Second order fully discrete energy stable methods on staggered grids for hydrodynamic phase field models of binary viscous fluids

Abstract

We present fully-discrete, energy stable methods on a spatially staggered grid for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. Based on the idea of traditional structure-preserving algorithms, a fully nonlinear implicit energy stable method is constructed. Afterward, we apply the newly developed energy quadratization (EQ) strategy to develop a linear-implicit scheme, where in each time step only a linear algebraic equation system needs to be solved. Then, we extend it to a decoupled, linear scheme by introducing an intermediate velocity term, where the phase variable, velocity field and pressure can be solved sequentially. The two new fully discrete linear schemes are then shown to be unconditionally energy stable and the linear systems resulted from the schemes are proved to be uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared with the coupled scheme. The code developed using the coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric solutions.