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High-order energy diminishing ALE-SAV finite element methods for two-phase Navier--Stokes flow
Shu Ma
TL;DR
Discrete energy decay is a central design target for sharp-interface two-phase Navier–Stokes flow, but the introduction identifies a lack of higher-order fully discrete ALE methods preserving that structure. The paper develops an ALE–SAV finite element scheme with stable geometric-energy treatment and kinetic-energy transfer, and reports a modified energy inequality with high-order convergence behavior.
Problem
The introduction identifies no higher-order fully discrete ALE method that preserves the discrete energy-diminishing structure of sharp-interface two-phase Navier–Stokes flow, for which energy inequalities are desirable in convergence analysis with limited-regularity solutions.
Source: §1 · p. 3Method
The method combines an SAV reformulation for surface and gravitational energies with an energy-stable ALE transfer operator controlling density-weighted kinetic energy across moving meshes.
Source: §1 · p. 3Results
The method satisfies a modified discrete energy inequality and exhibits approximately second-order convergence in time and arbitrarily high-order convergence in space.
Source: §1 · p. 3Takeaways & Limitations
The proposed scheme provides an energy-diminishing ALE finite element approach for sharp-interface two-phase Navier–Stokes flow on moving meshes.
Source: §1 · p. 3The reported h behavior is an observed numerical property rather than a stated theoretical guarantee.
Source: §4 · p. 13
Abstract
from arXiv · showhide
We develop a high-order fully discrete ALE-SAV finite element method for sharp-interface two-phase Navier--Stokes flow that satisfies a discrete energy dissipation law. We introduce an SAV reformulation to ensure the stable treatment of the surface and gravitational potential energies, together with an energy-stable ALE transfer operator that controls the density-weighted kinetic energy across moving meshes. The resulting scheme is linear and satisfies a modified energy-dissipation inequality. Numerical experiments, including rising-bubble benchmarks, demonstrate the robustness of the method and confirm second-order convergence in time and arbitrarily high-order convergence in space.
1 Introduction
The introduction motivates discrete energy decay for sharp-interface two-phase flow and identifies a gap in higher-order fully discrete ALE methods. It proposes an ALE–SAV finite element method that treats geometric energies and kinetic-energy transfer stably on moving meshes.
1 Introduction
- Sharp-interface models represent two immiscible fluids with a zero-thickness interface across which density and viscosity may be discontinuous.The introduction distinguishes these from diffuse-interface models, which use a thin transition layer with rapidly but smoothly varying material parameters.Source: §1 · p. 1
- In an ALE method, the mesh follows the moving interface while its interior velocity can differ from the fluid velocity, allowing adjustment of the bulk mesh as the flow evolves.Source: §1 · p. 2
- A discrete energy inequality is desirable because the continuous system’s total energy decays through viscosity, and convergence analysis for limited-regularity solutions often relies on such an inequality.Source: §1 · p. 3
- The introduction identifies the absence, to the authors’ knowledge, of a higher-order fully discrete ALE method preserving the two-phase Navier–Stokes energy-diminishing structure.The cited first-order ALE method uses a square-root Jacobian factor, while related higher-order methods omit fluid coupling or inertia.Source: §1 · p. 3
- The proposed method combines an SAV treatment of surface and gravitational energies with an energy-stable ALE transfer operator for density-weighted kinetic energy, and satisfies a discrete energy inequality.The reported experiments indicate approximately second-order time convergence and arbitrarily high-order spatial convergence.Source: §1 · p. 3
2 The ALE–SAV formulation of the sharp-interface model
The formulation augments the sharp-interface model with a scalar auxiliary variable and uses an ALE description designed to preserve energy structure. Its weak form is equivalent to the shifted model for sufficiently regular solutions and has a non-increasing total energy.
2.1 Geometric energy and SAV reformulation
- The SAV reformulation introduces an independent scalar r determined by the interface quantity R(X), and scales the shifted interfacial traction by r/R(X).The geometric energy is shifted by a sufficiently large C0 so that F(X) + C0 is positive.Source: §2.1 · p. 4
- The gravitational force is absorbed into the pressure by shifting p± by ρ±gxd, after which the shifted pressure and stress are used in the reformulated system.Source: §2.1 · p. 4
2.2 The ALE formulation
- The ALE mesh map follows the interface, with phasewise harmonic extension of its velocity and boundary conditions preserving the fixed boundary partition.The harmonic extension is a mesh-motion choice and contributes no physical dissipation.Source: §2.2 · p. 5
- The half-density ALE derivative is introduced to recover the kinetic-energy chain rule in the moving-mesh formulation.Source: §2.2 · p. 5
2.3 The ALE–SAV weak formulation
- In the weak formulation, the left-hand-side transport terms are skew-symmetric and vanish when tested with the velocity itself, without an additional divergence constraint.Source: §2.3 · p. 6
2.4 Equivalence and continuous energy law
- Every sufficiently regular solution of the augmented system satisfies r(t) = R(X(t)), and conversely the shifted sharp-interface model yields the SAV system when r = R(X).Source: §2.4 · pp. 6–7
- With that identity, the energy in the continuous law is the system’s total energy E(t), which is non-increasing.Source: §2.4 · pp. 6–7
3 The fully discrete ALE–SAV method
The method uses evolving degree-k isoparametric fitted meshes, an energy-stable ALE transfer, and a fully discrete Crank–Nicolson scheme. For every geometrically accepted step, its solution satisfies a discrete energy inequality.
3.1 Evolving isoparametric finite element spaces
- Degree-k isoparametric element maps define the evolving fitted mesh, with interface geometry error of order O(h^(k+1)).Admissible configurations preserve phase labels, fix Γ1 pointwise, and map Γ2 onto itself.Source: §3.1 · p. 7
3.2 Energy-stable ALE transfer operator
- The ALE transfer uses half-density composition and density-weighted projection to control kinetic energy across changing meshes.The half-density composition is an isometry, and the transfer is defined through projection onto the target velocity space.Source: §3.2 · p. 8
3.3 Fully discrete Crank–Nicolson scheme
- The fully discrete Crank–Nicolson scheme constructs a midpoint mesh, solves a coupled system there, and then updates the mesh and velocity.For smooth solutions, its one-time startup has O(τ^2) error and does not reduce formal second-order temporal accuracy.Source: §3.3 · pp. 10–11
3.4 Discrete energy decay
- For every geometrically accepted time step, the fully discrete solution satisfies the discrete energy inequality.The proof combines the carrier-mesh balance with the energy-stable incoming and outgoing ALE transfers.Source: §3.4 · pp. 12–13
4 Numerical results in two dimensions
The two-dimensional tests examine capillary relaxation and rising-bubble benchmarks, reporting convergence and energy behavior for the fully discrete scheme. BP-1 shows second-order temporal self-convergence before remeshing, while BP-2 is tested at only one resolution.
4 Numerical results in two dimensions
- The tests use monolithic solves in NGSolve without remeshing in refinement studies, and report self-differences rather than errors against an exact solution.Capillary tests use degree-k isoparametric elements for k = 1, 2, 3; rising-bubble tests use k = 1.Source: §4 · p. 13
4.1 Force-free capillary relaxation
- For capillary relaxation, temporal order is two for velocity and interface at k = 1, 2, 3; spatial orders are two, three, and four, respectively.The modified energy decreases for each degree, with maximum balance residual below 2.4×10−14; physical energy also decreases, though Theorem 3.3 does not imply this.Source: §4.1 · p. 14, Figure 4 · p. 15, Figure 5 · p. 15
4.2 Benchmark problem 1
- BP-1 shows second-order temporal self-convergence on [0, T0] before remeshing, and both modified and physical energies decrease.The full benchmark histories include remeshing, but are not used for the convergence-rate estimate.Source: Table 2 · p. 16, §4.2 · pp. 16, 18
4.3 Benchmark problem 2
- BP-2 is evaluated at one resolution, so no convergence order is reported; small late-time variations occur in circularity and rise velocity.The maximum accepted-step balance residual is 3.829 × 10−11.Source: §4.3 · p. 19, Figure 9 · p. 19