Efficient Numerical Methods
for Phase-Field Equations
Tao Tang
Hong Kong Baptist University
September 11-13， 2013
Russian-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 (u ) 
t
Microstructural evolution under the Cahn–Hilliard equation, demonstrating
distinctive coarsening and phase separation.
PHYSICS
Cahn-Hilliard equation
u
   u  f (u ) 
t
Phase separation in a binary alloy (metal, liquid,
…)
 Spinodal decomposition
 Mass conservation
 Interface minimization

MATHEMATICS

2


E (u )    u  F (u ) dx
 2


d
u (t )   H 1 E (u (t ))
dt
d
d
E (u (t ))   u (t )
dt
dt
2
0
H 1
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 ]
2
ht   2 h   (1  h )h 



Inpaiting with Cahn-Hilliard Equation
[A.L. Bertozzi, etc]
ut    u  1 W ' (u )    ( x )( f  u )

Phase field crystal equation [Lowengrub, Wang, Wise etc]
tt  t  (3    2  2)

Thin Film epitaxy [J. Shen, X.M. Wang, Wise, etc]
 t    (   )- - 2  2
2
EXAMPLE

Cahn-Hilliard impainting
1 ' 

ut    u  F (u )    ( f  u )



[Bertozzi etc. IEEE Tran. Imag. Proc. 2007, Commun. Math. Sci, 2011]
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?

Thin-film epitaxy without
slope selection
Energy
Cahn-Hilliard equation
Energy
Energy
Allen-Cahn equation
Energy curves of three different models
EXAMPLE

Consider the IBV problem for the Cahn-Hilliard Eq.
 t u  (u  u 3  u )  0,
u ( x,0)  u0 ( x),
( x, t )    R 
x

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)
A stable first-order method:

t  0, in 
ut  E (u )

u (t  0)  u0 in 
E (u)  0, u
E(u)  , as u  
J (E)(u)u, u  , u,
J (E )(u)
dE (u )
  E (u )
dt
E
2
Eyre’s method
 Convexity splitting

E(u)  Ec (u)  Ee (u)
where Ec , Ee  C 2 and are strictly convex. The
semi-implicit discretization is given by
U k 1 U k  t (Ec (U k 1 ) Ee (U k ))
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 semistable method?
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]
t
y(t )  y(tn )   f (s, y(s))ds, t [tn , tn1 ].
tn
Collocation Method:
Y (t )  U n  K mY
tn ,i
 m

 m    f (tn, j , j )  ln , j ( s)ds 
tn
 j 1
1i m
Algorithm (SDC method)
1 (Prediction).
[0]
[0]
[0] T
Use a k0-th order numerical method to compute   [1 , ,m ]
2 (Correction). For j=1,…,J

Compute the residual for  [ j 1]
 [ j 1]  un  Km [ j 1]  [ j 1]

Define the error function for  [ j 1]
e[ j 1] (t )  ym (t )  Lm ( [ j 1] )

Form the error equation
e(t )  Km (e(t )  Lm ( [ j 1] ))  Km [ j 1]  Lm ( [ j 1] )

Use an k-th order method to compute i[ j 1]  e(tn,i )
at the grid points tn,i on [tn,tn+1].

Define a new approximation solution  [ j ]   [ j 1]   [ j 1]
Convergence analysis
[T., H.-H. Xie, X.-B. Yin, JSC 2012]
Theorem 1: Let  [ J ] be computed in the Correction
step of the SDC Algorithm. If the step-size h is
sufficiently small, then the following error estimate
holds:
ym  
[J ]
 Ch
k0  J
y
k0  J
 Ch
m
y
m 1
Outline
Ym  [ j ]  Km [ j 1]  Km [ j ]  KmYm  Km [ j 1]

 Ch  Y
 Ym   [ j ]  Ch  [ j 1]   [ j ]  Ym   [ j 1]
 Ym   [ j ]
[ j]
[ j 1]



Y


m
m
 Ym   [ j ]  Ch Ym   [ j 1]
 Ym   [ J ]  Ch J Ym   [0]
Note
K m1  K m 2  Ch 1   2


Algorithm (Modified SDC method) :
Same as Algorithm 1, except that after
Picard smoothing
 [  ]  un  K m [  1] ,   1,
 [ j 1] is computed we use some
, k  1,
( S1)
 [0]   [ j 1] .
After the above k-1 iterations, let
( S 2)
 [ j 1]   [k 1].
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
[J ]
Ym   [ J ]  Ch Jk  k0 y
k0
2 m 1

Ch
y
1, 
m 1, 
.
Proof. It follows from (S1) that
 [  ]  Ym  KmYm  Km [  1]
  [ k 1]  Ym  Chk 1  [0]  Ym  Chk 1  [ j 1]  Ym
 Ym   [ j ]  Chk  [ j 1]  Ym
 Ym   [ J ]  Ch Jk Ym   [0]
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 t n and tn 1 steps
Eh (u n )  Eh (u n 1 )
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

coarse mesh
with
correction
u0 ( x, y)  0.05sin x sin y  0.001
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
 N max
n
n 1
0,
if Eh (u )  Eh (u )  


 N max  k
 N max  k 1
n
n 1
N p  k ,
if 
 Eh (u )  Eh (u )  
,

n
n 1
1

 N max , if Eh (u )  Eh (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

t  max(tmin ,

tmax
1   | E(t ) |2
)
∆tmin corresponds to quick evolution of the solution,
while ∆tmax to slow evolution.
[Qiao, T., Zhang, SISC, 2010]
Time adaptivity via energy variation
(Xie; T., Luo)

Numerical Scheme for C-H eqn:
U nj ,k1  U nj ,k
t

n  12
j ,k

  h 
U nj ,k1 U nj ,k
Eh (U ) 
2

2

 hU
n  12
j ,k
,
U nj ,k1 U nj ,k (U nj ,k1 )2  (U nj ,k )2
2
2
2
1

U
1
4
h
  h
2
h.
Stability: Eh (U n1 )  Eh (U n ).
 Discrete energy identity:

Eh (U n 1 )  Eh (U n )
n  12
 h 
t
2
h
0
U nj ,k1 U nj ,k
2
Equi-energy:
• It follows from the numerical scheme and the energy identity that
n 1
h  2
U n 1  U n

n 1
n  12
 Eh
h 
2
h
• The prescribed energy decrease ( E ) equation
n 1
h  2
U n 1  U n

n  12
E
h 
2
h
•Time stepping formula
t 
E
h 
n  12
2
h
•One step fix-point iteration to solve the prescribed energy
decrease equation
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:
d
E ( h )  ht  0
dt
where
2
1

2
E (h)  h  1  h
4
2
2
ARTIFICIAL DISSIPATION:
Remedy:
h n1  h n
 2 h n1  Ah n1    [(1 | h n |2  A)h n ]
t
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
3
1
2
A  h  ,
2
2
T & C.Xu: [SINUM, 2006]
a.e. in   (0, T ]
MORE ON REGULARITY:

Consider the nonlinear 2-D model for epitaxial growth
of thin films:
ht   [(| h |2 1)h]  2h,
  0,
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.

Conclusions/Remarks
High-order time discretization is needed for highorder 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