Implicit time schemes for simulating two-phase flows using diffuse interface methods on adaptive octree grids

The length scales needed to resolve for correct interfacial dynamics governed by diffuse interface methods and the associated velocity scales (governed by NS) are disparate. Specifically, for applications like contact line dynamics, where a small interface thickness parameter is needed, requiring fine mesh resolution to resolve the diffuse interface. We use fast octree-based adaptive meshes to resolve the diffuse interface to overcome this challenge. However, such a mesh adaption strategy leads to a prohibitively restrictive stability condition for explicit time schemes. Therefore, designing efficient and scalable implicit time schemes for diffuse interface methods becomes critical. Such implicit time schemes achieve stability for much larger time steps than explicit time schemes. This study assesses amenability and strategy constraints for designing implicit time schemes for second-order diffuse interface models compared to the fourth-order Cahn-Hilliard-based models. The time schemes are implemented with continuous Galerkin Finite Elements on adaptive octree meshes (AMR). The resulting framework with AMR and implicit time scheme is massively parallel and fast, potentially providing a leeway into simulating interface resolved two-phase flows.