2,820 research outputs found
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
A new class of efficient and robust energy stable schemes for gradient flows
We propose a new numerical technique to deal with nonlinear terms in gradient
flows. By introducing a scalar auxiliary variable (SAV), we construct efficient
and robust energy stable schemes for a large class of gradient flows. The SAV
approach is not restricted to specific forms of the nonlinear part of the free
energy, and only requires to solve {\it decoupled} linear equations with {\it
constant coefficients}. We use this technique to deal with several challenging
applications which can not be easily handled by existing approaches, and
present convincing numerical results to show that our schemes are not only much
more efficient and easy to implement, but can also better capture the physical
properties in these models. Based on this SAV approach, we can construct
unconditionally second-order energy stable schemes; and we can easily construct
even third or fourth order BDF schemes, although not unconditionally stable,
which are very robust in practice. In particular, when coupled with an adaptive
time stepping strategy, the SAV approach can be extremely efficient and
accurate
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
Stability and convergence analysis of A linear, fully decoupled and unconditionally energy stable scheme for magneto-hydrodynamic equations
In this paper, we consider numerical approximations for solving the nonlinear
magneto-hydrodynamical system, that couples the Navier-Stokes equations and
Maxwell equations together. A challenging issue to solve this model numerically
is about the time marching problem, i.e., how to develop suitable temporal
discretizations for the nonlinear terms in order to preserve the energy
stability at the discrete level. We solve this issue in this paper by
developing a linear, fully decoupled, first order time-stepping scheme, by
combining the projection method and some subtle implicit-explicit treatments
for nonlinear coupling terms. We further prove that the scheme is unconditional
energy stable and derive the optimal error estimates rigorously. Various
numerical experiments are implemented to demonstrate the stability and the
accuracy in simulating some benchmark simulations, including the
Kelvin-Helmholtz shear instability and the magnetic-frozen phenomenon in the
lid-driven cavity
Numerical approximation of a phase-field surfactant model with fluid flow
Modelling interfacial dynamics with soluble surfactants in a multiphase
system is a challenging task. Here, we consider the numerical approximation of
a phase-field surfactant model with fluid flow. The nonlinearly coupled model
consists of two Cahn-Hilliard-type equations and incompressible Navier-Stokes
equation. With the introduction of two auxiliary variables, the governing
system is transformed into an equivalent form, which allows the nonlinear
potentials to be treated efficiently and semi-explicitly. By certain subtle
explicit-implicit treatments to stress and convective terms, we construct first
and second-order time marching schemes, which are extremely efficient and
easy-to-implement, for the transformed governing system. At each time step, the
schemes involve solving only a sequence of linear elliptic equations, and
computations of phase-field variables, velocity and pressure are fully
decoupled. We further establish a rigorous proof of unconditional energy
stability for the first-order scheme. Numerical results in both two and three
dimensions are obtained, which demonstrate that the proposed schemes are
accurate, efficient and unconditionally energy stable. Using our schemes, we
investigate the effect of surfactants on droplet deformation and collision
under a shear flow, where the increase of surfactant concentration can enhance
droplet deformation and inhibit droplet coalescence
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
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
Diffuse-Interface Two-Phase Flow Models with Different Densities: A New Quasi-Incompressible Form and a Linear Energy-Stable Method
While various phase-field models have recently appeared for two-phase fluids
with different densities, only some are known to be thermodynamically
consistent, and practical stable schemes for their numerical simulation are
lacking. In this paper, we derive a new form of thermodynamically-consistent
quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a
two-phase flow of incompressible fluids with different densities. The
derivation is based on mixture theory by invoking the second law of
thermodynamics and Coleman-Noll procedure. We also demonstrate that our model
and some of the existing models are equivalent and we provide a unification
between them. In addition, we develop a linear and energy-stable
time-integration scheme for the derived model. Such a linearly-implicit scheme
is nontrivial, because it has to suitably deal with all nonlinear terms, in
particular those involving the density. Our proposed scheme is the first linear
method for quasi-incompressible two-phase flows with nonsolenoidal velocity
that satisfies discrete energy dissipation independent of the time-step size,
provided that the mixture density remains positive. The scheme also preserves
mass. Numerical experiments verify the suitability of the scheme for two-phase
flow applications with high density ratios using large time steps by
considering the coalescence and break-up dynamics of droplets including
pinching due to gravity.Comment: 38 page
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