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We present a fully-coupled, implicit-in-time framework for solving athermodynamically-consistent Cahn-Hilliard Navier-Stokes system that modelstwo-phase flows. In this work, we extend the block iterative method presentedin Khanwale et al. [\textit{Simulating two-phase flows with thermodynamicallyconsistent energy stable Cahn-Hilliard Navier-Stokes equations on paralleladaptive octree based meshes}, J. Comput. Phys. (2020)], to a fully-coupled,provably second-order accurate scheme in time, while maintainingenergy-stability. The new method requires fewer matrix assemblies in eachNewton iteration resulting in faster solution time. The method is based on afully-implicit Crank-Nicolson scheme in time and a pressure stabilization foran equal order Galerkin formulation. That is, we use a conforming continuousGalerkin (cG) finite element method in space equipped with a residual-basedvariational multiscale (RBVMS) procedure to stabilize the pressure. We deploythis approach on a massively parallel numerical implementation using paralleloctree-based adaptive meshes. We present comprehensive numerical experimentsshowing detailed comparisons with results from the literature for canonicalcases, including the single bubble rise, Rayleigh-Taylor instability, andlid-driven cavity flow problems. We analyze in detail the scaling of ournumerical implementation.

computer physics communications

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes  equations: Second-order, energy-stable numerical methods on adaptive octree  based meshes