Weakly Compressible Subcycling for Accelerating Simulations of Surface-Tension-Dominated Incompressible Two-Phase Flows

Abstract: Simulations of surface-tension-dominated incompressible two-phase flows are computationally expensive due to the severe capillary time-step constraint. Although many studies have proposed time-implicit discretizations of surface tension to allow larger time-step sizes and accelerate simulations, these methods suffer from either artificial dissipation or complex implementation. Here, we propose a simple and novel approach: an incompressible solver with weakly compressible subcycling. The proposed approach relaxes the capillary time-step constraint, thereby accelerating simulations by more than $8.6\times$ without relying on artificially dissipative stabilization or requiring complex implementation. The key idea is to introduce lightweight substeps using a weakly compressible solver to assist the main incompressible solver. These substeps enable the main incompressible solver to use accurately computed fluxes and surface tension force, even with large time-step sizes. Numerical tests demonstrate the effectiveness of the proposed approach for practical problems, including the Rayleigh--Plateau instability and two-phase flows in porous media. This study paves the way for a new paradigm in which a weakly compressible solver serves as an assistant to an incompressible solver.