Energy Stable Finite Difference Scheme Research Articles

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

Linear and fully decoupled scheme for a hydrodynamics coupled phase-field surfactant system based on a multiple auxiliary variables approach

We propose a linear, fully decoupled, and energy stable finite difference scheme for solving a phase-field surfactant fluid system. Inspired by the idea of multiple scalar auxiliary variables (MSAV) approach, two scalar auxiliary variables are used to transform the original governing equations into their equivalent forms. Based on the equivalent system, a highly efficient scheme can be developed. In one time cycle, the proposed algorithm can be efficiently performed, i.e., the surfactant ψ is explicitly updated, then the phase-field function ϕ, velocity field u, and pressure field p can be computed by solving linear systems with constant coefficients. The energy dissipation law for a modified energy can be estimated by using the proposed method. Various computational simulations confirm that the proposed method is not only accurate and energy stable but also works well for simulating surfactant-laden droplet dynamics.

A fractional phase-field model using an infinitesimal generator of α stable Lévy process

We study a space-fractional Allen-Cahn phase-field model in which the fractional Laplacian is defined to be an infinitesimal generator of an α-stable Lévy process with 0<α<2. Compared with the Allen-Cahn model using the Riemann-Liouville (R-L) fractional derivatives in the literature, the current model enjoys the following advantages: (1) it can control the sharpness and the decaying behavior of the interface via the tunable order parameter α as the Riemann-Liouville fractional model does; (2) it generates numerical solutions with sharper interfaces than the Riemann-Liouville fractional Allen-Cahn model with the same 1<α<2, and it can generate numerical solutions with even sharper interfaces as it also applies to the case for 0<α<1; (3) as α→2−, it provides good approximations to the classical Allen-Cahn model as the fractional Riemann-Liouville model does; (4) it is rotational invariant; and (5) it has a much slower mass decaying rate than the Riemann-Liouville fractional model. We also propose and analyze a first order in time and second order in space energy stable finite difference scheme.

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

In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

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.

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

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 particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

An Adaptive Time-Stepping Strategy for the Molecular Beam Epitaxy Models

This paper is concerned with the numerical simulations for the dynamics of the molecular beam epitaxy (MBE) model. The numerical simulations of the MBE models require long time computations, and therefore large time-stepping methods become necessary. In this work, we consider some unconditionally energy stable finite difference schemes, which will be used in the time adaptivity strategies. It is found that the use of the time adaptivity cannot only resolve the steady-state solutions but also the dynamical changes of the solution accurately and efficiently. The adaptive time step is selected based on the energy variation or the change of the roughness of the solution. The numerical experiments demonstrated that the CPU time is significantly saved for long time simulations.

An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation

We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.