[Paper Review] On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes Phase Field Model

Xiaoli Li, Jie Shen|arXiv (Cornell University)|May 21, 2019

Solidification and crystal growth phenomena27 references4 citations

TL;DR

This paper presents a novel SAV-MAC finite difference scheme for the Cahn-Hilliard-Navier-Stokes phase field model, combining the scalar auxiliary variable (SAV) approach in time with the MAC (marker and cell) finite difference method in space. The scheme is linear, second-order accurate in both time and space, unconditionally energy stable, and efficiently implementable, with the first rigorous second-order error estimates established for phase field variable, chemical potential, velocity, and pressure without requiring a uniform Lipschitz condition on the nonlinear potential.

ABSTRACT

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and error analysis. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase field variable, chemical potential, velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the our scheme.

Motivation & Objective

• To develop a fully discrete numerical scheme that preserves the energy law at the discrete level for the Cahn-Hilliard-Navier-Stokes phase field model.
• To overcome limitations of existing convex splitting and stabilized linearly implicit schemes, which either lead to nonlinear systems or lack second-order energy stability.
• To construct a linear, second-order, unconditionally energy-stable scheme that is computationally efficient and suitable for long-time simulations.
• To provide the first rigorous second-order error estimates in both time and space for a fully discrete linear scheme applied to the Cahn-Hilliard-Stokes system, without assuming a uniform Lipschitz condition on the nonlinear potential.

Proposed method

• The scheme employs the scalar auxiliary variable (SAV) approach in time to decouple the nonlinear system and achieve linear, second-order, unconditionally energy-stable time discretization.
• Spatial discretization is performed using the MAC (marker and cell) finite difference method on staggered grids, ensuring discrete divergence-free velocity and consistent pressure-velocity coupling.
• The SAV approach transforms the original system into a new system with constant coefficients, enabling efficient solution via decoupled linear systems at each time step.
• Discrete energy stability is proven by constructing a discrete energy law that mimics the continuous energy dissipation structure.
• Error estimates are derived using a combination of energy techniques, discrete Sobolev inequalities, and stability estimates in various discrete norms.
• The scheme is implemented using uniform Cartesian grids with consistent treatment of boundary conditions and staggered variable placement.

Experimental results

Research questions

• RQ1 Can a linear, second-order, unconditionally energy-stable scheme be constructed for the Cahn-Hilliard-Navier-Stokes phase field model using the SAV approach?
• RQ2 Can rigorous second-order error estimates be established for the phase field variable, chemical potential, velocity, and pressure in different discrete norms?
• RQ3 Does the proposed SAV-MAC scheme maintain unconditional energy stability while ensuring computational efficiency?
• RQ4 Can the error analysis be carried out without assuming a uniform Lipschitz condition on the nonlinear potential in the free energy functional?

Key findings

• The proposed SAV-MAC scheme achieves second-order accuracy in both time and space for the phase field variable, chemical potential, velocity, and pressure in appropriate discrete norms.
• The scheme is unconditionally energy stable, preserving the continuous energy dissipation law at the discrete level without requiring time-step restrictions.
• The method is linear and decoupled due to the SAV reformulation, enabling efficient solution via successive solves of linear systems at each time step.
• The first rigorous second-order error estimates for a fully discrete linear scheme are established for the Cahn-Hilliard-Stokes system, without assuming a uniform Lipschitz condition on the nonlinear potential.
• Numerical experiments confirm the theoretical convergence rates and demonstrate the scheme’s robustness and accuracy in simulating phase separation and fluid flow.
• The scheme is applicable to the Cahn-Hilliard-Navier-Stokes model, though error estimates for the full Navier-Stokes case require further development involving higher-order upwind schemes for nonlinear terms.