827 research outputs found
Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation
Phase-field model is a powerful mathematical tool to study the dynamics of
interface and morphology changes in fluid mechanics and material sciences.
However, numerically solving a phase field model for a real problem is a
challenge task due to the non-convexity of the bulk energy and the small
interface thickness parameter in the equation. In this paper, we propose two
stabilized second order semi-implicit linear schemes for the Allen-Cahn
phase-field equation based on backward differentiation formula and
Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force
is treated explicitly with two second-order stabilization terms, which make the
schemes unconditional energy stable and numerically efficient. By using a known
result of the spectrum estimate of the linearized Allen-Cahn operator and some
regularity estimate of the exact solution, we obtain an optimal second order
convergence in time with a prefactor depending on the inverse of the
characteristic interface thickness only in some lower polynomial order. Both
2-dimensional and 3-dimensional numerical results are presented to verify the
accuracy and efficiency of proposed schemes.Comment: keywords: energy stable, stabilized semi-implicit scheme, second
order scheme, error estimate. related work arXiv:1708.09763, arXiv:1710.0360
Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation
Efficient and unconditionally stable high order time marching schemes are
very important but not easy to construct for nonlinear phase dynamics. In this
paper, we propose and analysis an efficient stabilized linear Crank-Nicolson
scheme for the Cahn-Hilliard equation with provable unconditional stability. In
this scheme the nonlinear bulk force are treated explicitly with two
second-order linear stabilization terms. The semi-discretized equation is a
linear elliptic system with constant coefficients, thus robust and efficient
solution procedures are guaranteed. Rigorous error analysis show that, when the
time step-size is small enough, the scheme is second order accurate in time
with aprefactor controlled by some lower degree polynomial of .
Here is the interface thickness parameter. Numerical results are
presented to verify the accuracy and efficiency of the scheme.Comment: 26 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:1708.0976
A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain
We propose a pure source transfer domain decomposition method (PSTDDM) for
solving the truncated perfectly matched layer (PML) approximation in bounded
domain of Helmholtz scattering problem. The method is a modification of the
STDDM proposed by [Z. Chen and X. Xiang, SIAM J. Numer. Anal., 51 (2013), pp.
2331--2356]. After decomposing the domain into non-overlapping layers, the
STDDM is composed of two series steps of sources transfers and wave expansions,
where truncated PML problems on two adjacent layers and truncated
half-space PML problems are solved successively. While the PSTDDM consists
merely of two parallel source transfer steps in two opposite directions, and in
each step truncated PML problems on two adjacent layers are solved
successively. One benefit of such a modification is that the truncated PML
problems on two adjacent layers can be further solved by the PSTDDM along
directions parallel to the layers. And therefore, we obtain a block-wise PSTDDM
on the decomposition composed of squares, which reduces the size of
subdomain problems and is more suitable for large-scale problems. Convergences
of both the layer-wise PSTDDM and the block-wise PSTDDM are proved for the case
of constant wave number. Numerical examples are included to show that the
PSTDDM gives good approximations to the discrete Helmholtz equations with
constant wave numbers and can be used as an efficient preconditioner in the
preconditioned GMRES method for solving the discrete Helmholtz equations with
constant and heterogeneous wave numbers.Comment: 31 pages, 7 figure
Numerical approximation of elliptic problems with log-normal random coefficients
In this work, we consider a non-standard preconditioning strategy for the
numerical approximation of the classical elliptic equations with log-normal
random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was
proposed by modeling the random flux through the Wick product. Due to the
lower-triangular structure of the uncertainty propagator, this model can be
approximated efficiently using the Wiener chaos expansion in the probability
space. Such a Wick-type model provides, in general, a second-order
approximation of the classical one in terms of the standard deviation of the
underlying Gaussian process. Furthermore, when the correlation length of the
underlying Gaussian process goes to infinity, the Wick-type model yields the
same solution as the classical one. These observations imply that the Wick-type
elliptic equation can provide an effective preconditioner for the classical
random elliptic equation under appropriate conditions. We use the Wick-type
elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin
finite element method. Numerical results are presented and discussed.Comment: 28 pages, 11 figures, 5 tables, to appear on International Journal
for Uncertainty Quantificatio
On Efficient Second Order Stabilized Semi-Implicit Schemes for the Cahn-Hilliard Phase-Field Equation
Efficient and energy stable high order time marching schemes are very
important but not easy to construct for the study of nonlinear phase dynamics.
In this paper, we propose and study two linearly stabilized second order
semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses
backward differentiation formula and the other uses Crank-Nicolson method to
discretize linear terms. In both schemes, the nonlinear bulk forces are treated
explicitly with two second-order stabilization terms. This treatment leads to
linear elliptic systems with constant coefficients, for which lots of robust
and efficient solvers are available. The discrete energy dissipation properties
are proved for both schemes. Rigorous error analysis is carried out to show
that, when the time step-size is small enough, second order accuracy in time is
obtained with a prefactor controlled by a fixed power of , where
is the characteristic interface thickness. Numerical results are
presented to verify the accuracy and efficiency of proposed schemes
Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number
A preasymptotic error analysis of the finite element method (FEM) and some
continuous interior penalty finite element method (CIP-FEM) for Helmholtz
equation in two and three dimensions is proposed. - and - error
estimates with explicit dependence on the wave number are derived. In
particular, it is shown that if is sufficiently small, then
the pollution errors of both methods in -norm are bounded by
, which coincides with the phase error of the FEM obtained
by existent dispersion analyses on Cartesian grids, where is the mesh size,
is the order of the approximation space and is fixed. The CIP-FEM extends
the classical one by adding more penalty terms on jumps of higher (up to -th
order) normal derivatives in order to reduce efficiently the pollution errors
of higher order methods. Numerical tests are provided to verify the theoretical
findings and to illustrate great capability of the CIP-FEM in reducing the
pollution effect
A Laguerre homotopy method for optimal control of nonlinear systems in semi-infinite interval
This paper presents a Laguerre homotopy method for optimal control problems
in semi-infinite intervals (LaHOC), with particular interests given to
nonlinear interconnected large-scale dynamic systems. In LaHOC, spectral
homotopy analysis method is used to derive an iterative solver for the
nonlinear two-point boundary value problem derived from Pontryagins maximum
principle. A proof of local convergence of the LaHOC is provided. Numerical
comparisons are made between the LaHOC, Matlab BVP5C generated results and
results from literature for two nonlinear optimal control problems. The results
show that LaHOC is superior in both accuracy and efficiency
A kind of infinite-dimensional Novikov algebras and its realization
In this paper, we construct a kind of infinite-dimensional Novikov algebras
and give its realization by hyperbolic sine functions and hyperbolic cosine
functions.Comment:
Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines
The sharp-interface limits of a phase-field model with a generalized Navier
slip boundary condition for moving contact line problem are studied by
asymptotic analysis and numerical simulations. The effects of the {mobility}
number as well as a phenomenological relaxation parameter in the boundary
condition are considered. In asymptotic analysis, we focus on the case that the
{mobility} number is the same order of the Cahn number and derive the
sharp-interface limits for several setups of the boundary relaxation parameter.
It is shown that the sharp interface limit of the phase field model is the
standard two-phase incompressible Navier-Stokes equations coupled with several
different slip boundary conditions. Numerical results are consistent with the
analysis results and also illustrate the different convergence rates of the
sharp-interface limits for different scalings of the two parameters
Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units
Deep neural networks with rectified linear units (ReLU) are getting more and
more popular due to their universal representation power and successful
applications. Some theoretical progress regarding the approximation power of
deep ReLU network for functions in Sobolev space and Korobov space have
recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H.
Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this
paper, we show that deep networks with rectified power units (RePU) can give
better approximations for smooth functions than deep ReLU networks. Our
analysis bases on classical polynomial approximation theory and some efficient
algorithms proposed in this paper to convert polynomials into deep RePU
networks of optimal size with no approximation error. Comparing to the results
on ReLU networks, the sizes of RePU networks required to approximate functions
in Sobolev space and Korobov space with an error tolerance , by
our constructive proofs, are in general
times smaller than the sizes of
corresponding ReLU networks constructed in most of the existing literature.
Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math.
1:61-80, 1993], our constructions use less number of activation functions and
numerically more stable, they can be served as good initials of deep RePU
networks and further trained to break the limit of linear approximation theory.
The functions represented by RePU networks are smooth functions, so they
naturally fit in the places where derivatives are involved in the loss
function.Comment: 28 pages, 4 figure
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