Time Integration Schemes for a Hydrodynamic Multiphase Density Gradient Model

In this work, we show how energy stable (also known as gradient stable) linearly implicit time integration schemes can be constructed for a density gradient theory (DGT) system to model hydrodynamics of a two-component two-phase isothermal compressible viscous mixture. Various combinations of the physical process splitting and the convex Eyre splitting are discussed. Their influence on time stepping accuracy is studied numerically. The local iteration modified (LIM) method, based on Chebyshev polynomials, is employed and shown to lead to efficient matrix-free and solution-free, yet stable, time integration schemes. Numerical tests are carried out to determine a largest acceptable time step size for each of the proposed schemes.