Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn-Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end. ]. However, in the case of a phase field description, when the phase variable represents the volume fraction of a material component, this model does not warrant the conservation of the volume of that component. In order to preserve the volume, the free energy functional has to be augmented by a volume preserving mechanism with a penalizing potential term or a Lagrange multiplier [20] [24]. This often results in a nonlocal term in the modified Allen-Cahn equation. In this paper, we call these nonlocal Allen-Cahn equations or Allen-Cahn equations with nonlocal constraints.The Cahn-Hilliard equation is an alternative model for the gradient flow. One feature of the Cahn-Hilliard equation is its volume preserving property. Rubinstein and Sternberg studied the Allen-Cahn model with the volume constraint analytically and compared it with the Cahn-Hilliard model [24]. Their result favors using the Allen-Cahn model with a volume constraint in place of the Cahn-Hilliard model when studying interfacial dynamics of incompressible, immiscible multi-component material systems. For the Allen-Cahn equation as well as the Cahn-Hilliard equation, there have been several popular numerical approaches to construct energy stable schemes for the equations, including the convex splitting approach [11-13, 18, 25, 28, 30] , the stabilizing approach [6, 10, 27], the energy quadratization (EQ) approach [14,15,33,39] and the scalar auxiliary variable approach, which is a special form of EQ method, [9,26,29]. Recently, the energy quadratization (EQ) and its reincarnation in the scalar auxiliary variable (SAV) method have been applied to a host of thermodynamical and hydrodynamic models owing to their simplicity, ease of implementation, computational efficiency, linearity, and most importantly their energy stabi...

Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints

“…Both Figure 5.2 and Figure 5.4 show that the free energy computed by the nonlocal Allen-Cahn models reach the steady state faster than that of the Cahn-Hillard model, which is different from our previous results on a different free energy functional [26]. Especially, the simulations computed by the models with mobility coefficient M = 1 and time steps 1 × 10 −2 , 1 × 10 −4 show the same time evolution behavior.…”

Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

“…3 is a linear, second order in time, unconditionally energy stable and the linear system resulting from the scheme is uniquely solvable [12]. It then follows that the norms of solutions of the linear system resulting from Scheme 3.3: φ n L 2 , r n L 2 , ζ n L 2 (n = 1, 2, .…”

“…This scheme is linear, second order in time, and the linear system resulting from it is uniquely solvable. The scheme obeys a discrete dissipation law, i.e., Scheme 3.1 is unconditionally energy stable [12], which implies F n ≤ F 0 , namely, φ n L 2 , q n L 2 , ζ n L 2 (n = 1, 2, . .…”

“…Scheme 3.4 is a linear, second order in time, and unconditionally energy stable [12]. The linear system resulting from the scheme is uniquely solvable.…”

“…In particular, we design the schemes so that they are uniquely solvable and unconditionally energy stable. We present an extended class of schemes, numerical strategies, numerical implementation issues as well as performance tests of the models in another paper [12]. In this one, we focus exclusively on error estimates for the four schemes.…”