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Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation
Arvind Baskaran, John Lowengrub, Cheng Wang, Steve Wise
TL;DR
The paper addresses the missing convergence analysis for a second-order convex-splitting finite-difference scheme for the hyperbolic MPFC equation. It uses higher-order asymptotic consistency analysis to handle the ψ-variable formulation and establishes second-order convergence in time and space in a discrete L∞(0,T; H^3) norm.
Problem
A detailed convergence analysis was missing for the second-order convex-splitting scheme for MPFC, whose hyperbolic formulation introduces a potentially accuracy-reducing O(s^2) consistency issue.
Method
The paper performs a higher-order consistency analysis by asymptotic expansion and projects the exact solution onto Fourier space to obtain optimal regularity.
Results
Second-order convergence in both time and space is established in a discrete L∞(0,T; H^3) norm.
Takeaways & Limitations
The analysis validates second-order accuracy for an unconditionally energy stable, semi-implicit finite-difference scheme based on second-order convex splitting of a discrete pseudo-energy.
Abstract
from arXiv · showhide
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 [2]. An introduction of a new variable ψ, corresponding to the temporal derivative of the phase variable φ, 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.
1. Introduction.
The paper develops a detailed convergence analysis for a second-order convex-splitting scheme for MPFC, extending prior stability and solvability results. Its analysis addresses hyperbolic consistency difficulties and establishes full second-order convergence in a discrete L∞(0,T; H^3) norm.
- Model and context: MPFC is a generalized damped wave equation, with the usual PFC equation recovered when β = 0.The paper restricts its analysis to β > 0 to avoid degeneracy.
- Research gap: The second-order convex-splitting scheme had previously been devised with unconditional solvability and energy-stability results, but without convergence analysis.The present paper fills that analysis gap for the scheme proposed in [2].
- Energy structure: The MPFC solution dissipates a pseudo-energy, motivating schemes that mimic this non-increasing quantity.The energy itself is not necessarily non-increasing along solution trajectories, whereas the pseudo-energy is.
- Consistency challenge: Introducing ψ as the temporal derivative of φ facilitates implementation but creates an O(s^2) discrepancy between centered φ differences and midpoint ψ averages.Because of the second-order time derivative, this discrepancy can appear to reduce temporal accuracy unless handled carefully.
- Analysis strategy: A higher-order asymptotic consistency analysis, together with Fourier projection for optimal regularity, overcomes the apparent accuracy reduction.The error analysis also uses discrete Sobolev inequalities and the discrete time derivative of the error equation.
- Main result: Full second-order convergence is established in a discrete L∞(0,T; H^3) norm.The paper presents the finite-difference scheme, convergence analysis, and technical consistency details in its main sections and appendices.
2. The Second-Order Scheme and its Properties.
The scheme uses a convex splitting of a discrete energy and retains mass conservation, unique solvability, and unconditional energy stability. The analysis establishes estimates for sufficiently regular periodic solutions, with constants independent of discretization parameters in stated results.
- Discrete energy and scheme: The fully discrete second-order scheme is built from a convex splitting F = Fc − Fe of the discrete energy.The convexity structure is used to support numerical stability and solvability.
- Discrete energy and scheme: The scheme can be written in an equivalent decoupled form: solve for φk+1 first, then update ψk+1.Its solvability therefore rests on the equation determining φk+1.
- Mass conservation and solvability: The scheme is uniquely solvable for every time-step size s > 0 and conserves mass.These properties are stated as unconditional results for the second-order MPFC scheme.
- Mass conservation and solvability: The second-order MPFC scheme is unconditionally energy stable under the stated periodic-solution assumptions.The stability result is formulated using a third fully discrete energy introduced for each time step.
- A priori estimates: For sufficiently regular periodic solutions with zero initial time derivative, the analysis gives estimates whose constants are independent of s and h.The stated theorem assumes ∂tΦ(x, y, 0) = 0 and includes constants independent of T in one estimate.
3. Error Estimate for the Second Order Scheme.
The convergence proof addresses nonlinear hyperbolic error terms through projection, asymptotic consistency correction, and discrete energy estimates. It concludes with global error bounds obtained by combining numerical-versus-projection estimates with Fourier approximation estimates.
- Error estimates: The nonlinear error requires careful decomposition and discrete Sobolev, Hölder, Green’s identity, and Cauchy estimates.The analysis separately controls convex diffusion, concave diffusion, and nonlinear contributions.
- Proof strategy: The proof proceeds in three steps: local truncation error for a finite Fourier projection, numerical-versus-projection error, and a global estimate by the triangle inequality.This separates consistency, discrete error stability, and projection approximation.
- Consistency construction: An O(s2) correction term for ΨN, derived from an asymptotic expansion, provides higher-order consistency needed for stability and convergence.The Fourier projection avoids aliasing error, while ΨN remains an O(s2) approximation to Ψ.
- Error estimates: The error analysis introduces modified energies and combines one-step estimates before summing over time and applying a discrete Gronwall inequality.The resulting constants may depend on T and the exact solution but are independent of h and s where stated.
- Global convergence: The final estimates are obtained by combining the discrete H3 error bound for ˜φ with the Fourier projection approximation estimate.The projection-to-exact-solution comparison uses the spectral accuracy of ΦN.
4. Conclusions.
The paper establishes convergence for an unconditionally energy stable, second-order accurate finite difference scheme for the sixth-order MPFC equation.
- The scheme uses a second-order convex splitting of a discrete pseudo-energy and is semi-implicit.
- The parabolic PFC equation is recovered as a special case of the MPFC equation.
Appendix A. Tools for Cell-Centered Finite Differences.
Appendix A develops the cell-centered finite-difference framework used to analyze the scheme, including grid functions, operators, inner products, norms, inequalities, and periodicity.
- The framework is formulated on a two-dimensional rectangular domain with cell-centered grid functions and spatial step size h.The framework has a straightforward extension to three space dimensions.
- Weighted grid inner products are introduced for cell-centered and edge-centered functions, together with one-dimensional inner products.
- The appendix defines edge-to-center difference operators, center-to-edge averages and differences, and associated periodicity relations.These operators support the summation-by-parts identities used in the analysis.
- Discrete L2, Sobolev-type, L4, and L∞ norms are defined for grid functions.
- Discrete Sobolev-type inequalities and negative-norm structures are supplied for later estimates.The appendix states that a bilinear form defines an inner product on the relevant space and that an associated expression defines a norm.
Appendix B. Consistency Analysis of the Second Order Numerical Scheme.
Appendix B derives the local truncation error estimate for the second-order scheme using vertex-centered grid functions to simplify indexing.
- The appendix gives a detailed derivation of the local truncation error estimate (3.25).
- The derivation is carried out for vertex-centered rather than cell-centered grid functions because the indexing is simpler.
B.1. Proof of Estimate (3.25).
The proof of estimate (3.25) combines temporal Taylor expansions, spatial truncation bounds, projection estimates, discrete Fourier analysis, and nonlinear estimates. An O(s2) correction to ΨN is used to preserve higher-order consistency.
- Temporal consistency estimates use integral Taylor expansions, while spatial discretization contributes an O(h2) truncation error bound.
- The projection solution ΦN is chosen in BN/2 so centered-difference approximation avoids an aliasing error.The consistency argument compares discrete Fourier coefficients of the relevant projected quantities.
- An O(s2) correction in ΨN produces higher-order consistency between ΨN at tk+1/2 and the centered time difference of ΦN.This addresses the apparent temporal-accuracy reduction associated with introducing ψ.
- The analysis estimates first- and second-order time derivatives, convex and concave diffusion terms, and nonlinear terms separately before combining them.The second-order derivative analysis uses the corrected approximate solution ΨN.
- The local truncation error estimate for τ1 follows by combining the component estimates with comparisons between the truncation equations and the original PDE.The associated constant estimate for M is also verified.