972 research outputs found
Decoupled, Energy Stable Scheme for Hydrodynamic Allen-Cahn Phase Field Moving Contact Line Model
In this paper, we present an efficient energy stable scheme to solve a phase
field model incorporating contact line condition. Instead of the usually used
Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field
model with the static contact line boundary condition that coupled with
incompressible Navier-Stokes equations with Navier boundary condition. The
projection method is used to deal with the Navier-Stokes equa- tions and an
auxiliary function is introduced for the non-convex Ginzburg-Landau bulk
potential. We show that the scheme is linear, decoupled and energy stable.
Moreover, we prove that fully discrete scheme is also energy stable. An
efficient finite element spatial discretization method is implemented to verify
the accuracy and efficiency of proposed schemes. Numerical results show that
the proposed scheme is very efficient and accurat
Second Order, linear and unconditionally energy stable schemes for a hydrodynamic model of Smectic-A Liquid Crystals
In this paper, we consider the numerical approximations for a hydrodynamical
model of smectic-A liquid crystals. The model, derived from the variational
approach of the modified Oseen-Frank energy, is a highly nonlinear system that
couples the incompressible Navier-Stokes equations and a constitutive equation
for the layer variable. We develop two linear, second-order time-marching
schemes based on the "Invariant Energy Quadratization" method for nonlinear
terms in the constitutive equation, the projection method for the Navier-Stokes
equations, and some subtle implicit-explicit treatments for the convective and
stress terms. Moreover, we prove the well-posedness of the linear system and
their unconditionally energy stabilities rigorously. Various numerical
experiments are presented to demonstrate the stability and the accuracy of the
numerical schemes in simulating the dynamics under shear flow and the magnetic
field
Linearly decoupled energy-stable numerical methods for multi-component two-phase compressible flow
In this paper, for the first time we propose two linear, decoupled,
energy-stable numerical schemes for multi-component two-phase compressible flow
with a realistic equation of state (e.g. Peng-Robinson equation of state). The
methods are constructed based on the scalar auxiliary variable (SAV) approaches
for Helmholtz free energy and the intermediate velocities that are designed to
decouple the tight relationship between velocity and molar densities. The
intermediate velocities are also involved in the discrete momentum equation to
ensure the consistency with the mass balance equations. Moreover, we propose a
component-wise SAV approach for a multi-component fluid, which requires solving
a sequence of linear, separate mass balance equations. We prove that the
methods preserve the unconditional energy-dissipation feature. Numerical
results are presented to verify the effectiveness of the proposed methods.Comment: 22 page
Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach
In this paper, we carry out stability and error analyses for two first-order,
semi-discrete time stepping schemes, which are based on the newly developed
Invariant Energy Quadratization approach, for solving the well-known
Cahn-Hilliard and Allen-Cahn equations with general nonlinear bulk potentials.
Some reasonable sufficient conditions about boundedness and continuity of the
nonlinear functional are given in order to obtain optimal error estimates.
These conditions are naturally satisfied by two commonly used nonlinear
potentials including the double-well potential and regularized logarithmic
Flory-Huggins potential. The well-posedness, unconditional energy stabilities
and optimal error estimates of the numerical schemes are proved rigorously
Efficient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard Equation using the stabilized-Scalar Augmented Variable (S-SAV) approach
In this paper, we consider numerical approximations for the anisotropic
Cahn-Hilliard equation. The main challenge of constructing numerical schemes
with unconditional energy stabilities for this model is how to design proper
temporal discretizations for the nonlinear terms with the strong anisotropy. We
propose two, second order time marching schemes by combining the recently
developed SAV approach with the linear stabilization approach, where three
linear stabilization terms are added. These terms are shown to be crucial to
remove the oscillations caused by the anisotropic coefficients, numerically.
The novelty of the proposed schemes is that all nonlinear terms can be treated
semi-explicitly, and one only needs to solve three decoupled linear equations
with constant coefficients at each time step. We further prove the
unconditional energy stabilities rigorously, and present various 2D and 3D
numerical simulations to demonstrate the stability and accuracy
Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential
The Allen--Cahn equation is one of fundamental equations of phase-field
models, while the logarithmic Flory--Huggins potential is one of the most
useful energy potentials in various phase-field models. In this paper, we
consider numerical schemes for solving the Allen--Cahn equation with
logarithmic Flory--Huggins potential. The main challenge is how to design
efficient numerical schemes that preserve the maximum principle and energy
dissipation law due to the strong nonlinearity of the energy potential
function. We propose a novel energy factorization approach with the stability
technique, which is called stabilized energy factorization approach, to deal
with the Flory--Huggins potential. One advantage of the proposed approach is
that all nonlinear terms can be treated semi-implicitly and the resultant
numerical scheme is purely linear and easy to implement. Moreover, the discrete
maximum principle and unconditional energy stability of the proposed scheme are
rigorously proved using the discrete variational principle. Numerical results
are presented to demonstrate the stability and effectiveness of the proposed
scheme
On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes Phase Field Model
We construct a numerical scheme based on the scalar auxiliary variable (SAV)
approach in time and the MAC discretization in space for the
Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and
error analysis. The scheme is linear, second-order, unconditionally energy
stable and can be implemented very efficiently. We establish second-order error
estimates both in time and space for phase field variable, chemical potential,
velocity and pressure in different discrete norms. We also provide numerical
experiments to verify our theoretical results and demonstrate the robustness
and accuracy of the our scheme
Numerical approximations for the binary Fluid-Surfactant Phase Field Model with fluid flow: Second-order, Linear, Energy stable schemes
In this paper, we consider numerical approximations of a binary
fluid-surfactant phase-field model coupled with the fluid flow, in which the
system is highly nonlinear that couples the incompressible Navier-Stokes
equations and two Cahn-Hilliard type equations. We develop two, linear and
second order time marching schemes for solving this system, by combining the
"Invariant Energy Quadratization" approach for the nonlinear potentials, the
projection method for the Navier-Stokes equation, and a subtle
implicit-explicit treatment for the stress and convective terms. We prove the
well-posedness of the linear system and its unconditional energy stability
rigorously. Various 2D and 3D numerical experiments are performed to validate
the accuracy and energy stability of the proposed schemes.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0744
A novel energy factorization approach for the diffuse-interface model with Peng-Robinson equation of state
The Peng-Robinson equation of state (PR-EoS) has become one of the most
extensively applied equations of state in chemical engineering and petroleum
industry due to its excellent accuracy in predicting the thermodynamic
properties of a wide variety of materials, especially hydrocarbons. Although
great efforts have been made to construct efficient numerical methods for the
diffuse interface models with PR-EoS, there is still not a linear numerical
scheme that can be proved to preserve the original energy dissipation law. In
order to pursue such a numerical scheme, we propose a novel energy
factorization (EF) approach, which first factorizes an energy function into a
product of several factors and then treats the factors using their properties
to obtain the semi-implicit linear schemes. We apply the EF approach to deal
with the Helmholtz free energy density determined by PR-EoS, and then propose a
linear semi-implicit numerical scheme that inherits the original energy
dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum
principle in both the time semi-discrete form and the cell-centered finite
difference fully discrete form under certain conditions. Numerical results are
presented to demonstrate the stability and efficiency of the proposed scheme.Comment: keywords: Diffuse interface model; Peng-Robinson equation of state;
Energy stability; Maximum principl
Parallel energy-stable phase field crystal simulations based on domain decomposition methods
In this paper, we present a parallel numerical algorithm for solving the
phase field crystal equation. In the algorithm, a semi-implicit finite
difference scheme is derived based on the discrete variational derivative
method. Theoretical analysis is provided to show that the scheme is
unconditionally energy stable and can achieve second-order accuracy in both
space and time. An adaptive time step strategy is adopted such that the time
step size can be flexibly controlled based on the dynamical evolution of the
problem. At each time step, a nonlinear algebraic system is constructed from
the discretization of the phase field crystal equation and solved by a domain
decomposition based, parallel Newton--Krylov--Schwarz method with improved
boundary conditions for subdomain problems. Numerical experiments with several
two and three dimensional test cases show that the proposed algorithm is
second-order accurate in both space and time, energy stable with large time
steps, and highly scalable to over ten thousands processor cores on the Sunway
TaihuLight supercomputer
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