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A Second Order Positivity-Preserving Scheme for the Nonlocal Cahn-Hilliard System with Variable Mobility and Logarithmic Flory-Huggins Potential

Yuanyi Sheng, Zhengru Zhang

arXiv:2609.06092v1math.NAcs.NA

TL;DR

Variable mobility coupled with a singular logarithmic potential makes second-order analysis challenging. The paper analyzes a modified Crank–Nicolson finite-difference scheme with nonlinear regularization and establishes optimal convergence alongside solvability, positivity, and energy stability.

  • Problem

    Variable mobility coupled with a singular logarithmic potential creates analytical challenges for second-order schemes, while the earlier positivity-preserving scheme has only first-order temporal accuracy.

    Source: §1 · p. 2

  • Method

    The scheme combines modified Crank–Nicolson discretization, explicit mobility extrapolation, and nonlinear logarithmic regularization.

    Source: §1 · p. 2, §6 · p. 23

  • Results

    The analysis obtains optimal convergence for a broad, physically relevant class of mobility functions.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Numerical experiments validate the theoretical results and demonstrate the scheme’s efficiency.

    Source: §6 · p. 24

    The convergence result assumes uniformly positive, bounded mobility with bounded derivative and sufficiently small steps satisfying C1h ≤ τ ≤ C2h.

    Source: §4 · p. 11, §4.3 · p. 17

Abstract

from arXiv · show

A second order finite difference scheme is analyzed for the nonlocal Cahn-Hilliard system with variable mobility and the singular Flory-Huggins logarithmic potential. The temporal discretization employs a modified Crank-Nicolson formulation, while the mobility is treated explicitly to guarantee ellipticity and reduce computational cost. To strictly enforce the positivity-preserving property at the discrete level, a nonlinear regularization term is added into the numerical scheme. Rigorous analysis shows that the scheme is uniquely solvable and stable under a modified numerical energy. The well-known analytical challenge associated with the variable mobility has to be handled carefully. To overcome this, a convergence framework is constructed based on two non-standard techniques: (1) a higher-order asymptotic expansion (extending up to the third order in time and fourth order in space) to retain a sufficiently order of accuracy; (2) a two-stage error strategy, wherein a rough error estimate first guarantees the discrete boundedness of the mobility and nonlinear terms, followed by a refined error analysis that yields the optimal convergence rate. Numerical experiments validate the theoretical results and demonstrate the robustness of the proposed scheme.

1 Introduction

The paper develops a second-order scheme for the nonlocal Cahn–Hilliard system with variable mobility and a singular logarithmic potential. Its convergence analysis addresses the nonlinear coupling through higher-order consistency and staged error estimates.

1 Introduction

  • For nonlocal interactions, the inverse fractional-Laplacian term gives the Ohta–Kawasaki free energy used to model diblock copolymer self-assembly and microscopic phase separation.At σ = 0, the energy reduces to the classical local Cahn–Hilliard definition.Source: §1 · p. 2
  • Variable mobility models concentration-dependent transport, but its coupling with the singular logarithmic potential complicates analysis.The mobility-weighted energy dissipation structure underlies the continuous analysis and positivity-preserving discrete approximations.Source: §1 · p. 2
  • The scheme combines modified Crank–Nicolson time discretization with explicitly extrapolated mobility and nonlinear logarithmic regularization.The explicit mobility treatment ensures strict ellipticity and reduces computational cost; the logarithmic regularization supports positivity preservation.Source: §1 · p. 2
  • Higher-order consistency and rough-then-refined error estimates establish an a priori discrete ℓ∞ bound and yield optimal convergence for a broad class of mobility functions.The bound provides regularity for the variable mobility and singular logarithmic terms; the article also reports numerical experiments validating its theoretical results.Source: §1 · p. 3

2 The second order accurate numerical scheme

The scheme uses centered finite differences on periodic grids and defines a variable-coefficient discrete divergence operator. Its second-order time treatment uses face-centered mobility with a positive lower bound and a ghost-level initialization.

2.1 The finite difference spatial discretization

  • The spatial discretization uses centered finite differences on a uniform periodic grid.The discrete Laplacian is formed by composing the discrete gradient and divergence.Source: §2.1 · pp. 3–4
  • A face-centered mobility coefficient defines the variable-coefficient discrete divergence operator used in the spatial scheme.The operator applies pointwise multiplication by the coefficient before taking the discrete divergence.Source: §2.1 · p. 4
  • Summation-by-parts identities provide the discrete conservation and dissipation structure, alongside a mobility-weighted inverse operator on mean-zero grid functions..Source: §2.1 · pp. 4–5

2.2 A second order numerical scheme

  • For the three-level scheme, initializing the ghost level with φ⁻¹ = φ⁰ simplifies energy stability analysis while preserving second-order temporal accuracy.Source: §2.2 · pp. 5–6
  • The face-centered discrete mobility definition guarantees second-order time accuracy and a pointwise positive lower bound.Source: §2.2 · p. 5

3 Positivity-preserving analysis and energy stability estimate

Section 3 proves positivity preservation and unique solvability for the discrete scheme, then establishes dissipation of a modified discrete energy. The energy bound is independent of spatial mesh size.

3 Positivity-preserving analysis and energy stability estimate

  • The scheme has a positivity-preserving property, with the theorem assuming mobility M(x) ≥ M0 > 0 and admissible prior values.The physical requirement is −1 < φ < 1 at each grid point.Source: §3 · p. 6
  • Strict convexity of the discrete functional yields a unique solution, after existence is established by locating its minimizer in the admissible interior.The boundary of the regularized admissible set is excluded for the minimizer.Source: §3 · pp. 6, 7, 9
  • The modified energy is nonincreasing across time steps and bounded above by C3, independent of h.The dissipation law is stated for the proposed second-order scheme.Source: §3 · p. 9

4 Optimal rate convergence analysis

The section develops a convergence framework for the scheme using higher-order consistency and successive rough and refined error analyses. It establishes the theorem’s estimate under stated bounds and refinement conditions on the time and space steps.

4 Optimal rate convergence analysis

  • The main convergence theorem holds for sufficiently small steps with C1h ≤ τ ≤ C2h, under the stated mobility and solution regularity assumptions.The bound's constant is independent of the time step, mesh size, and time index.Source: §4 · p. 11

4.1 Higher-order consistency analysis

  • A higher-order corrected profile supplies consistency beyond the standard second-order truncation error, enabling inverse inequalities to establish separation and control nonlinear terms under variable mobility.The correction functions depend only on the exact solution; the extrapolated mobility approximation remains strictly positive, so consistency analysis requires no regularization.Source: §4.1 · p. 12

4.2 Rough error estimate

  • The rough error estimate first yields boundedness and separation of the numerical solution, supporting control of nonlinear terms and a positive mobility bound under the linear refinement constraint.Under its a priori assumptions, the rough error bound has a constant independent of τ and h; the analysis also shows mobility regularization can be omitted in this case.Source: §4.2 · pp. 14–15

4.3 A refined error estimate

  • The refined error analysis uses the established regularity, separation, and temporal-derivative bounds to control nonlinear terms by the squared discrete gradient error.Source: §4.3 · p. 17
  • For sufficiently small τ and h, the discrete Gronwall inequality yields a higher-order convergence estimate that closes the induction and proves Theorem 4.1.The proof also uses boundedness of the constructed fields Φτ and Φh and the projection estimate.Source: §4.3 · p. 20

5 Numerical experiments

The numerical experiments test convergence and examine spinodal-decomposition dynamics, phase-variable extrema, energy decay, and mass conservation. They report second-order accuracy in space and time and show the scheme’s stated stability and conservation properties in the tested cases.

5 Numerical experiments

  • The numerical experiments use a nonlinear FAS multigrid solver and are restricted to two dimensions, although the theory covers both two and three dimensions.The restriction is stated as balancing computational focus and clarity.Source: §5 · p. 20

5.1 Convergence test

  • The convergence test at T=0.1 confirms second-order accuracy in both space and time.The refinement path is τ = 0.01h; errors are computed between solutions on successively finer meshes.Source: §5.1 · p. 21, Table 1 · p. 21

5.2 Spinodal decomposition

  • The σ = 5 phase-field evolution exhibits numerous microstructures from the start until the energy stabilizes.The snapshots use the specified mobility and initial condition; the text attributes the microstructures to the nonlocal term.Source: §5.2 · p. 21, Figure 1 · p. 22
  • The phase-variable extrema remain clearly separated from the singular limits −1 and 1 during spinodal decomposition.Table 2 reports the minimum and maximum at specified time instants.Source: §5.2 · p. 21, Table 2 · p. 22
  • For σ = 5, the discrete energy decays and the mass error is measured relative to the initial state; the results confirm energy stability and mass conservation.Figure 2 plots energy versus time on log-log axes and reports mass error.Source: §5.2 · p. 21, Figure 2 · p. 22
  • With φ0 = 0.3 and enhanced nonlocal interactions, the coarsening dynamics exhibit notably distinct structures.The accompanying extrema remain separated from the singular limits, while Figure 4 shows energy decay and mass conservation.Source: §5.2 · pp. 21–22, Table 2 · p. 22, Figure 3 · p. 23

6 Conclusions

The paper develops a second-order finite difference scheme for the nonlocal Cahn-Hilliard system with variable mobility and a singular logarithmic potential, and establishes its core analytical properties and convergence framework.

6 Conclusions

  • The scheme combines modified Crank-Nicolson discretization and nonlinear artificial regularization, while treating mobility explicitly for strict ellipticity and reduced computational cost.The analysis establishes unique solvability, positivity preservation, and discrete energy stability.Source: §6 · pp. 23–24
  • A convergence framework combines higher-order consistency analysis with rough and refined error estimates to address difficulties from variable mobility and the singular potential.Numerical experiments confirm the theoretical results and demonstrate the scheme’s efficiency.Source: §6 · p. 24

Appendix A: Proof of Proposition 4.1

The appendix constructs temporal and spatial corrections to obtain higher-order consistency estimates for the scheme.

Appendix A: Proof of Proposition 4.1

  • The time-correction function Φτ solves a linear PDE, depends on the projection solution ΦN, and is bounded with a mass-conservative property.Trivial zero initial data may be imposed for Φτ.Source: Appendix A · p. 26
  • A semi-implicit discretization and Fourier projection yield a third-order temporal consistency estimate for the corrected function.The temporal correction uses linearized expansions.Source: Appendix A · pp. 26–27
  • For spatial discretization, the correction Φh is introduced to enhance accuracy; with a difference approximation, the O(h3) truncation-error term vanishes, and Φh is obtained by solving a linear PDE system.Source: Appendix A · pp. 27–28

Appendix B: Proof of Lemma 4.1

The appendix establishes estimates for nonlinear terms using separation properties and mean-value arguments.

Appendix B: Proof of Lemma 4.1

  • The logarithmic artificial-regularization term is estimated using a Taylor expansion and positivity of the corrected and numerical phase values.The pointwise bound includes 0 < 1 + φn+1 < 2.Source: Appendix B · pp. 28–30
  • Mean-value decompositions and the resulting inner-product estimates combine to establish the bounds stated in Lemma 4.1.The proof derives estimate (4.23); it omits the analogous argument for (4.24).Source: Appendix B · pp. 29–30
  • Separation properties place the intermediate values ξ(n) and ξ(n+1) between the corrected and numerical phase values, supporting boundedness estimates.The proof invokes separation properties for both corrected and numerical solutions.Source: Appendix B · p. 29

Appendix C: Proof of Lemma 4.3

The appendix derives discrete norm estimates for nonlinear terms and uses them to obtain a pointwise estimate.

Appendix C: Proof of Lemma 4.3

  • Combining estimates for the decomposed terms yields an integrated bound.The proof combines estimates (C.4)–(C.6).Source: Appendix C · pp. 30–32
  • A pointwise estimate follows from bounds on intermediate values between the corrected and numerical phase variables.The indices include neighboring grid points.Source: Appendix C · p. 31
  • Directional derivative bounds are combined to bound the discrete gradient norm of the nonlinear term, with Poincaré and discrete Sobolev inequalities used in the proof.The proof also derives the gradient estimate for the term’s derivative.Source: Appendix C · p. 32
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