Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation

Abstract

In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In * Currently at Haerbin Inst. Techn., China 1 arXiv:1208.2643v1 [math.NA] 8 Aug 2012 particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.which is significantly different from schemes we propose and analyze. In particular, theirs is not expected to be unconditionally solvable or energy stable. The authors did, however, give some evidence of the efficiency of their multigrid solver. We will conduct a similar study for our multigrid solver in this paper. The MPFC scheme in [10] is more or less the same as the first-order convex-splitting that we devised earlier in [18,19]. The first-order convex splitting scheme in our work [18,19] has two fundamental properties. It is unconditionally energy stable and unconditionally uniquely solvable. We rigorously analyzed this convex splitting scheme in [18,19], but we did not provide a practical solution strategy. That gap is filled here.In [8,21] we presented first and second-order accurate (in time) finite difference schemes for the purely parabolic PFC model, based on a convex splitting framework applied to the physical energy. The convexity splitting idea -in the context of first order (in time) convex splitting schemes -is generally credited to [7]. One of the main advances in [8] -and the more recent work in [14] -was the demonstration that the convex splitting framework can be extended to second-order (in time) methods as well, in a natural way. While the motivation in [8] was the application to the PFC model, the second-order convex splitting idea that we developed is, in fact, rather general.The main goal of this paper is to apply the first and second-order convex splitting framework to the MPFC equation. The principal challenge in doing so is that the extension to damped wave dynamics appears, at first sight, not so straightforward. This obstacle was first surmounted in [18,19], as already mentioned, where we extended the first-order convexity splitting idea for the MPFC model. Specifically, we showed the existence of a pseudo energy (different from the physical energy),...

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

“…In more detail, a convex splitting numerical scheme, which treats the terms of the variational derivative implicitly or explicitly according to whether the terms corresponding to the convex or concave parts of the energy, was formulated in [19], with a mixed finite element approximation in space. Such a numerical approach assures two mathematical properties: unique solvability and unconditional energy stability; also see the related works for various PDE systems, including the phase field crystal (PFC) equation [4,5,27,34,35,39], epitaxial thin film growth model [8,10,31,33], and others [21,22]. Moreover, for a gradient system coupled with fluid motion, the idea of convex splitting can still be applied and these distinguished mathematical properties are retained, as given by a few recent works [9,12,13,19,38].…”

Two finite difference schemes for the phase field crystal equation

“…Throughout this article, we suppose that (1.1)-(1.3) is subject to periodic boundary conditions and has smooth solution u(x, y, t) ∈ C 8,8,3 x,y,t (Ω × [0, T ]). Without loss of generality, we suppose Ω = [0, 2π] × [0, 2π].…”

“…Gomez and Nogueira [8] also presented some unconditionally energy-stable method with the truncation error of order two both in time and in space for PFC model, but did not discuss the global convergence. Galenko et al [7] and Baskaran et al [3] provided some energy stable second order nonlinear difference schemes without the convergence proof for the modified phase-field crystal (MPFC) equation. Baskaran et al [4] gave a detailed convergence analysis for the nonlinear difference scheme derived in [3].…”

“…Galenko et al [7] and Baskaran et al [3] provided some energy stable second order nonlinear difference schemes without the convergence proof for the modified phase-field crystal (MPFC) equation. Baskaran et al [4] gave a detailed convergence analysis for the nonlinear difference scheme derived in [3]. The analysis method in that paper can be easily modified to prove the convergence of the second-order scheme for the PFC scheme in Hu et al [9].…”

An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

“…The framework of convex splitting, popularized by Eyre [16], has been successfully applied to various PFC models [5,20,25,45], but its extension to the present setting is, we will see, rather complicated. Another important feature of our scheme is that we perform the spatial discretization on hexagonal mesh, which has been employed extensively in global climate models [22,40] and elsewhere [29,50].…”