Efficient Numerical Methodsfor Phase-Field Equations

Tao Tang Hong Kong Baptist University

September 11-13 ， 2013Russian-Chinese Workshop on Numer Analysis and Scientific Computing

John W. Cahn (1928 -- ) John E. Hilliard (1826-1987)

The Cahn-Hilliard equation: describes the process of phase separation:

( )u

u f ut

Microstructural evolution under the Cahn–Hilliard equation, demonstrating distinctive coarsening and phase separation.

PHYSICS

Cahn-Hilliard equation

Phase separation in a binary alloy (metal, liquid, …)

Spinodal decomposition Mass conservation Interface minimization

( )u

u f ut

MATHEMATICS

2( ) ( )

2E u u F u dx

1( ) ( ( ))H

du t E u t

dt

1

2

( ( )) ( ) 0H

d dE u t u t

dt dt

High-order nonlinear diffusion equations:

How to do time integration? If both dynamics and steady state are

required ， how to do efficient time discretization?

Higher order methods vs. efficiency; Adaptivity

Examples of high order nonlinear diffusion equation

Molecular Beam Epitaxy (MBE) Model [T., Xu; WB Chen et ]

Inpaiting with Cahn-Hilliard Equation [A.L. Bertozzi, etc]

Phase field crystal equation [Lowengrub, Wang, Wise etc]

Thin Film epitaxy [J. Shen, X.M. Wang, Wise, etc]

2 2(1 )th h hh

'1 ( ) ( )( )tu u W u x f u

3 2( 2 )tt t

2 2 2( )- -t

EXAMPLE Cahn-Hilliard impainting

[Bertozzi etc. IEEE Tran. Imag. Proc. 2007, Commun. Math. Sci, 2011]

'1( ) ( )tu u F u f u

Monotonic decrease of the energy functional during the coarsening process

NUMERICAL CHALLENGES

Interior layers (i.e. thin interface) see a figure

Time discretization: Lower order (good for stability)? “h”-adaptivity; “p”-adaptivity?

Allen-Cahn equation Cahn-Hilliard equationThin-film epitaxy without slope selection

Ene

rgy

Ene

rgy

Ene

rgy

Energy curves of three different models

EXAMPLE Consider the IBV problem for the Cahn-Hilliard Eq.

Explicit Euler’s scheme (10-7)

Semi-implicit Euler’s scheme (10-4)

Implicit Euler’s scheme (10-6)!

Non-linearly stabilized scheme (implicit for the biharmonic and non-linear terms : -- 10-3)

Linearly stabilized splitting scheme: introducing two splitting functionals; one is contractive and the other is expansive: (10-2 ~ 10-3)

xxuxu

txuuuut ),()0,(

R),( ,0)(

0

3

A stable first-order method:

0

( ) 0, in

( 0) in tu E u t

u t u

( ) 0,E u u ( ) , as E u u

( )( ) , , ,J E u u u u

2( )( )

dE uE u

dt

( )( )J E u E

Eyre’s method Convexity splitting

where and are strictly convex. The semi-implicit discretization is given by

Various Eyre’s type or various extension: Inpaiting problem [Schönlieb & Bertozzi]

Coarsening simulations [Vollmayr-Lee & Rutenberg]

Second-order convex splitting [J. Shen, C. Wang, Wise]

Question:Given Eyre’s GS scheme, can we use some iterative ideas to obtain a higher order semi-stable method?

( ) ( ) ( )c eE u E u E u 2,c eE E C

1 1( ( ) ( ))k k k kc eU U t E U E U

Spectral deferred Correction (SDC) for y’=f(t,y) The method is introduced by Dutt, Greengard and

Rokhlin (BIT, 2000) Multi-implicit SDC method (Layton and Minion, 2004) SDC with high-order RK schemes [Christliek, Qiu and

Ong, 2010]

1( ) ( ) ( , ( )) , [ , ].n

t

n n nty t y t f s y s ds t t t

,

, ,1 1

( )

( , ) ( )n i

n

n m

m t

m n j j n jtj i m

Y t U K Y

f t l s ds

Collocation Method:

[ 1] [ 1]( ) ( ) ( )j jm me t y t L

Use an k-th order method to compute [ 1],( )j

i n ie t

at the grid points tn,i on [tn,tn+1].

Define a new approximation solution [ ] [ 1] [ 1]j j j

[ 1] [ 1] [ 1]j j jn mu K

[ 1]j Compute the residual for

Define the error function for

Form the error equation

[ 1] [ 1] [ 1]( ) ( ( ) ( )) ( )j j jm m m me t K e t L K L

Algorithm (SDC method)1 (Prediction).

Use a k0-th order numerical method to compute [0] [0] [0]

1[ , , ]Tm

2 (Correction). For j=1,…,J

[ 1]j

Convergence analysis[T., H.-H. Xie, X.-B. Yin, JSC 2012]

Theorem 1: Let be computed in the Correction step of the SDC Algorithm. If the step-size h is sufficiently small, then the following error estimate holds:

0

0[ ]

1

JJ mm k J m

ky Ch y Ch y

[ ]J

[ ] [ 1] [ ] [ 1]j j j jm m m m m mY K K K Y K

[ ] [ 1] [ ] [ 1]

[ ] [ ]

[ ] [ 1

[ 1]

[ ] [0]

]j j

j j j jm m

j j jm m m

J Jm

m m

m

Y Ch Y

Y Ch Y Y

Y C

C

h

Y Y

Y

h

Outline

1 2 1 2m mK K Ch

Note

Algorithm (Modified SDC method) :

Same as Algorithm 1, except that after is computed we use some Picard smoothing

After the above k-1 iterations, let

[ ] [ 1]

[0] [ 1]

, 1, , , ( 1)

. ( 2)

1n m

j

ku K S

S

[ 1] [ 1].j k

Theorem 2

Let be computed in the Correction step of the modified SDC Algorithm. If the step size h is sufficiently small, and if special collocation points (e.g. Gauss) are used, then

0

0[

,

2]

1, 1

1 .Jm k m

Jk k mY Ch y Ch y

[ ]J

[ 1]j

Proof. It follows from (S1) that

[ ] [ 1]m m m mY K Y K

[ ] [ 1]

[ 1] 1 [0] 1 [ 1]

[ ] [0]

j k j

k k k jm m m

J J

m

m

m

km

Y

Y Ch Y Ch Y

Y Ch Y

Ch Y

Efficiency comparison using Legendre-Gauss quadrature nodes

Energy evolutions with different time steps

FIRST-ORDER LINEAR SCHEME

Simple, linear discretization in time; First-order with energy decreasing; In space, central differencing FFT is used

STABILTY+ACCURACY VIA P-ADAPTIVITY

Use Energy difference at and steps

If the difference is small, no correction; If the difference is large, judge how many

SDC corrections are needed. Note most of time regimes, no corrections

are needed

nt 1nt

1( ) ( )n nh hE u E u

fine mesh

coarse mesh with correction

0 ( , ) 0.05sin sin 0.001u x y x y

Energy evolutions with different time steps and different numbers of corrections for the Cahn-Hilliard equation

Blowing up phenomenon of semi-implicit spectral deferred correction with uniform number of corrections

HOW MANY ITERATIONS NEEDED

max

max max

1

11

1 1max

0, if ( ) ( )

, if ( ) ( ) ,

, if ( ) ( )

Nn nh h

N k N kn np h h

n nh h

E u E u

N k E u E u

N E u E u

Energy curves of the thin film model without slope selection and number of corrections

CPU time comparison

Adaptive Time Stepping:

Energy is an important physical quantity to reflect the structure evolution.

Adaptive time step is determined by

∆tmin corresponds to quick evolution of the solution, while ∆tmax to slow evolution.

[Qiao, T., Zhang, SISC, 2010]

)|)(|1

,max(2

maxmin

tE

ttt

Time adaptivity via energy variation (Xie; T., Luo)

Numerical Scheme for C-H eqn:

Stability: Discrete energy identity:

12

12

1 1 1 2 2 1, , , , , , , ,

1, ,

,

( ) ( )

, 2 2 2 2

2 212 4 .

,

( ) 1

n n n n n n n nj k j k j k j k j k j k j k j k

n nj k j k

h j k

U U U U U U U U

j k h

h h h

n

h

nU U

t

E U U U

1( ) ( ).n nh hE U E U

12

1 2( ) ( )0

n nnh h

hh

E U E U

t

Equi-energy:

• It follows from the numerical scheme and the energy identity that12

12

1

21

nn nh

nn

hh

h

U U

E

• The prescribed energy decrease ( ) equation

•Time stepping formula

•One step fix-point iteration to solve the prescribed energy decrease equation

E

12

12

1

2

nn nh

nh

h

U U

E

12

2n

hh

Et

The initial condition is random in [-0.1,0.1], with periodic boundary condition and

0.001

Example [artificial dissippation] Molecular Bean Epitaxy (MBE) Model:

Model eqn:

ht = -2h - [ (1 - |h|2)h ]

Energy identity:

where

0 )( thhEdt

d

222

21

4

1)( hhhE

ARTIFICIAL DISSIPATION: Remedy:

i.e. an O(t) is added, where A > 0 is an O(1) constant.

Property: If the constant A is sufficiently large, then

E(hn+1) E(hn)?

If the numerical solution is convergent, then the condition for A is

T & C.Xu: [SINUM, 2006]

])||1[( 21121

nnnnnn

hAhhAht

hh

],0( ,2

1

2

3 2Tina.e.hA

MORE ON REGULARITY: Consider the nonlinear 2-D model for epitaxial

growth of thin films:

Here, we prove an a-priori bound on the L-norm of h in the 2-D case with =0.

The proof heavily relies on the maximum principle. It is hard to see how it can be extended to the case

of (small) positive . However, it is intuitively clear that in the case of

positive , the solution should be more regular, and one may expect that the similar bound on h still holds.

,0 ,])1|[(| 22 hhhht

Conclusions/Remarks

High-order time discretization is needed for high-order nonlinear diffusion equations.

The use of the SDC method seems a useful way.

Analysis of nonlinear stability and convergence require deep understanding of the relevant PDEs and numerical methods. [local estimates … T. & Xu SINUM 2006, Bertozzi etc]

The analysis for adaptive schemes is highly nontrivial. Most of the existing numerical methods are lack of rigorous mathematical justification.

Thanks!

http://www.math.hkbu.edu.hk/~ttang