A positivity-preserving scheme for fluctuating hydrodynamics

A finite-difference hybrid numerical method for the solution of the isothermal fluctuating hydrodynamic equations is proposed. The primary focus is to ensure the positivity-preserving property of the numerical scheme, which is critical for its functionality and reliability especially when simulating fluctuating vapour systems. Both cases of single- and two-phase flows are considered by exploiting the van der Waals' square-gradient approximation to model the fluid (often referred to as “diffuse-interface” model). The accuracy and robustness of the proposed scheme is verified against several benchmark theoretical predictions for the statistical properties of density, velocity fluctuations and liquid-vapour interface, including the static structure factor of the density field and the spectrum of the capillary waves excited by thermal fluctuations at interface. Finally, the hybrid scheme is applied to the challenging bubble nucleation process, and is shown to capture the salient features of the phenomenon, namely nucleation rate and subsequent bubble-growth dynamics.