共查询到20条相似文献,搜索用时 31 毫秒
1.
Craig Collins Jie Shen & Steven M. Wise 《Communications In Computational Physics》2013,13(4):929-957
We present an unconditionally energy stable and uniquely solvable finite
difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised
of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model
and describes two phase very viscous flows in porous media. The scheme is based on
a convex splitting of the discrete CH energy and is semi-implicit. The equations at the
implicit time level are nonlinear, but we prove that they represent the gradient of a
strictly convex functional and are therefore uniquely solvable, regardless of time step
size. Owing to energy stability, we show that the scheme is stable in the time and space
discrete$ℓ^∞$(0,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard
part, and a method based on the Vanka smoothing strategy for the Brinkman part – for
solving these equations. In particular, we provide evidence that the solver has nearly
optimal complexity in typical situations. The solver is applied to simulate spinodal
decomposition of a viscous fluid in a porous medium, as well as to the more general
problems of buoyancy- and boundary-driven flows. 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.
Wenbin Chen Jianyu Jing Cheng Wang Xiaoming Wang & Steven M. Wise 《Communications In Computational Physics》2022,31(1):60-93
In this paper we propose and analyze a second order accurate numerical
scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,
which ensures the positivity-preserving property, i.e., the numerical value of the phase
variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form
of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized
stability analysis. A few numerical results, including both the constant-mobility and
solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme. 相似文献
4.
An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems 下载免费PDF全文
Ye Li 《Communications In Computational Physics》2016,19(2):442-472
In this paper, we propose a uniformly convergent adaptive finite element
method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed
nonlinear eigenvalue problems under constraints with applications in Bose-Einstein
condensation (BEC) and quantum chemistry. We begin with the time-independent
Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed
nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations
for the problem are reviewed to confirm the asymptotic behaviors of the solutions
in the boundary/interior layer regions. By using the normalized gradient flow, we
propose an adaptive finite element with hybrid basis to solve the singularly perturbed
nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively
to the small parameter ε. Extensive numerical results are reported to show the
uniform convergence property of our method. We also apply the AFEM-HB to compute
the ground and excited states of BEC with box/harmonic/optical lattice potential
in the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis of
our AFEM-HB to a simpler singularly perturbed two point boundary value problem,
show that our method has a minimum uniform convergence order $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$). 相似文献
5.
H. Yan K. Li R. Khatiwada E. Smith W. M. Snow C. B. Fu P.-H. Chu H. Gao & W. Zheng 《Communications In Computational Physics》2014,15(5):1343-1351
We present a high precision frequency determination method for digitized
NMR FID signals. The method employs high precision numerical integration rather
than simple summation as in many other techniques. With no independent knowledge of the other parameters of a NMR FID signal (phase $ϕ$, amplitude $A$, and transverse relaxation time $T_2$) this method can determine the signal frequency $f_0$ with a
precision of 1/(8$π^2$$f^2_0$$T^2_2$) if the observation time $T$ ≫$T_2$. The method is especially
convenient when the detailed shape of the observed FT NMR spectrum is not well
defined. When $T_2$ is +∞ and the signal becomes pure sinusoidal, the precision of the
method is 3/(2$π^2$$f^2_0$$T_2$) which is one order more precise than the ±1 count error induced precision of a typical frequency counter. Analysis of this method shows that the
integration reduces the noise by bandwidth narrowing as in a lock-in amplifier, and
no extra signal filters are needed. For a pure sinusoidal signal we find from numerical
simulations that the noise-induced error in this method reaches the Cramer-Rao Lower
Band (CRLB) on frequency determination. For the damped sinusoidal case of most interest, the noise-induced error is found to be within a factor of 2 of CRLB when the
measurement time $T$ is 2 or 3 times larger than $T_2$. We discuss possible improvements
for the precision of this method. 相似文献
6.
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
7.
Chaoyu Quan Tao Tang Boyi Wang & Jiang Yang 《Communications In Computational Physics》2023,33(4):962-991
In this article, we study the energy dissipation property of time-fractional
Allen–Cahn equation. On the continuous level, we propose an upper bound of energy
that decreases with respect to time and coincides with the original energy at $t=0$ and
as $t$ tends to $ ∞.$ This upper bound can also be viewed as a nonlocal-in-time modified
energy which is the summation of the original energy and an accumulation term due
to the memory effect of time-fractional derivative. In particular, the decrease of the
modified energy indicates that the original energy indeed decays w.r.t. time in a small
neighborhood at $t=0.$ We illustrate the theory mainly with the time-fractional Allen–Cahn equation but it could also be applied to other time-fractional phase-field models
such as the Cahn–Hilliard equation. On the discrete level, the decreasing upper bound
of energy is useful for proving energy dissipation of numerical schemes. First-order L1
and second-order L2 schemes for the time-fractional Allen–Cahn equation have similar
decreasing modified energies, so that stability can be established. Some numerical
results are provided to illustrate the behavior of this modified energy and to verify our
theoretical results. 相似文献
8.
A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations 下载免费PDF全文
Huangxin Chen Amiya K. Pani & Weifeng Qiu 《Communications In Computational Physics》2022,31(5):1434-1466
In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along
mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme
can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis
of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its
analysis is further generalized to the von Kármán equations. Finally, numerical results
verifying the theoretical estimates of the proposed algorithms are also presented. 相似文献
9.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 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.
This paper generalizes the exponential Runge-Kutta asymptotic preserving
(AP) method developed in [G. Dimarco and L. Pareschi, SIAM Numer. Anal., 49 (2011),
pp. 2057–2077] to compute the multi-species Boltzmann equation. Compared to the
single species Boltzmann equation that the method was originally applied to, this
set of equation presents a new difficulty that comes from the lack of local conservation
laws due to the interaction between different species. Hence extra stiff nonlinear
source terms need to be treated properly to maintain the accuracy and the AP property.
The method we propose does not contain any nonlinear nonlocal implicit solver,
and can capture the hydrodynamic limit with time step and mesh size independent of
the Knudsen number. We prove the positivity and strong AP properties of the scheme,
which are verified by two numerical examples. 相似文献
12.
Marco Caliari & Simone Zuccher 《Communications In Computational Physics》2021,29(5):1336-1364
We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced
4th-order finite differences. This results in a free boundary approach, which can be
solved in time by the Strang splitting method. The linear part is solved by a new, fast
approximation of the action of the matrix exponential at machine precision accuracy,
while the nonlinear part can be solved exactly. Numerical results confirm existing
ones based on the Fourier pseudospectral method and point out some weaknesses of
the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the
vortical structures, of their reconnections, and of integral quantities like mass, energy,
and momentum. The free boundary approach reproduces them correctly, also in finite
subdomains, at low computational cost. We show the versatility of this method by
carrying out one- and three-dimensional simulations and by using it also in the case of
Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity. 相似文献
13.
Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System 下载免费PDF全文
Yong Zhang 《Communications In Computational Physics》2013,13(5):1357-1388
We study compact finite difference methods for the Schrödinger-Poisson
equation in a bounded domain and establish their optimal error estimates under proper
regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with mesh
size $h$ and time step $τ$. For the errors of compact finite difference approximation to
the second derivative and Poisson potential are nonlocal, thus besides the standard
energy method and mathematical induction method, the key technique in analysis is
to estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discrete
maximum principle of elliptic equation and properties of some related matrix. Also
some useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods. 相似文献
14.
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
15.
Natural Convection Heat Transfer in a Porous Cavity with Sinusoidal Temperature Distribution Using Cu/Water Nanofluid: Double MRT Lattice Boltzmann Method 下载免费PDF全文
Hasan Sajjadi Amin Amiri Delouei Rasul Mohebbi Mohsen Izadi & Sauro Succi 《Communications In Computational Physics》2021,29(1):292-318
In this study, natural convection flow in a porous cavity with sinusoidal
temperature distribution has been analyzed by a new double multi relaxation time
(MRT) Lattice Boltzmann method (LBM). We consider a copper/water nanofluid filling a porous cavity. For simulating the temperature and flow fields, D2Q5 and D2Q9
lattices are utilized respectively, and the effects of different Darcy numbers (Da) (0.001-0.1) and various Rayleigh numbers (Ra) ($10^3$-$10^5$) for porosity ($ε$) between 0.4 and 0.9
have been considered. Phase deviation ($θ$) changed from 0 to $π$ and the volume fraction
of nanoparticles (Ø) varied from 0 to 6%. The present results show a good agreement
with the previous works, thus confirming the reliability the new numerical method
proposed in this paper. It is indicated that the heat transfer rate increases at increasing
Darcy number, porosity, Rayleigh number, the volume fraction of nanoparticles and
phase deviation. However, the most sensitive parameter is the Rayleigh number. The
maximum Nusselt deviation is 10%, 32% and 33% for Ra=$10^3$, $10^4$ and $10^5$, respectively, with $ε = 0.4$ to $ε = 0.9$. It can be concluded that the effect of Darcy number on
the heat transfer rate increases at increasing Rayleigh number, yielding a maximum
enhancement of the average Nusselt number around 12% and 61% for Ra=$10^3$ and
Ra=$10^5$, respectively. 相似文献
16.
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
17.
On Invariant-Preserving Finite Difference Schemes for the Camassa-Holm Equation and the Two-Component Camassa-Holm System 下载免费PDF全文
The purpose of this paper is to develop and test novel invariant-preserving
finite difference schemes for both the Camassa-Holm (CH) equation and one of its
2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting
soliton-like peakon solutions which are characterized by a slope discontinuity
at the peak in the wave shape, and therefore suitable for modeling both short wave
breaking and long wave propagation phenomena. The proposed numerical schemes
are shown to preserve two invariants, momentum and energy, hence numerically producing
wave solutions with smaller phase error over a long time period than those
generated by other conventional methods. We first apply the scheme to the CH equation
and showcase the merits of considering such a scheme under a wide class of initial
data. We then generalize this scheme to the 2CH equation and test this scheme under
several types of initial data. 相似文献
18.
Chenguang Duan Yuling Jiao Yanming Lai Dingwei Li Xiliang Lu & Jerry Zhijian Yang 《Communications In Computational Physics》2022,31(4):1020-1048
Using deep neural networks to solve PDEs has attracted a lot of attentions
recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz
method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM
using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a
theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of
number of training samples. Technically, we derive bound on the approximation error
of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the
non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of
independent interest. 相似文献
19.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation 下载免费PDF全文
Nadir-Alexandre Messaï Sebastien Pernet & Abdesselam Bouguerra 《Communications In Computational Physics》2022,31(5):1585-1635
This study aimed to specialise a directional $\mathcal{H}^2
(\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular
equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency
problems. We introduced the necessary special optimisations to make this algorithm
more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter
tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable
for an auto-adaptive context in integral equations. 相似文献
20.
Reliability Investigation of BiCGStab and IDR Solvers for the Advection-Diffusion-Reaction Equation 下载免费PDF全文
Chris Schoutrop Jan ten Thije Boonkkamp & Jan van Dijk 《Communications In Computational Physics》2022,32(1):156-188
The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated. The resulting discretization matrices have real eigenvalues. We consider BiCGStab, IDR$(S),$ BiCGStab$(L)$ and various modifications of BiCGStab, where $S$ denotes the dimension of
the shadow space and $L$ the degree of the polynomial used in the polynomial part. Several implementations of BiCGStab exist which are equivalent in exact arithmetic, however, not in finite precision arithmetic. The modifications of BiCGStab we consider are;
choosing a random shadow vector, a reliable updating scheme, and storing the best intermediate solution. It is shown that the Local Minimal Residual algorithm, a method
similar to the "minimize residual" step of BiCGStab, can be interpreted in terms of
a time-dependent advection-diffusion-reaction equation with homogeneous Dirichlet
boundary conditions for the residual, which plays a key role in the convergence analysis. Due to the real eigenvalues, the benefit of BiCGStab$(L)$ compared to BiCGStab
is shown to be modest in numerical experiments. Non-sparse (e.g. uniform random)
shadow residual turns out to be essential for the reliability of BiCGStab. The reliable
updating scheme ensures the required tolerance is truly achieved. Keeping the best intermediate solution has no significant effect. Recommendation is to modify BiCGStab
with a random shadow residual and the reliable updating scheme, especially in the
regime of large Péclet and small Damköhler numbers. An alternative option is IDR($S$),
which outperforms BiCGStab for problems with strong advection in terms of the number of matrix-vector products. The MATLAB code used in the numerical experiments is available on GitLab: https://gitlab.com/ChrisSchoutrop/krylov-adr, a
C++ implementation of IDR$(S)$ is available in the Eigen linear algebra library: http:
//eigen.tuxfamily.org. 相似文献