an adaptive time-stepping strategy for epitaxial growth...
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An adaptive time-stepping strategy for epitaxial growth models
Zhonghua QiaoDepartment of Mathematics
Hong Kong Baptist [email protected]
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!