Taming the stiff stiffness of pressure work and equilibration in numerical schemes for compressible multi-fluid flows

Designing models and schemes for compressible multi-fluid flow is often considered as challenging when dealing with contrasted equations of state, low volume fractions, or high compression or expansion ratios. This is due to the potentially severe stiffness of pressure couplings, aggravated by (i) their quadratic multiplicity as they connect all energy reservoirs (ii) the entropy conditions to be preserved on each fluid, (iii) their potentially unknown signs, and (iv) their potential stiff stiffness , defined in the present work. Whether captured under pressure-equilibration or pressure-relaxation models-where fluids respectively share a common pressure or undergo a damped evolution towards a common pressure-the stiff terms of the evolution equations have always required dedicated numerical approaches, with sometimes mixed results depending on specific applications. Some broad general guidelines are here proposed to tame the stiff stiffness issues of the pressure terms. The framework is that of pressure-equilibrated average-field (or Euler-Euler) models, discretized with a splitting of momentum and internal energy equations: this makes the stiff terms behave locally as simple ODEs, amenable to integration by simple schemes. The present approach resorts to (partly) exponential integrators to provide explicit estimations of implicit pressures. The final explicit scheme, though not as robust as would its fully implicit version, displays a vastly improved resilience to stiffness. Numerical tests were carried out on strenuous versions of usual 1D shock tubes on two-fluid mixtures of air and water (described by ideal and stiffened gases). In view of the excellent results, further 1D tests in more extreme conditions were considered: free expansion to vacuum and shocks on zero pressure state (Noh's test). At usual acoustic CFL values, expansion was robustly simulated for air volume fractions from less than 10(-12) to up to nearly 1. Similarly, shocks on zero pressure states were simulated to different final states for air volume fractions as low as 10(-8), with air density factors across shocks ranging between a few hundreds and hundredths depending on the choice of dissipation in each fluid. All tests were carried out on a previously developed Geometry, Energy, and Entropy Consistent (GEEC) scheme (Vazquez-Gonzalez et al., 2020).

Automated summary

Designing models and schemes for compressible multi-fluid flow is challenging due to the stiffness of pressure couplings and the need to preserve entropy conditions. This study proposes general guidelines to address the stiffness issues of pressure terms and presents a numerical approach that uses exponential integrators to estimate implicit pressures explicitly. The results show improved resilience to stiffness and successful simulations of different flow conditions.