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1.
Cahn-Hilliard vs Singular Cahn-Hilliard equations in simulations of immiscible binary fluids
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Lizhen Chen 《Journal of Applied Analysis & Computation》2018,8(4):1050-1060
An efficient semi-implicit spectral method is implemented to solve the Cahn-Hilliard equation with a variable mobility in this paper. We compared the kinetics of bulk-diffusion-dominated and interface-diffusion-dominated coarsening in two-phase systems. As expected, the interface-diffusion-controlled coarsening evolves much slower. Also we find that the velocity field will be caused different greatly by using Singular Cahn-Hilliard equation and using Cahn-Hilliard in the simulation of immiscible binary fluids. 相似文献
2.
Numerical solutions for the viscous Cahn-Hilliard equation are considered using the crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution inH 1-norm are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13]. 相似文献
3.
Ahmad Makki 《Applications of Mathematics》2016,61(6):685-725
We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation. 相似文献
4.
Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows
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Anass Bouchriti Morgan Pierre Nour Eddine Alaa 《Journal of Applied Analysis & Computation》2020,10(5):2198-2219
We introduce a time semi-discretization of a damped wave equation by a SAV scheme with second order accuracy. The energy dissipation law is shown to hold without any restriction on the time step. We prove that any sequence generated by the scheme converges to a steady state (up to a subsequence). We notice that the steady state equation associated to the SAV scheme is a modified version of the steady state equation associated to the damped wave equation. We show that a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and we compare numerically the two steady state equations. 相似文献
5.
Finite element scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient 总被引:1,自引:0,他引:1
A finite element scheme is considered for the viscous Cahn-Hilliard equation with the nonconstant gradient energy coefficient. The scheme inherits energy decay property and mass conservation as for the classical solution. We obtain the corresponding error estimate using the extended Lax-Richtmyer equivalence theorem. 相似文献
6.
Jie Shen 《计算数学(英文版)》1990,8(3):276-288
Theoretical time step constraints of semi-implicit schemes are known to be more restrictive than should be in practice. We intend to alleviate the constraints with more smoothness assumptions on the solutions. By introducing a new scheme with modification on the treatment of the nonlinear term, we are able to prove that the scheme is unconditionally stable and convergent. Furthermore, we show that the modified scheme and the original semi-implicit one are equivalent under a weak condition on the time step and the number of space discretization points. 相似文献
7.
Summary Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation - a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.Research supported by DFG Priority Program Analysis, Modeling and Simulation of Multiscale Problems under AR234/5-2 and GA695/2-2 相似文献
8.
DGH方程作为一类重要的非线性水波方程有着许多广泛的应用前景.基于Hamilton系统的多辛理论研究了一类强色散DGH方程的数值解法,利用多辛普雷斯曼方法构造了一种典型的半隐式的多辛格式.分析了该格式的局部能量和动量守恒律误差,并给出了数值算例.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性. 相似文献
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11.
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term. 相似文献
12.
This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=um(1-u)m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility. 相似文献
13.
Analytical solutions for the viscous Cahn-Hilliard equation are considered. Existence and uniqueness of the solution are shown. The exponential decay of the solution inH 2-norm, which is an improvement of the result in Elliott and Zheng[5]. We also compare the early stages of evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation, which has been given as an open question in Novick-Cohen[8]. 相似文献
14.
The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. Like solutions of the Cahn-Hilliard equation converging to two main values during the phase separation process, the grayvalues inside the missing part of the image are oriented towards the binary states black and white. We present stability/instability results for solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. In particular we will consider the Willmore functional as a quantity to find the optimal scale of the inpainting result. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
15.
We present a mixed finite element method for the thin film epitaxy problem. Comparing to the primal formulation which requires $C^2$ elements in the discretization, the mixed formulation only needs to use $C^1$ elements, by introducing proper dual variables. The dual variable in our method is defined naturally from the nonlinear term in the equation, and its accurate approximation will be essential for understanding the long-time effect of the nonlinear term. For time-discretization, we use a backward-Euler semi-implicit scheme, which involves a convex–concave decomposition of the nonlinear term. The scheme is proved to be unconditionally stable and its convergence rate is analyzed. 相似文献
16.
《Journal of Computational and Applied Mathematics》2002,145(1):31-48
This paper deals with development and analysis of finite volume schemes for a one-dimensional nonlinear, degenerate, convection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze and compare three families of numerical schemes corresponding to explicit, semi-implicit, and implicit discretization of the diffusion term and a Godunov scheme for the advection term. L∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the schemes satisfy a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem. Results of numerical experiments using the present approach are reported. 相似文献
17.
The authors discuss the W1,p-solutions and the interior regularity of weak solutions for the Keldys-Fichera boundary value problem using the acute angle principle,the reversed Hlder inequality and the generalized poincar'e inequalities. 相似文献
18.
We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized,
in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based
on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives
the solution in an efficient and unconditionally stable way. 相似文献
19.
Daisuke Furihata 《Numerische Mathematik》2001,87(4):675-699
Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes
a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly
ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The
new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by
discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate
for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme.
Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000 相似文献
20.
Hongling Hu Xianlin Jin Dongdong He Kejia Pan Qifeng Zhang 《Numerical Methods for Partial Differential Equations》2022,38(1):4-32
In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second-order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings. 相似文献