212 research outputs found
Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach
In this paper, we propose and analyze high order efficient schemes for the
time fractional Allen-Cahn equation. The proposed schemes are based on the L1
discretization for the time fractional derivative and the extended scalar
auxiliary variable (SAV) approach developed very recently to deal with the
nonlinear terms in the equation. The main contributions of the paper consist
in: 1) constructing first and higher order unconditionally stable schemes for
different mesh types, and proving the unconditional stability of the
constructed schemes for the uniform mesh; 2) carrying out numerical experiments
to verify the efficiency of the schemes and to investigate the coarsening
dynamics governed by the time fractional Allen-Cahn equation. Particularly, the
influence of the fractional order on the coarsening behavior is carefully
examined. Our numerical evidence shows that the proposed schemes are more
robust than the existing methods, and their efficiency is less restricted to
particular forms of the nonlinear potentials
Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model
An implicit variable-step BDF2 scheme is established for solving the space
fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived
from a gradient flow in the negative order Sobolev space ,
. The Fourier pseudo-spectral method is applied for the spatial
approximation. The proposed scheme inherits the energy dissipation law in the
form of the modified discrete energy under the sufficient restriction of the
time-step ratios. The convergence of the fully discrete scheme is rigorously
provided utilizing the newly proved discrete embedding type convolution
inequality dealing with the fractional Laplacian. Besides, the mass
conservation and the unique solvability are also theoretically guaranteed.
Numerical experiments are carried out to show the accuracy and the energy
dissipation both for various interface widths. In particular, the
multiple-time-scale evolution of the solution is captured by an adaptive
time-stepping strategy in the short-to-long time simulation
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Recent results in the literature provide computational evidence that
stabilized semi-implicit time-stepping method can efficiently simulate phase
field problems involving fourth-order nonlinear dif- fusion, with typical
examples like the Cahn-Hilliard equation and the thin film type equation. The
up-to-date theoretical explanation of the numerical stability relies on the
assumption that the deriva- tive of the nonlinear potential function satisfies
a Lipschitz type condition, which in a rigorous sense, implies the boundedness
of the numerical solution. In this work we remove the Lipschitz assumption on
the nonlinearity and prove unconditional energy stability for the stabilized
semi-implicit time-stepping methods. It is shown that the size of stabilization
term depends on the initial energy and the perturba- tion parameter but is
independent of the time step. The corresponding error analysis is also
established under minimal nonlinearity and regularity assumptions
On Power Law Scaling Dynamics for Time-fractional Phase Field Models during Coarsening
In this paper, we study the phase field models with fractional-order in time.
The phase field models have been widely used to study coarsening dynamics of
material systems with microstructures. It is known that phase field models are
usually derived from energy variation so that they obey some energy dissipation
laws intrinsically. Recently, many works have been published on investigating
fractional-order phase field models, but little is known of the corresponding
energy dissipation laws. We focus on the time-fractional phase field models and
report that the effective free energy and roughness obey a universal power-law
scaling dynamics during coarsening. Mainly, the effective free energy and
roughness in the time-fractional phase field models scale by following a
similar power law as the integer phase field models, where the power is
linearly proportional to the fractional order. This universal scaling law is
verified numerically against several phase field models, including the
Cahn-Hilliard equations with different variable mobilities and molecular beam
epitaxy models. This new finding sheds light on potential applications of time
fractional phase field models in studying coarsening dynamics and crystal
growths
A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
We develop a robust solver for a second order mixed finite element splitting
scheme for the Cahn-Hilliard equation. This work is an extension of our
previous work in which we developed a robust solver for a first order mixed
finite element splitting scheme for the Cahn-Hilliard equaion. The key
ingredient of the solver is a preconditioned minimal residual algorithm (with a
multigrid preconditioner) whose performance is independent of the spacial mesh
size and the time step size for a given interfacial width parameter. The
dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text
overlap with arXiv:1709.0400
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