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A linear, fully decoupled, and unconditionally energy-stable SAV-FEM for the Cahn--Hilliard--Navier--Stokes model

This paper proposes a linear, fully decoupled, and unconditionally energy-stable SAV-FEM scheme for the Cahn–Hilliard–Navier–Stokes system that utilizes a single scalar auxiliary variable to reformulate nonlinearities, enabling efficient solution via decoupled linear subproblems while guaranteeing energy stability and optimal-order error estimates.

Original authors: Jinting Yang, Nianyu Yi, Peimeng Yin

Published 2026-07-16

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CC BY 4.0

Original authors: Jinting Yang, Nianyu Yi, Peimeng Yin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a drop of oil mix with water. At first, they are distinct, but as they swirl together, they don't just blend into a uniform gray soup; they form intricate, shifting patterns, like swirling clouds or marbled cake. This isn't just a pretty picture; it's a complex dance of physics where the fluid's movement (flow) and the separation of materials (phase) constantly influence each other. Scientists call this the Cahn–Hilliard–Navier–Stokes (CHNS) system. It's the mathematical rulebook for how two fluids that hate to mix behave when they are forced to interact. The challenge is that the math behind this dance is incredibly messy. It involves "nonlinear" terms, which are like equations that change their own rules as the game progresses, making them a nightmare to solve on a computer. If you try to solve them exactly, your computer might crash or take longer than the age of the universe. So, scientists build "numerical schemes"—simplified, step-by-step recipes that approximate the solution. The holy grail of these recipes is finding one that is fast, simple to run, and, most importantly, "energy-stable." This means the recipe never accidentally creates energy out of thin air, ensuring the simulation doesn't explode into nonsense as time goes on.

This paper introduces a new, clever recipe for solving the CHNS system, designed by Jinting Yang, Nianyu Yi, and Peimeng Yin. Think of the existing methods as a team of mechanics trying to fix a car with a dozen different tools, each handling a specific part of the engine separately. They often need multiple "helper variables" (like extra wrenches or sensors) to keep the math from getting tangled. The authors of this paper say, "What if we could do it with just one tool?" They developed a method using a single "Scalar Auxiliary Variable" (SAV)—imagine this as a magical, universal remote control that can tune the entire system at once. By introducing this single variable and a special way to update it, they managed to rewrite the messy, nonlinear equations into a form that is linear and fully decoupled. In plain English, this means they broke the giant, tangled knot of equations into three tiny, independent loops that can be solved one after another, rather than all at once.

The paper proves that this new method is not only simpler but also rock-solid reliable. They showed mathematically that the method is "unconditionally energy-stable," meaning no matter how big the time steps are or how wild the simulation gets, the total energy will always behave correctly, just like in the real world. They also proved that the method is accurate, deriving "optimal-order error estimates," which is a fancy way of saying they calculated exactly how close their approximation is to the true answer and showed it gets better at the expected rate as they use finer grids. To test their theory, they ran several simulations. In one, they watched a bubble of fluid evolve from a square shape into a circle, just as physics predicts. In another, they simulated random blobs of fluid merging and growing, a process called "coarsening." In every test, the energy in their simulation dropped steadily over time, exactly as it should, confirming that their single-variable "universal remote" works perfectly. The authors conclude that while their method is a major step forward for speed and stability, the next challenge is to see if this same clever trick can handle even more complex scenarios, like fluids with changing densities, without losing its stability.

Problem Statement
The paper addresses the numerical simulation of the Cahn–Hilliard–Navier–Stokes (CHNS) system, a coupled nonlinear partial differential equation model describing two-phase incompressible flows. The system involves significant challenges for numerical discretization, including:

Nonlinear Coupling: The velocity field and phase variable are coupled through convective terms in the phase equation and stress terms in the momentum equation.
Nonlinearity and Stiffness: The system features a nonlinear energy potential (Ginzburg–Landau free energy) and stiffness induced by small interfacial width parameters.
Energy Stability: Designing schemes that preserve the physical energy dissipation law at the discrete level is difficult, particularly when seeking linear and fully decoupled formulations.

Existing methods often rely on multiple scalar auxiliary variables (MSAV) or combinations of auxiliary variables with zero-energy-contribution (ZEC) techniques to handle these nonlinearities. While effective, these approaches increase computational complexity by requiring the decomposition of the nonlinear system multiple times and solving multiple auxiliary ordinary differential equations.

Methodology
The authors propose a linear, fully decoupled, and unconditionally energy-stable fully discrete finite element scheme based on the Scalar Auxiliary Variable (SAV) approach combined with an Implicit–Explicit (IMEX) Euler time discretization.

Reformulation: A single scalar auxiliary variable, $r(t) = \sqrt{E_1[\phi(t)]}$ , is introduced to reformulate all nonlinear terms arising from both the Cahn–Hilliard and Navier–Stokes equations simultaneously. Unlike standard SAV approaches, the authors introduce a novel update equation for this single variable that includes specific correction terms. This allows the treatment of multiple nonlinearities without splitting them into separate auxiliary variables.
Discretization:
- Temporal: The IMEX Euler scheme treats linear terms implicitly and nonlinear terms explicitly.
- Spatial: A standard finite element method (FEM) is used. The velocity and pressure spaces satisfy the inf-sup condition (e.g., MINI elements or Taylor-Hood elements), and the phase field uses continuous piecewise linear elements.
Algorithmic Structure: The resulting fully discrete scheme decomposes into three independent subproblems per time step:
1. A linear Cahn–Hilliard-type system.
2. A linear Stokes-type system.
3. A scalar quadratic algebraic equation for the auxiliary variable.
  Crucially, the velocity and pressure are solved as a coupled Stokes system at each step (rather than using a pressure-correction projection), which simplifies the error analysis.

Key Contributions and Results

Single Auxiliary Variable Efficiency: The primary contribution is the demonstration that a single scalar auxiliary variable, coupled with a specific update strategy, is sufficient to handle all nonlinearities in the CHNS system. This reduces the computational overhead compared to MSAV or SAV-ZEC methods that require multiple auxiliary variables.
Unconditional Energy Stability: The authors prove that the proposed scheme satisfies an unconditional discrete energy dissipation law. The modified discrete energy decreases monotonically at each time step, regardless of the time-step size.
Optimal Error Estimates: Under standard regularity assumptions ( $u, \phi \in H^2$ $u, ϕ \in H^{2}$ ), the paper establishes optimal-order $L^2$ error estimates for the fully discrete approximation.
- The phase field, chemical potential, and pressure achieve optimal convergence rates.
- The velocity achieves optimal-order $L^2$ convergence.
- A significant technical achievement is handling the non-globally Lipschitz nature of the nonlinear potential $g(\phi) = F'(\phi)$ . The authors prove the uniform $L^\infty$ boundedness of the numerical solution, which allows the error analysis to proceed without restrictive assumptions on the free energy.
Numerical Verification: Numerical experiments confirm the theoretical convergence rates (observing $O(h^2)$ in $L^2$ and $O(h)$ in $H^1$ for spatial discretization) and demonstrate the monotonic decay of the modified discrete energy in various test cases, including phase separation and coarsening dynamics.

Significance
The paper claims that its approach offers a more efficient and simpler alternative to existing SAV-based methods for the CHNS system. By utilizing a single auxiliary variable and a unified update strategy, the method avoids the complexity of decomposing the system multiple times. Furthermore, the direct solution of the Stokes subsystem (without projection steps) simplifies the theoretical error analysis, allowing for optimal error estimates under weaker regularity assumptions ( $\phi \in H^2$ ) compared to some projection-based methods that may require higher regularity. The work provides a robust framework for simulating two-phase flows that is both computationally efficient (linear and decoupled) and theoretically rigorous (unconditionally stable with optimal error bounds).

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