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Convergence analysis of a numerical scheme for the Cahn-Hilliard-Navier-Stokes system with dynamical boundary condition and its application to moving contact line problem
Yunzhuo Guo, Cheng Wang, Xianmin Xu, Zhengru Zhang
TL;DR
The paper addresses the lack of theoretical convergence analysis and error estimates for numerical schemes related to the moving contact line problem. It analyzes a fully discrete finite-difference CHNS scheme with a relaxed dynamical boundary condition. The analysis establishes unique solvability, modified energy stability, and optimal-rate convergence.
Problem
Theoretical convergence analysis and error estimates for moving-contact-line schemes have remained open despite existing energy-stable methods, amid nonlinear coupling between bulk and boundary.
Source: §1 · p. 4Method
The paper analyzes a fully discrete finite-difference CHNS scheme using convex splitting for bulk and boundary chemical potentials and rough and refined error estimates.
Source: §6 · pp. 27–28Results
The scheme has a unique solution, modified total-energy stability, and theoretically established optimal-rate convergence.
Source: §3 · p. 13, §6 · p. 28Takeaways & Limitations
The analysis provides a theoretical convergence and error-estimate result for this scheme in the moving-contact-line setting.
Source: §1 · p. 2, §6 · p. 28The convergence analysis uses finite differences; boundary-trace arguments with other spatial discretizations are expected to be more complicated and are left for future work.
Source: §4.3 · p. 24
Abstract
from arXiv · showhide
A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, combined with a dynamical boundary condition. Such a physical system has potential applications in the moving contact line problem. The boundary profile is governed by a lower-dimensional energy potential, coupled with a non-homogeneous boundary condition for the phase variable. In the numerical design, a convex-splitting approach is applied to the chemical potential in both bulk and surface levels, which leads to a highly coupled nonlinear system. A semi-implicit discretization is taken in the nonlinear fluid convection, as well as the coupled terms between the fluid motion and phase variable evolution. A careful finite difference approximation and convexity analysis reveals that such a numerical system could be represented as a non-symmetric and monotone mapping associated with the fluid convection. In turn, the unique solvability is valid based on the monotonicity argument. The total energy stability analysis is obtained through a careful summation-by-part calculation. In particular, an optimal rate convergence analysis is theoretically established in this work. The discrete mass conservation of the exact solution is required to preserve the mean-zero property of the error function, so that the associated discrete H_h^{-1} norm is well-defined. A combination of the Fourier projection and an auxiliary function is applied to overcome this difficulty. Furthermore, an approach of rough and refined error estimates concludes the desired convergence result. Some numerical results are presented in this article, which demonstrate the robustness of the proposed numerical scheme. In our knowledge, this work provides a theoretical proof of convergence analysis and error estimate for a numerical scheme to the moving contact line problem, for the first time in the literature.
1 Introduction
The paper studies a CHNS model with a relaxed dynamical boundary condition, motivated by the moving contact line problem. It develops a fully discrete scheme and analyzes its solvability, energy stability, and convergence.
1 Introduction
- Dynamical boundary models differ in mass exchange: the Gal model does not conserve total or boundary mass, while Liu–Wu and GMS models conserve at their respective levels.The paper selects an Allen–Cahn-type dynamical boundary condition.Source: §1 · p. 2
- The model uses an Allen–Cahn-type relaxed boundary condition for the moving contact line, coupling the boundary profile to the bulk through the normal derivative.The paper uses no-slip velocity boundary conditions for simplicity, omitting interaction between motion and the surface.Source: §1 · pp. 2–3
- Fully discrete energy-stable design is challenging because the normal derivative is undefined at square-domain vertices, prompting prior work to use temporal semi-discretizations or modified domains.The cited fully discrete approaches use periodicity in one direction and the physical boundary in the other.Source: §1 · p. 3
- The scheme convex-splits the chemical potential in the bulk and on the boundary, while coupling the next boundary profile to the phase-variable boundary condition.A wide-stencil normal derivative is used to retain second-order spatial accuracy and support unconditional total-energy dissipation.Source: §1 · p. 3
- The fluid momentum step uses a projection method, with semi-implicit treatment of convection and phase-coupling terms before projecting velocity onto a divergence-free field.Kinematic diffusion is implicit and the pressure gradient explicit in the intermediate-velocity step.Source: §1 · p. 3
- The convective term makes the numerical system non-symmetric, so unique solvability is established through monotonicity rather than convex minimization.The Browder–Minty lemma is applied, and a discrete total-energy dissipation law follows from an energy estimate using summation-by-parts formulas.Source: §1 · pp. 3–4
- The paper addresses the long-standing lack of theoretical convergence and error estimates for moving-contact-line schemes despite existing energy-stable methods.The analysis combines rough and refined error estimates with a Fourier projection and auxiliary function to secure discrete mass conservation and the Poincaré inequality.Source: §1 · p. 4
2 Numerical scheme
This section presents a finite-difference convex-splitting scheme and analyzes its discrete operators and properties, including positivity and symmetry of the Neumann Laplacian. It establishes unique solvability while accounting for the dynamical boundary condition, for free-energy densities with bounded second derivatives.
2.1 Finite difference spatial discretization
- The discretization uses a uniform finite-difference mesh on a two-dimensional domain, periodicity in x, and physical top and bottom boundaries with ghost layers for Neumann conditions.The three-dimensional extension is described as straightforward but omitted.Source: §2.1 · pp. 4–6
- The discrete operator Lh incorporating homogeneous Neumann conditions is positive and symmetric on the mean-zero grid-function space.The analysis also uses discrete Poincaré and inverse-style inequalities, including an estimate identified as essential for a later maximum-norm bound.Source: §2.1 · pp. 7–8
2.2 The fully discrete numerical scheme
- The fully discrete scheme applies convex splitting with F'(ϕ) = G'(ϕ) = ϕ^3 − ϕ, using F(ϕ) = G(ϕ) = R(ϕ) = 1/4ϕ^4 − 1/2ϕ^2 for the energy densities.The text notes that the analysis could also apply to other choices of energy densities.Source: §2.2 · p. 9
2.3 Unconditionally unique solvability
- The operator Gn_h is monotone, and its inverse is also monotone, supporting the section’s unique-solvability analysis.For the inverse operator, equality holds if and only if the two mean-zero grid functions are equal.Source: §2.3 · p. 10
- Because the mapping’s monotonicity analysis does not incorporate the dynamical boundary, the proof addresses the boundary condition in its coercivity analysis.The Laplacian term is treated using the dynamical boundary condition.Source: §2.3 · pp. 10–11
- Given ϕⁿ with ϕⁿ=β₀, there exists a unique solution ϕⁿ⁺¹ to the numerical scheme (2.38)–(2.43), equivalently (2.52), with ϕⁿ⁺¹=β₀.An application of the Browder–Minty lemma implies a unique solution to (2.52).Source: §2.3 · pp. 11–12
- The solvability, energy-stability, and convergence analyses hold for free-energy densities with bounded second derivatives; quartic F and G are used for presentation.The passage gives trigonometric functions as an example of admissible densities.Source: §2.3 · p. 12
3 Unconditional total energy stability
The scheme satisfies a modified discrete energy dissipation law. The proof combines discrete inner products, convexity estimates, and boundary evolution identities.
3 Unconditional total energy stability
- The modified energy includes kinetic energy and bulk, top-boundary, and bottom-boundary energy contributions.The discrete energy is defined with R(c) = 1.Source: §3 · p. 13
- Theorem 3.1 establishes modified energy stability for the numerical solution, yielding total energy dissipation.The proof concludes by establishing inequality (3.3).Source: §3 · pp. 13–14
- The proof takes discrete inner products of the momentum equation with velocity and the phase equation with chemical potential.The resulting estimates include a kinetic-energy difference and a viscous term.Source: §3 · p. 13
- Convexity estimates and the boundary evolution equation control the boundary energy terms.The proof applies the boundary equation and considers convexity of the boundary energy.Source: §3 · p. 13
- A discrete pressure identity and an orthogonality relation help combine the estimates into the energy inequality.The pressure term is rewritten as an inner product of discrete gradients.Source: §3 · pp. 13–14
4 Convergence analysis
The section analyzes convergence of the scheme under regularity and time–space refinement conditions, using mass-corrected profiles and progressing from rough and preliminary error bounds to a refined estimate. It establishes optimal-rate convergence for sufficiently small time step and mesh size, and notes boundary-treatment choices and model extensions for future analysis.
4 Convergence analysis
- Under regularity and sufficiently small steps with C1h ≤ ∆t ≤ C2h, Theorem 4.1 establishes convergence; the discrete error is mean-zero, so its discrete negative norms are well-defined.A cosine correction preserves discrete bulk mass without changing the overall convergence order and leaves the normal derivative unaffected.Source: §4 · p. 16
4.1 The consistency analysis
- The consistency analysis constructs an approximate chemical-potential profile with O(∆t + h^2) truncation error, while high-order ghost-point extrapolations avoid reducing truncation accuracy.The consistency estimate addresses projection-related boundary challenges for the no-slip velocity condition.Source: §4.1 · p. 17
4.2 A rough error estimate
- Under regularity assumption (4.41) for Vn+1 and a-priori assumption (4.42), the rough error estimate for system (4.33)–(4.40) is bounded by C(∆t^(3/4)+h^(11/4)) when ∆t and h are sufficiently small.Source: §4.2 · p. 18
4.3 A refined error estimate
- The refined-estimate analysis first bounds the nonlinear error terms using the regularity and functional bounds for the constructed and numerical solutions.It treats bulk and boundary nonlinear errors, with constants independent of ∆t and h.Source: §4.3 · p. 21
- C(∆t + h^2) bounds the combined bulk phase, boundary phase, velocity, and time-step-weighted pressure errors in the refined estimate.The estimate follows from a discrete Gronwall argument using higher-order truncation accuracy; induction then recovers the a-priori assumption and completes the optimal-rate convergence proof.Source: §4.3 · pp. 23–24
- Finite differences simplify boundary-trace analysis by evaluating bulk and boundary profiles pointwise at boundary grid points; mixed finite elements are expected to make those arguments more complicated.Convergence analysis with other spatial discretizations is left for future work.Source: §4.3 · p. 24
- The analysis is expected to extend to three dimensions with higher-order consistency expansion to control the numerical solution’s discrete maximum norm by an inverse inequality.Technical details are deferred to future work.Source: §4.3 · p. 24
- A second-order-in-time scheme based on modified Crank–Nicolson discretization is proposed as a possible design, with convergence analysis and error estimates left for future work.Related homogeneous-boundary-condition CHNS works are cited as [6].Source: §4.3 · p. 24
- The analysis assumes no-penetration, no-slip velocity conditions; free-slip or Navier conditions introduce boundary convection and make numerical design and analysis more challenging.For free-slip or Navier conditions, total-energy stability and convergence analysis are deferred [38].Source: §4.3 · p. 24
- Density variation is omitted, but convergence analysis is expected to extend to variable-density CHNS if the total-energy dissipation law holds.The authors leave this challenging issue for future work.Source: §4.3 · p. 24
5 Numerical results
The numerical examples examine convergence, spinodal decomposition, and moving-contact-line evolution. They report convergence orders consistent with theory and illustrate mass conservation, energy dissipation, and contact-angle evolution.
5 Numerical results
- The solver combines a nonlinear FAS multigrid method for the phase field with Newton iteration on the boundary, Picard iteration for the fluid, and FFT for the Poisson equation.The examples use a two-dimensional unit-square domain and periodic boundary conditions in the x-direction, with physical boundaries at the top and bottom.Source: §5 · p. 24, §5.1 · pp. 24–25
5.1 Convergence test
- Spatial accuracy orders approach the theoretical value of 2, while temporal errors verify the theoretical convergence order of 1.The spatial and temporal tests use separate mesh and time-step refinements, respectively.Source: §5.1 · p. 25, Table 1 · p. 25, Table 2 · p. 25
5.2 Spinodal decomposition
- The spinodal-decomposition example confirms mass conservation and energy dissipation, while phase snapshots show a near-wall transition layer and non-perpendicular interface.The reported boundary behavior differs from that under homogeneous Neumann conditions and suggests a short-range wetting-type interaction.Source: §5.2 · p. 25, Figure 1 · p. 26, Figure 2 · p. 26
5.3 Moving contact line
- In the moving-contact-line example, a droplet extends outward and its contact angle changes over time; the simulations also show a slow approach to equilibrium.The evolution is described in two stages: bulk-interface diffusion first drives rapid energy decrease, followed by slower boundary evolution.Source: §5.3 · p. 27, Figure 3 · p. 27
6 Conclusions
The conclusion presents a fully discrete finite-difference scheme for CHNS with a relaxed dynamical boundary condition and potential moving-contact-line applications. It reports theoretical solvability, energy stability, and optimal-rate convergence results, alongside numerical experiments.
6 Conclusions
- A fully discrete finite-difference scheme treats CHNS with a relaxed dynamical boundary condition and is proposed for potential moving-contact-line applications.Convex splitting is applied to the chemical potential in both the interior and boundary regions.Source: §6 · p. 27
- The scheme is uniquely solvable, theoretically energy stable, and supported by an optimal-rate convergence analysis and error estimate.The convergence analysis uses higher-order boundary-extrapolation consistency with rough and refined error estimates; the boundary equation's lack of diffusion requires rearranged summations and an auxiliary mass function.Source: §6 · p. 28
- The numerical experiments demonstrate the scheme's theoretical properties, with figures showing phase evolution and the evolution of energies and mass changes in two cases.The supplied captions identify the plotted quantities but do not report specific numerical outcomes.Source: §6 · p. 28, Figure 4 · p. 28, Figure 5 · p. 28