An adaptive time-stepping strategy for epitaxial growth models

Zhonghua QiaoDepartment of Mathematics

Hong Kong Baptist Universityzqiao@math.hkbu.edu.hk

www.math.hkbu.edu.hk/~zqiao

Joint work with Dr. Zhengru Zhang and Prof. Tao Tang

Outline

Introduction An energy stable finite difference scheme An adaptive time stepping strategy Numerical Experiments Conclusions

We consider the nonlinear two-dimensional (2-D) equation designed to model epitaxial growth of thin films:

(1.1)

Large computational domain is necessary (to minimize the effect of periodicity assumption and to collect enough statistical information)

Long integration time is necessary(to detect the epitaxy growth behaviors and to reach the physical

scaling regime)

.0 ,])1|[(| 22 t

Molecular Bean Epitaxy (MBE) model

Energy identity: (Li and Liu, 2003)

where || || is the L2-norm and

Semi-implicit discretization

Difficulty: if << 1, then t has to be very small

0 )( tEdtd

dxE ]2

141[)( 222

])||1[( 2121

nnnnn

t

Remedy:

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

Property: If the constant A is sufficiently large, then

How large is A?C. Xu and T. Tang. Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Math., 44 (2006), 1759-1779.

])||1[( 21121

nnnnnn

AAt

)( )E( n1n E

Proof: Energy estimate gives:

Seems no room for improvement of the condition for A The problem is that A is dependent on

Second and third order semi-implicit schemes were also presented in C. Xu and T. Tang’s paper.

.0,41)1(

21

1141,1

2212

2222112

nt

nnn

nnnt

nnt

A

t

1 n

22 | |41

21||

21max nn

xA 1 n

A second-order energy stable finite difference scheme

Theorem: This second-order scheme is unconditionally energy stable. For any large time step ∆t >0, there holds

where Eh and ||•||h is the discrete energy and L2-norm, respectively.

Proof:

This can be equivalently written as

This form completely matches the Energy Identity: (Li & Liu, 2003)without any modification.

Since the scheme is unconditionally energy stable, any large ∆t >0 is allowed.

For the sake of accuracy, too large time step is not acceptable.

Adaptively choosing time step is a good idea for considering both stability and accuracy.

An adaptive time stepping strategy

MBE model may involve rapid transition of structure, which requires much smaller time steps.

The coarsening process may involve some extremely slow evolution, then much larger time steps (e.g., of order O(1)) may have to be used to significantly enhance the computational efficiency.

Adaptive time-stepping strategies have to be implemented.

This issue is of practical importance and has many technical and theoretical difficulties.

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

Adaptive time step is determined by

∆tmin corresponds to the stage of quick evolution of the solution, while ∆tmax is corresponding to the time interval of slow evolution.

αis a positive constant.

)|)(|1

,max(2

maxmin

tEttt

Numerical ExperimentsExample 1: 1D MBE model equation

Different time steps give different accuracy, but lead to the same steady state solution.

solution at t=40, adaptive time steps energy evolution using adaptive time steps, 0≤t ≤ 120

Energy evolution using adaptive time steps, 0 ≤ t ≤ 1.2.

Energy evolution using adaptive time steps, 40 ≤ t ≤ 60.

Time step evolution.

∆tmin=0.01, ∆tmax=0.5, α=105 is the best choice for this problem

There is no general rule to choose the parameter α.

Example 2: 2D MBE model equation

with initial condition

CPU time (seconds) comparison, constant ∆t=0.001,

)|)(|1

,max(2

maxmin

tEttt

with ∆tmin=0.001,∆tmax=0.5,α=105.

adaptive

After t=10, computational efficiency is significantly improved because the solution gradually reaches steady state and ∆t≈∆tmax is used.

0≤ t ≤ 120 4.8 ≤ t ≤ 12Energy evolution with different parameter settings.

Similar to the 1D case, different parameter settings willlead to the same steady state due to the unconditionallyenergy stability,

though the processing numerical simulation may have significant difference caused by accuracy.

For this problem, the good choice of parameter is∆tmax=0.5, ∆tmin=0.001, α=105.

time step evolution with different parameter settings

∆tmin is corresponding the transition stage (reflected from energy evolution).

∆tmax is corresponding the slow motion stage between two consecutive state.

∆t variying from ∆tmin to ∆tmax with appropriate parameters almost does not cause any loss of accuracy.

)|)(|1

,max(2

maxmin

tEttt

Example 3: coarsening dynamics

Ω=[0,200]X[0,200], Nx=Ny=400, the initial condition is a random state by a random number varying from -0.001 to 0.001 on each grid pint. This problem is subject to periodic boundary conditions.

The effective free energy is defined by

It needs very long time simulations, constant time steps such as ∆t=0.5 or ∆t=1 have been often used in the previous numerical simulations.

.2

141 222 freeF

Contour lines of the free energy Ffree (left) and solution (right) at t=2000,5000.

Power law for height and width growth

Numerical simulations parameters: ∆tmax=10, ∆tmin=0.5, α=1.

Energy evolution. Adaptive time step evolution.

Conclusions An energy stable finite difference scheme for nonlinear

diffusion equations modeling epitaxial growth of thin filmsis developed and analyzed.

Large time steps possible: good implicit methods are orders of magnitude more efficient than the conventional methods.

The schemes and analysis can be extended to several phase field models, such as Ginzburg-Landau, Cahn-Hilliard eqn, Allen-Cahn, Phase field crystal etc.

Adaptive time stepping is suitable for large time simulations without loss of accuracy. Once the stability is allowed, the time step can vary based on the dynamical behavior of some physical quantity.

On-going and Future Work Convergence Analysis of the energy stable schemes for

the MBE model.

(Z. Qiao, Z. Sun and Z. Zhang. The stability and convergence analysis of linearized finite-difference schemes for the nonlinear epitaxial growth model. Submitted to Numer. Meth. PDE.)

Energy stable schemes and adaptive time-stepping method for the Cahn-Hilliard equation.

Z. Zhang and Z. Qiao. An Adaptive Time-stepping Strategy for the Cahn-Hilliard Equation. Accepted by CICP, 2011.

.0 ,0)( 3 t

Energy stable schemes and adaptive time-stepping method for the phase field crystal model.

Here

Z. Zhang and Z. Qiao. An Adaptive Time-Stepping Strategy for Solving the Phase Field Crystal Model. Submitted to Journal of Computational Physics.

Error-estimate-based monitor function for the adaptive time-stepping method.

.0 ),)(( Mt

.2)1( 23

Thank You!