defined as in(13)but maps c_i

∈ [

0

,

¹

]

toF^1t

(

^ci

) ∈ [

⁰

,

¹

]

andH^1t

(

^ci

)

is defined in(7). Here the evolution operatorF^1t

(

^ci

)

whose range is

[

0

,

¹

]

can be written explicitly as follows [38]:

F^1t

(

^ci

) =

¹

2

+

^2ci

−

1

2



(

^2ci

−

1

)

²

+ (

¹

− (

^2ci

−

1

)

²

)

^e−21ϵt2

.

⁽¹³⁾

We also define the ‘‘nonlinear constraint evolution operator’’N_β^1tas follows:

N_β^1t

(

^c

(

^tn

)) :=

^c

(

^tn

+

₁t

),

where c

(

^tn

+

₁t

)

is a solution of the first order nonlinear differential equation

∂

^c

∂

^t

= − β(

^c

) ϵ

^{2 1}

with an initial condition c

(

^tn

)

and the variable Lagrange multiplier

β(

^c

(

^t

))

. Since there is no analytic form of the solution for this nonlinear evolution equation, we define a numerical approximation of the evolution operatorN_β¹_m^tbased on the Euler method,

N_β¹_m^t

(

^c

(

^tn

)) =

^c

(

^tn

) − β(

^c

(

^tm

))

1^t

ϵ

^{2 1}

.

⁽¹⁴⁾

Note thatN_β¹_n^t

(

^cn

)

is a first order forward Euler approximation,N_β¹_n^t₊₁

(

^cn

)

is a backward Euler approximation, andN_β^1t

n+¹₂

(

^cn

)

is a second order mid-point approximation of c

(

^tn

+

₁t

)

^.

A first order stable operator splitting method for Eq.(12)is cⁿ⁺¹

= 

F^1tH^1tN_β¹_n^t



cⁿ

.

⁽¹⁵⁾

This is an operator splitting method combining two semi-analytic evolution operators and a first order approximation of the nonlinear evolution operator. Note that the free-energy evolution operatorF^1tmakes the result to be bounded by B

(

1^t

)

as in(6)regardless of size of







^c

∗

=



H^1tN_β¹_n^t



cⁿ







^, max

x∈_Ω









F^1t



c^∗_i

(

^x

) −

¹ 2









≤

B

(

1^t

) :=

¹ 2



1

−

e^−1t/2ϵ2



−₁/2

(16)

for each i. This provides a reasonably good bound for a larger time step1^{t and lim}1^t→∞





F^1t



c_i^∗

(

^x

) −

¹

/

²



 =

1

/

^{2. As}1^t approaches to 0, B

(

1^t

)

is no longer useful to get a tight bound and we may apply another inequality,









^F

1t



c_i^∗

(

^x

) −

¹ 2









≤

max



1 2

,









c^∗_i

(

^x

) −

¹ 2











(17)

where

|

c^∗_i

−

c_iⁿ

|

is linearly bounded by smaller time step1t. Instead of developing a tighter bound for the solution c_i^T/1t at any given time T , we just want to conclude this argument with a comment that





_F^1t



c_i^∗

(

^x

) −

¹

/

²





is always bounded by B

(

1^t

)

for any fixed1t. For this reason, it is important to evaluateF^1tat the end to guarantee the boundedness of the numerical solution for any fixed1^t.

We now propose a second order operator splitting method for Eq.(12). It is a symmetric combination of two half step evolution operatorsH^1t/2,F^1t/2and a full step nonlinear evolution operatorN_β^1tin between. In order to make the nonlinear evolution operatorN_β^1tto be second order accurate, we use a mid-point rule using

β

n+¹₂

:= β(

^cn+12

)

^.

STEP 1. cⁿ⁺¹²

=



N_β¹_n^t/2H^1t/2F^1t/2



cⁿ

,

⁽¹⁸⁾

STEP 2. cⁿ⁺¹

=



F^1t/2H^1t/2N_β^1t

n+¹₂H^1t/2F^1t/2



cⁿ

,

⁽¹⁹⁾

where

β

n+¹₂ is computed using cⁿ⁺¹² given in the first step of the operator splitting method. It is worth remarking that



H^1t/2F^1t/2



cⁿappeared in STEP 1 can be reused in STEP 2 and the boundedness of cⁿ⁺¹(or stability of the method) is obtained ifF^1t/2is applied at the end.

Example 1 (Phase Separation of a Quaternary System). We consider a quaternary system in one-dimensional space,Ω

= [

0

,

¹

]

. An estimate of the convergence rate is obtained by performing a number of simulations for a sample initial problem on a set of increasingly finer time steps. The initial conditions are

c_i

(

^x

,

⁰

) =

⁰

.

²⁵

+

0

.

^{05 sin}

((

ⁱ

+

3

) √

8x

)

^{for i}

=

1

,

²

,

³

.

Note that we only solve c₁

,

^c2

,

^c3since c₄

=

1

−

c₁

−

c₂

−

c₃by the constraint(9). The numerical solutions are computed with h

=

1

/

^128,

ϵ =

⁰

.

^{01, and}1^t

=

T_f

/

^{2kfor k}

=

2

, . . . ,

10. For each time step, we integrate to time T_f

=

0

.

^{005. The} error is computed by comparison with a quadruply over-resolved numerical reference solution shown inFig. 1(a).Fig. 1(b) shows relative l₂errors of the solutions for different time step sizes by the first and second order methods described in(15) and(18)–(19), respectively. The numerical results show that the proposed methods provides the expected order accuracy in time.

In order to see the effect of interface thickness parameter

ϵ

on the numerical error, we consider the same problem in Fig. 1with

ϵ =

⁰

.

⁰⁰²

(

^Tf

=

0

.

⁰⁰⁰²

)

^,

ϵ =

⁰

.

⁰¹

(

^Tf

=

0

.

⁰⁰⁵

)

^{, and}

ϵ =

⁰

.

⁰⁵

(

^Tf

=

0

.

¹

)

^{.Fig. 2}shows relative l₂errors of (a) the first and (b) the second order methods for different

ϵ

values, respectively. The numerical results show that both first and second order accuracy is achieved for1^t

≤

5

ϵ

²(marked by arrows in the figures) and the solutions are stable even when 1^t

≫

5

ϵ

^2.

4. A phase-field model for dendritic crystal growth

Another example of the AC type equations with nonlinear source terms is a phase-field model for dendritic crystal growth.

The purpose of this section is not developing a best possible numerical method but demonstrating the feasibility of the

Fig. 1. (a) Reference solution c_ref(x,Tf)obtained by the second order method at Tf =0.005 with h=1/128,ϵ =0.01, and1t=Tf/2¹². (b) Relative l2

errors of the first and second order methods for different time step sizes. The dashed blue lines indicate the first or the second order convergence slope.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a) First order method. (b) Second order method.

Fig. 2. Relative l2errors of (a) the first and (b) the second order methods withϵ =⁰.^{002 (T}f =0.^{0002, ‘}∗’),ϵ =⁰.^{01 (T}f =0.^{005, ‘}◦’), andϵ =⁰.⁰⁵ (Tf =0.^{1, ‘}’) for different time step sizes, respectively. The dashed blue lines indicate the first or the second order convergence slope. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

simple scheme described in the previous section for this complex system of nonlinear evolution equations of AC type. The two-dimensional phase-field equation for dendritic crystal growth is given by

ϵ

²

(φ) ∂φ

∂

^t

= [ φ − λ

^U

(

¹

− φ

²

)](

¹

− φ

²

) + ∇ · (ϵ

²

(φ)∇φ) +



|∇ φ|

²

ϵ(φ) ∂ϵ(φ)

∂φ

x



x

+



|∇ φ|

²

ϵ(φ) ∂ϵ(φ)

∂φ

y



y

,

⁽²⁰⁾

∂

^U

∂

^t

=

D1^U

+

¹ 2

∂φ

∂

^t

,

⁽²¹⁾

where

φ

is an order parameter, which varies from negative one in the liquid to positive one in the solid,

ϵ(φ)

is an anisotropy function,

λ

is a dimensionless parameter that controls the strength of the coupling between the phase and diffusion fields, U is a dimensionless temperature field, and D is the dimensionless thermal diffusivity. For the four-fold symmetry,

ϵ(φ)

^is defined as

ϵ(φ) = (

¹

−

3

ϵ

4

)



1

+

⁴

ϵ

4

1

−

3

ϵ

4

φ

⁴x

+ φ

y⁴

|∇ φ|

⁴



,

⁽²²⁾

where

ϵ

4is a parameter for the anisotropy strength in the interface energy. For F

(φ) =

⁰

.

²⁵

(φ

²

−

1

)

^{2and F′}

(φ) = φ

³

− φ

^, Eq.(20)can be rewritten as follows:

∂φ

∂

^t

= −

¹

ϵ

²

(φ)

^F

′

(φ) +

1

φ + ω(φ,

^U

),

⁽²³⁾

where the nonlinear source term

ω(φ,

^U

)

is defined as

ω(φ,

^U

) = ϵ

²¹

(φ)



2

ϵ(φ)∇ϵ(φ) · ∇φ −

⁴

λ

^UF

(φ) +



|∇ φ|

²

ϵ(φ) ∂ϵ(φ)

∂φ

x



x

+



|∇ φ|

²

ϵ(φ) ∂ϵ(φ)

∂φ

y



y

.

⁽²⁴⁾

Before we present our method, we redefine the ‘‘free-energy evolution operator’’F_ϵ^1tdepending on

ϵ(φ)

as follows:

F_ϵ^1t

(φ(

^tn

)) := φ(

^tn

+

₁t

),

where

φ(

^tn

+

₁t

)

is a solution of the first order nonlinear differential equation,

∂φ

∂

^t

= −

^F

′

(φ) ϵ

²

(φ)

with an initial condition

φ(

^tn

)

. Assume that

ϵ(φ(

^t

))

is given as a fixed value

ϵ

m

= ϵ(φ(

^tm

))

, then we have an analytical formula for the evolution operatorF_ϵ¹_m^tin the physical space,

F_ϵ¹_m^t

(φ) = φ



φ

²

+ (

¹

− φ

²

)

^e−

21^t ϵm²

for

ϵ

m

= ϵ(φ(

^tm

)).

⁽²⁵⁾

We also define the ‘‘nonlinear anisotropic evolution operator’’N_ω^1tas follows:

N_ω^1t

(φ(

^tn

)) := φ(

^tn

+

₁t

),

where

φ(

^tn

+

₁t

)

is a solution of the first order nonlinear differential equation

∂φ

∂

^t

= ω(φ,

^U

)

with an initial condition

φ(

^tn

)

. Again there is no analytic form of the nonlinear evolution operator so we approximate the evolution operatorN_ω^1tby the Euler method using

ω

m

:= ω(φ(

^tm

),

^U

(

^tm

))

^,

N_ω¹_m^t

(φ) = φ + ω(φ(

^tm

),

^U

(

^tm

))

1^t

.

⁽²⁶⁾

And we define the ‘‘thermal evolution operator’’M^1tas follows:

M^1t

(

^U

(

^tn

)) :=

^U

(

^tn

+

₁t

),

where U

(

^tn

+

₁t

)

is a solution of the first order differential equation

∂

^U

∂

^t

=

¹ 2

∂φ

∂

^t

with an initial condition U

(

^tn

)

and the order parameter

φ(

^t

)

. Then, we have a formula for the evolution operatorM^1t M^1t

(

^U

) =

^U

+ φ(

^tn

+

₁t

) − φ(

^tn

)

2

,

⁽²⁷⁾

whose numerical accuracy is governed by the accuracy of

φ(

^t

)

^.

A first order operator splitting method for the solution

φ

ⁿ

≈ φ(

^tn

)

^{and Un}

≈

U

(

^tn

)

^{from Eqs.(21)and(23)can be} described as follows:

φ

ⁿ⁺¹

= 

F_ϵ¹_n^tH^1tN_ω¹_n^t



φ

^{n and Un+1}

= 

H^D1tM^1t



Uⁿ

.

⁽²⁸⁾

We propose a second order stable operator splitting method for Eqs.(21)and(23):

STEP 1.

φ

ⁿ⁺¹²

= 

N_ω¹_n^t/2H^1t/2F_ϵ¹_n^t/2



φ

ⁿ

,

⁽²⁹⁾

Uⁿ⁺¹²

= 

M^1t/2H^D1t/2



Uⁿ

,

⁽³⁰⁾

STEP 2.

φ

ⁿ⁺¹

=



F_ϵ¹_∗^t/2H^1t/2N_ω^1t

n+¹ 2

H^1t/2F_ϵ¹_n^t/2



φ

ⁿ

,

⁽³¹⁾

Uⁿ⁺¹

= 

H^D1t/2M^1tH^D1t/2



Uⁿ

,

⁽³²⁾

where

ω

n+¹₂

= ω(φ

ⁿ⁺¹²

,

^Un+12

)

can be computed from the results in STEP 1 and

ϵ

^∗

= ϵ(φ

^∗

)

is given by a predictor step for the computation of

φ

^∗,

φ

^∗

=



F_ϵ^1t/2

n+¹₂H^1t/2N_ω^1t

n+¹₂H^1t/2F_ϵ¹_n^t/2



φ

ⁿ

.