Efficient Hybridizable Discontinuous Galerkin Methods for Phase Field Fluid Models

Multiphase flow is ubiquitous in natural phenomena and industrial applications. Common examples include wave-breaking and sloshing, contaminant transport in aquifers, oil recovery in petroleum engineering, drug delivery in blood flow, gas-particle flow in combustion reactors, exhaust management in Polymer Electrolyte Membrane fuel cell technology, and so forth. The diffuse interface fluid models have become increasingly popular in the numerical modeling of interfacial phenomena associated with multiphase flows. They are able to capture smooth transitions of fluid interface, and simulations can be carried out on a fixed grid without explicit interface tracking. A particular challenge in solving diffuse interface models is that the diffusive interface of small width often exhibits instability such as bubble merging or splitting. Traditional high order methods are prone to spurious oscillations around diffusive interfaces which can pollute the numerical solution beyond the interface region and even cause blow-up of the code due to negative viscosity, density or mobility. The aim of this project is to develop high order numerical methods that can accurately capture moving interfaces of multiphase flow. Real-world applications see both diffusion dominated flows and advection dominated flows. The investigator first develops and analyzes provably superconvergent hybridizable discontinuous Galerkin methods (HDG) for solving diffuse interface fluid models in the diffusion-dominated regime. The key idea in the design is to approximate solution variables by higher order polynomials than those for the numerical traces and gradient variables, and to explore local projection based stabilization. The PI then designs stabilized high order HDG methods effected with the Scalar Auxiliary Variable (SAV) time-stepping schemes for advection-dominated flows. The methods stabilize advection in the nonlinear fourth order advection-diffusion equation while preserve the underlying energy laws. The stabilized SAV-HDG algorithms enable diffuse interface methods to accurately capture sharp fronts and unstable interfaces in the advection-dominated regime, and allow efficient parallel computation of smaller systems at each time step. Finally the PI develops and implements fast nonlinear HDG multigrid solvers for diffuse interface fluid models. The practical solvers will further address the lack of efficient iterative solvers/preconditioners for HDG methods Graduate students participate in the work of this project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.