Modified Phase Field Crystal Equation Research Articles

A new space-fractional modified phase field crystal equation and its numerical algorithm

In this paper, we develop a new space-fractional modified phase field crystal equation which has some similar properties including the mass conservation and energy dissipation. Then, we propose a second-order scheme based on a new Lagrange multiplier method that conserves the mass and dissipates the energy. For the new method, there are only two decoupled linear equations with constant coefficients and one nonlinear algebraic system to be solved at each time step which makes it efficient. Finally, we give some numerical experiments to verify the accuracy and stability of the proposed methods.

A linear second-order convex splitting scheme for the modified phase-field crystal equation with a strong nonlinear vacancy potential

The modified phase-field crystal (MPFC) equation was extended to the MPFC equation with a strong nonlinear vacancy potential (VMPFC) to include vacancies. In this paper, we develop a linear, second-order, and unconditionally energy stable scheme for solving the VMPFC equation by proposing a new linear convex splitting, using the Crank–Nicolson formula to the contractive part and the second-order Adams–Bashforth formula to the expansive part, and adding a second-order Douglas–Dupont regularization. The unconditional energy stability of the developed scheme is proved. By solving the VMPFC equation, vacancies which exist in real materials can be observed in numerical simulations.

Linear and unconditionally energy stable schemes for the modified phase field crystal equation

In this paper, two temporal discrete numerical schemes of the modified phase field crystal equation are presented by introducing a Lagrange multiplier U and two auxiliary functions ψ, W, which reduce the order of the original equation in time and space. For these schemes, we prove their mass conservation and energy stability. Error estimates of the BDF2 scheme are given. In the end, some numerical experiments are used to verify the correctness of our theoretical results.

A New L2-Gradient Flow-Based Fractional-in-Space Modified Phase-Field Crystal Equation and Its Mass Conservative and Energy Stable Method

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L2-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L2-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H−1-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework.

Error estimates for the Scalar Auxiliary Variable (SAV) schemes to the modified phase field crystal equation

We design first-order and second-order time-stepping schemes for the modified phase field crystal model based on the scalar auxiliary variable method in this work. The model is a nonlinear sixth-order damped wave equation that includes both elastic interactions and diffusive dynamics. Our schemes are linear and satisfy the unconditional energy stability with respect to pseudo energy. We also rigorously estimate the errors of the numerical schemes. Finally, some numerical tests are presented to validate our theoretical results.

Energy quadratization Runge–Kutta method for the modified phase field crystal equation

In this paper, we propose high order and unconditionally energy stable methods for a modified phase field crystal equation by applying the strategy of the energy quadratization Runge–Kutta methods. We transform the original model into an equivalent system with auxiliary variables and quadratic free energy. The modified system preserves the laws of mass conservation and energy dissipation with the associated energy functional. We present rigorous proofs of the mass conservation and energy dissipation properties of the proposed numerical methods and present numerical experiments conducted to demonstrate their accuracy and energy stability. Finally, we compare long-term simulations using an indicator function to characterize the pattern formation.

The convergence of two linearized finite difference schemes for the modified phase field crystal equation

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Two linearized difference schemes are presented based on the method of order reduction. One scheme is second order both in time and space, and the other scheme is convergent with the second temporal order and fourth spatial order. A theoretical analysis is carried out by the energy argument and mathematical induction. The uniqueness of the numerical solution and unconditional convergence in discrete -norm are proved rigorously. Numerical results demonstrate that the presented schemes for the modified phase field crystal equation can achieve the theoretical convergence order.

A linearly second-order, unconditionally energy stable scheme and its error estimates for the modified phase field crystal equation

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme, which is based on the Lagrange Multiplier approach and could also be viewed as a special case of the invariant energy quadratization approach, for the modified phase field crystal equation. At each time step, the proposed scheme only requires solving several linear equations. Rigorous proofs are presented to demonstrate the unique solvability, mass conservation, and unconditional energy stability of the scheme. Various numerical experiments in 2D and 3D are performed to validate the accuracy, unconditional energy stability and mass conservation of the proposed numerical strategy.

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

In this paper, we construct efficient schemes based on the scalar auxiliary variable (SAV) block-centered finite difference method for the modified phase field crystal (MPFC) equation, which is a sixth-order nonlinear damped wave equation. The schemes are linear, conserve mass and unconditionally dissipate a pseudo energy. We prove rigorously second-order error estimates in both time and space for the phase field variable in discrete norms. We also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy.

Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we adopt the “Invariant Energy Quadratization” approach. By using the first-order backward Euler, the second-order Crank–Nicolson, and the second-order BDF2 formulas, we obtain three linear and symmetric positive definite schemes. We rigorously prove their unconditional energy stabilities and implement a number of 2D and 3D numerical experiments to demonstrate the accuracy, stability, and efficiency.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

First- and second-order energy stable methods for the modified phase field crystal equation

The phase field crystal (PFC) model was extended to the modified phase field crystal (MPFC) model, which is a sixth-order nonlinear damped wave equation, to include not only diffusive dynamics but also elastic interactions. In this paper, we present temporally first- and second-order accurate methods for the MPFC equation, which are based on an appropriate splitting of the energy for the PFC equation. And we use the Fourier spectral method for the spatial discretization. The first- and second-order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the MPFC equation, respectively. Numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods.

Energy stable and convergent finite element schemes for the modified phase field crystal equation

We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.

Three dimensional structures predicted by the modified phase field crystal equation

We present the first numerical results on three dimensional structures predicted by the modified phase field crystal equation. The computations are performed using parallel algorithms based on isogeometric analysis, a generalization of the finite element method. The evolution of crystal structures to their steady equilibrium state is predicted for various atomic densities and temperatures. These steady structures are consistent with the phase diagram predicted earlier using one-mode approximations of analytical solutions to the classical parabolic phase-field crystal equation.

Robust exponential attractors for the modified phase-field crystal equation

We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponentialattractors $\mathcal{M}_\beta$ that are Hölder continuous with respect to $\beta$.

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.

Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation

In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second order accurate convex splitting 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 fully discrete, fully second order finite difference scheme in question was derived in a recent work [A. Baskaran et al., J. Comput. Phys., 250 (2013), pp. 270--292]. An introduction of a new variable $\psi$, corresponding to the temporal derivative of the phase variable $\phi$, could bring an accuracy reduction in the formal consistency estimate, because of the hyperbolic nature of the equation. A higher order consistency analysis by an asymptotic expansion is performed to overcome this difficulty. In turn, second order convergence in both time and space is established in a discrete $L^\infty (0,T; H^3)$ norm.

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.

Global Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equation

Global Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equation