Numerical Analysis of First and Second Order Unconditional Energy Stable Schemes for Nonlocal Cahn-Hilliard and Allen-Cahn Equations

This PhD dissertation concentrates on the numerical analysis of a family of fully discrete, energy stable schemes for nonlocal Cahn-Hilliard and Allen-Cahn type equations, which are integro-partial differential equations (IPDEs). These two IPDEs -- along with the evolution equation from dynamical density functional theory (DDFT), which is a generalization of the nonlocal Cahn-Hilliard equation -- are used to model a variety of physical and biological processes such as crystallization, phase transformations, and tumor growth. This dissertation advances the computational state-of-the-art related to this field in the following main contributions: (I) We propose and analyze a family of two-dimensional unconditionally energy stable schemes for these IPDEs. Specifically, we prove that the schemes are (a) uniquely solvable, independent of time and space step sizes; (b) energy stable, independent of time and space step sizes; and (c) convergent, provided the time step sizes are sufficiently small. (II) We develop a highly efficient solver for schemes we propose. These schemes are semi-implicit and contain nonlinear implicit terms, which makes numerical solutions challenging. To overcome this difficulty, a nearly-optimally efficient nonlinear multigrid method is employed. (III) Via our numerical methods, we are able to simulate crystal nucleation and growth phenomena, with arbitrary crystalline anisotropy, with properly chosen parameters for nonlocal Cahn-Hilliard equation, in a very efficient and straightforward way. To our knowledge these contributions do not exist in any form in any of the previous works in the literature

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes

On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity

A Linear Energy Stable Scheme for a Thin Film Model Without Slope Selection

We present a linear numerical scheme for a model of epitaxial thin film growth without slope selection. the PDE, which is a nonlinear, fourth-order parabolic equation, is the L2 gradient flow of the energy ∫Ω(-1/2 ln(1 + |ø|2) + ε2 2 |Ø(x)|2) dx. the idea of convex-concave decomposition of the energy functional is applied, which results in a numerical scheme that is unconditionally energy stable, i.e., energy dissipative. the particular decomposition used here places the nonlinear term in the concave part of the energy, in contrast to a previous convexity splitting scheme. as a result, the numerical scheme is fully linear at each time step and unconditionally solvable. Collocation Fourier spectral differentiation is used in the spatial discretization, and the unconditional energy stability is established in the fully discrete setting using a detailed energy estimate. We present numerical simulation results for a sequence of values ranging from 0.02 to 0.1. in particular, the long time simulations show the -log(t) decay law for the energy and the t 1/2 growth law for the surface roughness, in agreement with theoretical analysis and experimental/numerical observations in earlier works. © Springer Science+Business Media, LLC 2011

Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D

In this paper we study a distributed optimal control problem for a nonlocal convective Cahn--Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem

On the rate of convergence of Yosida approximation for rhe nonlocal Cahn-Hilliard equation

It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda}$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda$ could be linked to the discretization parameters, yielding appropriate error estimates