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In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments. 

Divergence-free vector fields and curl-free vector fields play an important role in many types of problems, including the incompressible Navier-Stokes equations, Maxwell's equations, the equations for magnetohydrodynamics, and surface reconstruction. In practice, these fields are often obtained by projection, resulting in a discrete approximation of the continuous field that is discretely divergence-free or discretely curl-free. This field can then be interpolated to non-grid locations, which is required for many algorithms such as particle tracing or semi-Lagrangian advection. This interpolated field will not generally be divergence-free or curl-free in the analytic sense. In this work, we assume these fields are stored on a MAC grid layout and that the divergence and curl operators are discretized using finite differences. This work builds on and extends [39] in multiple ways: (1) we design a divergence-free interpolation scheme that preserves the discrete flux, (2) we adapt the general construction of divergence-free fields into a general construction for curl-free fields, (3) we extend the framework to a more general class of finite difference discretizations, and (4) we use this flexibility to construct fourth-order accurate interpolation schemes for the divergence-free case and the curl-free case. All of the constructions and specific schemes are explicit piecewise polynomials over a local neighborhood.