Comparison of Structure Preserving Schemes for the Cahn-Hilliard-Navier-Stokes Equations with Degenerate Mobility and Adaptive Mesh Refinement

The Cahn-Hilliard-Navier-Stokes (CHNS) system utilizes a diffusive phase-field for interface tracking of multi-phase fluid flows. Recently structure preserving methods for CHNS have moved into focus to construct numerical schemes that, for example, are mass conservative or obey initial bounds of the phase-field variable. In this work decoupled implicit-explicit formulations based on the Discontinuous Galerkin (DG) methodology are considered and compared to existing schemes from the literature.
For the fluid flow a standard continuous Galerkin approach is applied. An adaptive conforming grid is utilized to further draw computational focus on the interface regions, while coarser meshes are utilized around pure phases. All presented methods are compared against each other in terms of bound preservation, mass conservation, and energy dissipation for different examples found in the literature, including a classical rising droplet problem.