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Energy Stable and Efficient Finite-Difference Nonlinear Multigrid Schemes for the Modified Phase Field Crystal Equation
Arvind Baskaran, Peng Zhou, Zhengzheng Hu, Cheng Wang, Steven M. Wise, John S. Lowengrub
TL;DR
The paper addresses the need for accurate, energy-stable solvers for the nonlinear sixth-order MPFC equation. It constructs first- and second-order finite-difference convex-splitting schemes and solves them with nonlinear multigrid methods. The schemes are unconditionally energy stable and uniquely solvable, while the multigrid solvers achieve optimal or near-optimal complexity.
Problem
MPFC requires practical numerical methods because it is a nonlinear sixth-order equation that cannot generally be solved analytically, while existing practical solution strategies were limited.
Method
The paper develops first- and fully second-order finite-difference convex-splitting schemes and nonlinear multigrid solvers for their nonlinear algebraic systems.
Results
The schemes are unconditionally pseudo-energy stable and uniquely solvable for any time-step size, and the nonlinear multigrid solvers have optimal or near-optimal complexity.
Takeaways & Limitations
The second-order scheme generally captures time evolution with greater accuracy and efficiency, while the methods support practical solidification and strained-crystal relaxation computations.
Takeaways & Limitations
The schemes use a multistep method, making adaptive time stepping more difficult; adaptive time and space stepping remain future work.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper develops energy-stable numerical methods for the MPFC equation, motivated by the need to model crystal defects and distinguish elastic relaxation from diffusion. It introduces a new second-order convex-splitting scheme alongside an existing first-order scheme and supplies practical multigrid solvers.
- The PFC model represents crystals with a continuous atomic-density field whose periodic solutions mimic the crystal lattice.
- The MPFC model extends PFC to address its failure to distinguish elastic relaxation and diffusion time scales.
- Efficient and accurate numerical methods are needed because MPFC is a sixth-order nonlinear equation that generally lacks practical analytic solutions.
- Nonlinear multigrid solvers fill the earlier practical solution gap, while numerical tests examine convergence, efficiency, solidification, and elastic relaxation.
- The paper applies first- and second-order convex splitting to MPFC, accounting for its wave-like dynamics and pseudo-energy stability.
- The new fully second-order scheme and the first-order scheme are nonlinear, mass conserving, unconditionally pseudo-energy stable, and uniquely solvable for arbitrary time and space steps.
2 The Modified Phase Field Crystal Model
The MPFC equation is formulated as a nonlinear damped-wave system driven by a chemical potential and equipped with a dissipating pseudo energy. Its energy admits a convex–concave splitting that underpins the paper’s first- and second-order schemes.
- The MPFC field φ is an atomic-density field on a periodic rectangular domain, with α defined as 1 − ϵ.
- The model uses a chemical potential µ and the equation µ = φ^3 + αφ + 2∆φ + ∆^2φ to express sixth-order nonlinear dynamics.
- The paper uses the standard MPFC form because it is common in the physics literature, although the physical energy can increase over some time intervals.
- Under the stated mean-zero condition for ψ = ∂tφ, a formal calculation shows that the pseudo energy is non-increasing.
- The energy is split as E = Ec − Ee, with Ec and Ee both convex functionals.
- This convex splitting provides the basis for constructing both the established first-order scheme and the new fully second-order scheme.
3 Finite Difference Schemes and Their Properties
The paper develops first- and second-order convex-splitting finite-difference schemes for MPFC that conserve mass, remain unconditionally energy stable, and are uniquely solvable for any time step. The schemes are nonlinear and are solved through nonlinear multigrid, with the second-order method using a two-step convex-splitting treatment.
- Scheme construction: Two finite-difference schemes are presented: a first-order convex-splitting method and a new fully second-order convex-splitting method.Both methods are designed for the MPFC equations and preserve the discrete convex-splitting structure.
- First-order scheme: The first-order scheme treats the convex energy contribution implicitly and the concave contribution explicitly, producing a mildly nonlinear scheme.After eliminating ψk+1, the remaining nonlinear equations are solved simultaneously before updating ψk+1.
- Second-order scheme: The second-order scheme uses an implicit second-order secant treatment for the convex contribution and explicit second-order extrapolation for the concave contribution.This produces a nonlinear two-step method whose coupled equations are solved by nonlinear multigrid.
- Mass conservation: Both schemes conserve discrete mass for any time step s > 0, provided the respective scheme has a solution.The second-order mass-conservation proof uses summation-by-parts and the initial condition ψ0 ≡ 0.
- Unique solvability: Both schemes are uniquely solvable for every time step s > 0 through strictly convex, coercive functional formulations.For the second-order scheme, minimizing the strictly convex functional G over the admissible set is equivalent to solving the scheme.
- Energy stability: The first- and second-order schemes are unconditionally pseudo-energy stable, with the second-order result holding for any s > 0 and h > 0.The stability statements concern the pseudo energy for the first-order scheme and the discrete energy for the second-order scheme.
4 Numerical Results
Numerical tests provide evidence that the convex-splitting schemes converge with their predicted spatial and temporal orders, remain energy stable, and support efficient multigrid solution. Applications demonstrate the framework's practical utility for crystallization and strained-crystal relaxation.
- 4.1 Convergence Tests: The refinement paths are chosen to verify the predicted O(h2) global error and are not CFL-type stability constraints because the schemes are unconditionally stable.The first-order scheme uses s = 0.025h2, while the second-order scheme uses s = 0.05h.
- 4.1 Convergence Tests: Both schemes are second-order accurate in space; the first-order scheme is first-order accurate in time, while the second-order scheme is second-order accurate in time.The observed error forms are E1(tf) = O(s) + O(h2) and E2(tf) = O(s2) + O(h2), respectively.
- 4.2 Multigrid Efficiency Test: At fixed s = 10.0 across increasingly fine grids, multigrid reduces the algebraic iteration error by roughly the same amount, indicating optimal or nearly optimal complexity.The reported error is the algebraic iteration error of the implicit scheme, not the discretization error.
- 4.3 Energy Stability Test: The pseudo energy is the quantity guaranteed to be non-increasing, whereas the physical energy may increase, especially in the wave-emphasizing small-β test.For β = 0.01, the physical energy increased while the pseudo energy was observed to decrease; the physical and pseudo energies were nearly identical.
- 4.5 Crystallization in an Undercooled Melt: The crystallization tests show grain nucleation, growth, grain-boundary formation, and point defects as differently oriented grains impinge.The application also compares damping-dependent crystal growth and microstructure evolution.
- 4.6 Effect of Applied Strain on Coherent Crystal: In the strained-crystal application, low damping produces oscillatory but rapid relaxation, while high damping yields slower, PFC-like relaxation without oscillatory overshoot.For β = 0.2, the strain rapidly relaxes with oscillations; for β = 9, relaxation is slow and lacks the observed overshoot.
5 Concluding Remarks
The paper presents first- and second-order unconditionally pseudo-energy-stable finite-difference methods for MPFC, with the second-order method achieving simultaneous energy stability and unique solvability. Numerical tests support accuracy, convergence, energy stability, and optimal or near-optimal nonlinear multigrid complexity, while adaptive stepping remains future work.
- Methods: The two MPFC methods are unconditionally pseudo-energy stable finite-difference schemes, with one first-order and one second-order accurate in time.The second-order method treats the second-order time derivative by introducing ψ := ∂tφ.
- Numerical results: Numerical evidence supports the schemes’ accuracy, convergence, efficiency, and pseudo-energy stability, while nonlinear FAS multigrid achieves optimal or near-optimal complexity.The second-order scheme generally captures time evolution more accurately and efficiently than the first-order scheme.
- Applications: The paper demonstrates practical applications in solidification and elastic relaxation in a strained crystal.
- Significance: The second-order convex splitting scheme simultaneously provides unconditional pseudo-energy stability and unconditional unique solvability.This combination distinguishes it from approaches that typically sacrifice solvability or stability.
- Limitation: The second-order scheme uses a multistep method, making adaptive time stepping more difficult; adaptive time and space stepping remain future research directions.
A Finite Difference Discretization on a Staggered Grid
The discretization is built on two-dimensional cell-centered and edge-centered grid functions, standard difference operators, and periodic or homogeneous Neumann boundary conditions. Summation-by-parts and discrete Green identities provide the supporting discrete calculus.
- Grid and function spaces: The framework uses a two-dimensional domain partitioned with Lx = m·h and Ly = n·h, together with cell-centered and edge-centered grid functions.
- Difference operators: The staggered-grid operators include edge-to-center differences, center-to-edge averages and differences, and the standard two-dimensional discrete Laplacian.
- Boundary conditions: The schemes allow periodic, homogeneous Neumann, or appropriate combinations of these boundary conditions.
- Discrete analysis: The appendix defines weighted grid inner products, grid-function norms, and function spaces used in the analysis.
- Discrete identities: Summation-by-parts and discrete Green identities hold, with boundary sums vanishing under periodic or homogeneous Neumann conditions.
B Nonlinear Multigrid Solvers
The appendix supplies implementation details for nonlinear multigrid time stepping of the proposed semi-implicit schemes, extending earlier first-order work that lacked implementation and simulations.
- Multigrid solvers: The appendix details nonlinear multigrid algorithms for time stepping the semi-implicit schemes.The first-order convex splitting scheme had previously been presented without an implementation strategy or simulations.
B.1 Multigrid Method for the First-Order Convex Splitting Scheme
The first-order convex splitting scheme is reduced to a nonlinear grid system whose coupled variables can be solved efficiently with a nonlinear FAS multigrid method. Local red-black Gauss–Seidel smoothing handles the remaining nonlinear algebra.
- Nonlinear system: Given φk and ψk, the first-order scheme requires solving for φ, ψ, μ, and κ on the grid.
- Nonlinear system: The first three equations decouple from the fourth, so φ, μ, and κ are solved first and ψ is updated afterward.
- Nonlinear formulation: The coupled equations are written as a nonlinear operator equation N(u) = S with u = (φ, μ, κ)^T.
- FAS multigrid: A nonlinear FAS multigrid method solves the operator equation efficiently using a standard V-cycle framework.
- Smoothing: Nonlinear red-black Gauss–Seidel smoothing uses local linearization because the operator’s nonlinearity comes from the cubic term φ^3_ij.The implementation may use either local Newton-type or simpler local Picard-type linearization.
- Smoothing: Each smoothing pass visits every grid point once, and smoothing ends after a prescribed maximum number of full passes.
B.2 Multigrid Method for the Second-Order Convex Splitting Scheme
The second-order convex-splitting scheme is formulated as a nonlinear system and solved through a nonlinear Gauss–Seidel smoothing procedure within the multigrid method.
- Nonlinear system: The scheme seeks periodic grid functions φ, ψ, µ, and κ from previously known solution values.The second-order formulation takes φk, ψk, and φk−1 as given.
- Solution update: The implementation first solves for φ, µ, and κ, then updates ψ.
- Nonlinear operator: The nonlinear operator N(u) is organized into three component functions.
- Source term: The source term S is likewise defined component-wise for the nonlinear system.
- Smoothing: Nonlinear Gauss–Seidel smoothing visits grid points using local updates, with a complete smoothing operation comprising ℓmax full passes.One full pass visits every grid point exactly once.
C Tables
The convergence tables confirm the expected spatial accuracy under refinement paths that balance temporal and spatial discretization errors for both convex-splitting schemes.
- First-order scheme: O(h2) global error is confirmed for the first-order scheme along the refinement path s = 0.025h2.The predicted error is O(s) + O(h2) = O(h2).
- Second-order scheme: O(h2) global error is confirmed for the second-order scheme along the refinement path s = 0.05h.The predicted error is O(s2) + O(h2) = O(h2).
D Figures
The figures test multigrid convergence, energy behavior, time-step effects, accuracy, polycrystal growth, and crystal response under shear and damping variations.
- Multigrid convergence: The residual norm decreases by about a factor of 5.5 per multigrid V-cycle independently of h.The test uses the second-order scheme with s = 10 and reports residuals from the final of 20 time steps.
- Energy stability: The modified energy eF is guaranteed to be non-increasing, whereas the physical energy F may increase and is nearly identical to eF in this test.
- Time-step effects: The second-order scheme compares density-field evolution at time steps s = 0.01, 1, 10, and 20 while matching plots within columns by pseudo energy F.
- Pseudo-energy evolution: The scaled pseudo energy is plotted over time for the fully second-order evolution using s = 0.01.
- Time-step comparison: The first-order scheme is evaluated at the same four time steps, with corresponding columns matched by pseudo energy F for comparison with the second-order scheme.
- Accuracy: The scaled solution difference D(s) is shown on a log-log plot against the baseline time step s1 = 0.01 at tf = 350.
- Polycrystal growth: The polycrystal density field is shown during freezing at t = 500, 1000, 2000, and 3000 for β = 0.9.
- Damping comparison: A second polycrystal simulation at β = 10 shows the density field at t = 3000 for comparison with β = 0.9.