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1.
A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters 下载免费PDF全文
Lixiu Dong Cheng Wang Hui Zhang & Zhengru Zhang 《Communications In Computational Physics》2020,28(3):967-998
We present and analyze a new second-order finite difference scheme for
the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation
combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level.
For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and
deal with all logarithmic terms directly by using the property that the nonlinear error
inner product is always non-negative. Moreover, we present the detailed convergent
analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$) norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various
numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties. 相似文献
2.
In this paper we propose and analyze a (temporally) third order accurate
backward differentiation formula (BDF) numerical scheme for the no-slope-selection
(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral
discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont
regularization term, in the form of $−A∆t^2∆^2_N
(u^{n+1}−u^n)$, is added in the numerical
scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the
optimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the
efficiency of the numerical scheme and the third order convergence. The long time
simulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law for
the energy decay, as well as the power laws for growth of the surface roughness and
the mound width. In particular, the power index for the surface roughness and the
mound width growth, created by the third order numerical scheme, is more accurate
than those produced by certain second order energy stable schemes in the existing
literature. 相似文献
3.
Decoupled,Energy Stable Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System with Logarithmic Flory-Huggins Potential 下载免费PDF全文
Hong-En Jia Ya-Yu Guo Ming Li Yunqing Huang & Guo-Rui Feng 《Communications In Computational Physics》2020,27(4):1053-1075
In this paper, a decoupling numerical method for solving Cahn-Hilliard-Hele-Shaw system with logarithmic potential is proposed. Combing with a convex-splitting of the energy functional, the discretization of the Cahn-Hilliard equation in
time is presented. The nonlinear term in Cahn-Hilliard equation is decoupled from
the pressure gradient by using a fractional step method. Therefore, to update the pressure, we just need to solve a Possion equation at each time step by using an incremental
pressure-correction technique for the pressure gradient in Darcy equation. For logarithmic potential, we use the regularization procedure, which make the domain for
the regularized functional $F$($ф$) is extended from (−1,1) to (−∞,∞). Further, the stability and the error estimate of the proposed method are proved. Finally, a series of
numerical experiments are implemented to illustrate the theoretical analysis. 相似文献
4.
An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes 下载免费PDF全文
Christophe Chalons Mathieu Girardin & Samuel Kokh 《Communications In Computational Physics》2016,20(1):188-233
We propose an all regime Lagrange-Projection like numerical scheme for the
gas dynamics equations. By all regime, we mean that the numerical scheme is able to
compute accurate approximate solutions with an under-resolved discretization with
respect to the Mach number M, i.e. such that the ratio between the Mach number M
and the mesh size or the time step is small with respect to 1. The key idea is to decouple
acoustic and transport phenomenon and then alter the numerical flux in the
acoustic approximation to obtain a uniform truncation error in term of M. This modified
scheme is conservative and endowed with good stability properties with respect
to the positivity of the density and the internal energy. A discrete entropy inequality
under a condition on the modification is obtained thanks to a reinterpretation of the
modified scheme in the Harten Lax and van Leer formalism. A natural extension to
multi-dimensional problems discretized over unstructured mesh is proposed. Then
a simple and efficient semi-implicit scheme is also proposed. The resulting scheme
is stable under a CFL condition driven by the (slow) material waves and not by the
(fast) acoustic waves and so verifies the all regime property. Numerical evidences are
proposed and show the ability of the scheme to deal with tests where the flow regime
may vary from low to high Mach values. 相似文献
5.
E. Abreu J. Douglas F. Furtado & F. Pereira 《Communications In Computational Physics》2009,6(1):72-84
We describe an operator splitting technique based on physics rather than
on dimension for the numerical solution of a nonlinear system of partial differential
equations which models three-phase flow through heterogeneous porous media. The
model for three-phase flow considered in this work takes into account capillary forces,
general relations for the relative permeability functions and variable porosity and permeability
fields. In our numerical procedure a high resolution, nonoscillatory, second
order, conservative central difference scheme is used for the approximation of the nonlinear
system of hyperbolic conservation laws modeling the convective transport of the
fluid phases. This scheme is combined with locally conservative mixed finite elements
for the numerical solution of the parabolic and elliptic problems associated with the
diffusive transport of fluid phases and the pressure-velocity problem. This numerical
procedure has been used to investigate the existence and stability of nonclassical shock
waves (called transitional or undercompressive shock waves) in two-dimensional heterogeneous
flows, thereby extending previous results for one-dimensional flow problems.
Numerical experiments indicate that the operator splitting technique discussed
here leads to computational efficiency and accurate numerical results. 相似文献
6.
Jiwei Zhang Zhizhong Sun Xiaonan Wu & Desheng Wang 《Communications In Computational Physics》2011,10(3):742-766
The paper is concerned with the numerical solution of Schrödinger equations
on an unbounded spatial domain. High-order absorbing boundary conditions
for one-dimensional domain are derived, and the stability of the reduced initial boundary
value problem in the computational interval is proved by energy estimate. Then a
second order finite difference scheme is proposed, and the convergence of the scheme
is established as well. Finally, numerical examples are reported to confirm our error
estimates of the numerical methods. 相似文献
7.
Fully Decoupled,Linear and Unconditionally Energy Stable Schemes for the Binary Fluid-Surfactant Model 下载免费PDF全文
Yuzhe Qin Zhen Xu Hui Zhang & Zhengru Zhang 《Communications In Computational Physics》2020,28(4):1389-1414
Here, we develop a first and a second order time stepping schemes for a binary fluid-surfactant phase field model by using the scalar auxiliary variable approach.
The free energy contains a double-well potential, a nonlinear coupling entropy and a
Flory-Huggins potential. The resulting coupled system consists of a Cahn-Hilliard
type equation and a Wasserstein type equation which leads to a degenerate problem.
By introducing only one scalar auxiliary variable, the system is transformed into an
equivalent form so that the nonlinear terms can be treated semi-explicitly. Both the
schemes are linear and decoupled, thus they can be solved efficiently. We further prove
that these semi-discretized schemes in time are unconditionally energy stable. Some
numerical experiments are performed to validate the accuracy and energy stability of
the proposed schemes. 相似文献
8.
A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations 下载免费PDF全文
In this paper, we propose, analyze, and numerically validate an adaptive
finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive
method is based on a linear, decoupled scheme introduced by Shen and Yang [30].
An unconditionally energy stable discrete law for the modified energy is shown for
the fully discrete scheme. A superconvergent cluster recovery based a posteriori error
estimations are constructed for both the phase field variable and velocity field function,
respectively. Based on the proposed space and time discretization error estimators, a
time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate
the reliability and efficiency of the proposed error estimators and the corresponding
adaptive algorithm. 相似文献
9.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes 下载免费PDF全文
Weijie Zhang Yulong Xing Yinhua Xia & Yan Xu 《Communications In Computational Physics》2022,31(3):771-815
In this paper, we propose a high-order accurate discontinuous Galerkin
(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and
provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation
on rectangular meshes, the extension to triangular meshes is conceptually plausible
but highly nontrivial. We first introduce a special way to recover the equilibrium state
and then design a group of novel variables at the interface of two adjacent cells, which
plays an important role in the well-balanced and positivity-preserving properties. One
main challenge is that the well-balanced schemes may not have the weak positivity
property. In order to achieve the well-balanced and positivity-preserving properties
simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity
property. A simple existing limiter can be applied to enforce the positivity-preserving
property, without losing high-order accuracy and conservation. Extensive one- and
two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness. 相似文献
10.
Andrea Thomann Markus Zenk Gabriella Puppo & Christian Klingenberg 《Communications In Computational Physics》2020,28(2):591-620
We present an implicit-explicit finite volume scheme for the Euler equations.
We start from the non-dimensionalised Euler equations where we split the pressure in
a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split
in an explicit part, solved using a Godunov-type scheme based on an approximate
Riemann solver, and an implicit part where we solve an elliptic equation for the fast
pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the
density and internal energy and asymptotic preserving towards the incompressible
Euler equations. For this first order scheme we give a second order extension which
maintains the positivity property. We perform numerical experiments in 1D and 2D to
show the applicability of the proposed splitting and give convergence results for the
second order extension. 相似文献
11.
An Explicit MUSCL Scheme on Staggered Grids with Kinetic-Like Fluxes for the Barotropic and Full Euler System 下载免费PDF全文
Thierry Goudon Julie Llobell & Sebastian Minjeaud 《Communications In Computational Physics》2020,27(3):672-724
We present a second order scheme for the barotropic and full Euler equations. The scheme works on staggered grids, with numerical unknowns stored at dual locations, while the numerical fluxes are derived in the spirit of kinetic schemes. We identify stability conditions ensuring the positivity of the discrete density and energy. We illustrate the ability of the scheme to capture the structure of complex flows with 1D and 2D simulations on MAC grids. 相似文献
12.
N. Anders Petersson & Bjö rn Sjö green 《Communications In Computational Physics》2012,12(1):193-225
We develop a stable finite difference approximation of the three-dimensional
viscoelastic wave equation. The material model is a super-imposition of N standard
linear solid mechanisms, which commonly is used in seismology to model a material
with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making
it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite
difference scheme for the elastic wave equation in second order formulation [SIAM J.
Numer. Anal., 45 (2007), pp. 1902–1936]. Our main result is a proof that the proposed
discretization is energy stable, even in the case of variable material properties. The
proof relies on the summation-by-parts property of the discretization. The new scheme
is implemented with grid refinement with hanging nodes on the interface. Numerical
experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used
to demonstrate how the number of viscoelastic mechanisms and the grid resolution
influence the accuracy. We find that three standard linear solid mechanisms usually
are sufficient to make the modeling error smaller than the discretization error. 相似文献
13.
A multigrid method is proposed to compute the ground state solution of
Bose-Einstein condensations by the finite element method based on the multilevel correction
for eigenvalue problems and the multigrid method for linear boundary value
problems. In this scheme, obtaining the optimal approximation for the ground state
solution of Bose-Einstein condensates includes a sequence of solutions of the linear
boundary value problems by the multigrid method on the multilevel meshes and some
solutions of nonlinear eigenvalue problems some very low dimensional finite element
space. The total computational work of this scheme can reach almost the same optimal
order as solving the corresponding linear boundary value problem. Therefore,
this type of multigrid scheme can improve the overall efficiency for the simulation of
Bose-Einstein condensations. Some numerical experiments are provided to validate
the efficiency of the proposed method. 相似文献
14.
This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the
Cahn-Hilliard model needs very long time to reach the steady state, and therefore large
time-stepping methods become useful. The main objective of this work is to construct
the unconditionally energy stable finite difference scheme so that the large time steps
can be used in the numerical simulations. The equation is discretized by the central
difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative.
An error estimate for the numerical solution is also obtained with second order in
both space and time. By using this energy stable scheme, an adaptive time-stepping
strategy is proposed, which selects time steps adaptively based on the variation of the
free energy against time. The numerical experiments are presented to demonstrate the
effectiveness of the adaptive time-stepping approach. 相似文献
15.
Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation 下载免费PDF全文
Teddy Pichard Denise Aregba-Driollet Sté phane Brull Bruno Dubroca & Martin Frank 《Communications In Computational Physics》2016,19(1):168-191
Because of stability constraints, most numerical schemes applied to hyperbolic
systems of equations turn out to be costly when the flux term is multiplied by
some very large scalar. This problem emerges with the $M_1$ system of equations in
the field of radiotherapy when considering heterogeneous media with very disparate
densities. Additionally, the flux term of the $M_1$ system is non-linear, and in order for
the model to be well-posed the numerical solution needs to fulfill conditions called
realizability. In this paper, we propose a numerical method that overcomes the stability
constraint and preserves the realizability property. For this purpose, we relax the
$M_1$ system to obtain a linear flux term. Then we extend the stencil of the difference
quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation
example. 相似文献
16.
Antonio Raudino Antonio Grassi Giuseppe Lombardo Giovanni Russo Clarissa Astuto & Mario Corti 《Communications In Computational Physics》2022,31(3):707-738
In this paper we propose a computational framework for the investigation
of the correlated motion between positive and negative ions exposed to the attraction
of a bubble surface that mimics the (oscillating) cell membrane. Specifically we aim
to investigate the role of surface traps with substances freely diffusing around the cell.
The physical system we want to model is an anchored gas drop submitted to a diffusive flow of charged surfactants (ions). When the diffusing surfactants meet the surface
of the bubble, they are reversibly adsorbed and their local concentration is accurately
measured. The correlated diffusion of surfactants is described by a Poisson-Nernst-Planck (PNP) system, in which the drift term is given by the gradient of a potential
which includes both the effect of the bubble and the Coulomb interaction between
the carriers. The latter term is obtained from the solution of a self-consistent Poisson
equation. For very short Debye lengths one can adopt the so called Quasi-Neutral
limit which drastically simplifies the system, thus allowing for much faster numerical
simulations. The paper has four main objectives. The first one is to present a PNP
model that describes ion charges in presence of a trap. The second one is to provide
benchmark tests for the validation of simplified multiscale models under current development [1]. The third one is to explore the relevance of the term describing the
interaction among the apolar tails of the anions. The last one is to quantitatively explore the validity of the Quasi-Neutral limit by comparison with detailed numerical
simulation for smaller and smaller Debye lengths. In order to reach these goals, we
propose a simple and efficient Alternate Direction Implicit method for the numerical
solution of the non-linear PNP system, which guarantees second order accuracy bothin space and time, without requiring solution of nonlinear equation at each time step.
New semi-implicit scheme for a simplified PNP system near quasi neutrality is also
proposed. 相似文献
17.
A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes 下载免费PDF全文
Yunlong Yu Yanzhong Yao Guangwei Yuan & Xingding Chen 《Communications In Computational Physics》2016,20(5):1405-1423
In this paper, a conservative parallel iteration scheme is constructed to solve
nonlinear diffusion equations on unstructured polygonal meshes. The design is based
on two main ingredients: the first is that the parallelized domain decomposition is
embedded into the nonlinear iteration; the second is that prediction and correction
steps are applied at subdomain interfaces in the parallelized domain decomposition
method. A new prediction approach is proposed to obtain an efficient conservative
parallel finite volume scheme. The numerical experiments show that our parallel
scheme is second-order accurate, unconditionally stable, conservative and has linear
parallel speed-up. 相似文献
18.
M. Holst J. A. McCammon Z. Yu Y. C. Zhou & Y. Zhu 《Communications In Computational Physics》2012,11(1):179-214
We consider the design of an effective and reliable adaptive finite element
method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first
complete solution and approximation theory for the Poisson-Boltzmann equation, the
first provably convergent discretization and also allowed for the development of a
provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this
regularization technique which can be shown to be less susceptible to such instability.
We establish a priori estimates and other basic results for the continuous regularized
problem, as well as for Galerkin finite element approximations. We show that the new
approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme
for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is
one of the first results of this type for nonlinear elliptic problems, is based on usingcontinuous and discrete a priori L∞ estimates. To provide a high-quality geometric
model as input to the AFEM algorithm, we also describe a class of feature-preserving
adaptive mesh generation algorithms designed specifically for constructing meshes of
biomolecular structures, based on the intrinsic local structure tensor of the molecular
surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages
of the new regularization scheme are demonstrated with FETK through comparisons
with the original regularization approach for a model problem. The convergence and
accuracy of the overall AFEM algorithm is also illustrated by numerical approximation
of electrostatic solvation energy for an insulin protein. 相似文献
19.
Guo-Dong Zhang & Xiaofeng Yang 《Communications In Computational Physics》2021,30(3):771-798
In this paper, we consider the numerical approximations of a magnetohy-drodynamic potential model that was developed in [15]. Several decoupled, linear, unconditionally energy stable schemes are developed by combining some subtle implicit-explicit treatments for nonlinear coupling terms and the projection-type method for the
Navier-Stokes equations. The divergence-free condition for the magnetic field is preserved in the fully-discrete level. We further establish the well-posedness and unconditional energy stabilities of the proposed schemes and present a series of numerical
examples in 3D, including accuracy/stability tests, benchmark simulations of driven
cavity flow and hydromagnetic Kelvin-Helmholtz instability. 相似文献
20.
A Numerical Scheme for the Quantum Fokker-Planck-Landau Equation Efficient in the Fluid Regime 下载免费PDF全文
We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a
scheme inevitably needs an implicit discretization of the nonlinear collision operator,
which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced by
the small Knudsen number. However, there is no suitable simple quantum operator
serving the purpose and for this kind of operators one has to solve the complicated
quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we
propose to penalize QqFPL by the "classical" linear Fokker-Planck operator. It is based
on the observation that the classical Maxwellian, with the temperature replaced by the
internal energy, has the same first five moments as the quantum Maxwellian. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the
scheme in both fluid and kinetic regimes. 相似文献