Séance Séminaire

Séminaire MACS

Structure-preserving numerical schemes for complex multiscale flows

Fluid flows often involve a wide range of interacting scales, arising from different time and length scales, stiff physical processes, or the coexistence of different flow regimes and phases. Accurately and efficiently capturing such multiscale behavior remains a central challenge in numerical computation. In this talk, we will see some numerical techniques to address these multiscale features in the simulation of complex flows, including compressible multiphase systems relevant to geophysical and energy applications, such as water-air flows and volcanic dynamics.

First, a semi-implicit IMEX scheme is introduced for the Baer-Nunziato equations, governing compressible two-phase flows. In this framework, a linearly implicit discretization is employed for both the pressure fluxes and the relaxation source terms, while nonlinear convective terms are treated explicitly. This formulation leads to a CFL-type stability condition on the maximum admissible time step that depends only on the mean flow velocity, rather than on the sound speed of each phase, allowing the scheme to operate uniformly across all Mach number regimes. Shock-capturing finite volume methods are used for the convective fluxes to ensure robustness in the presence of strong compressibility effects. The resulting scheme is well-balanced, and asymptotic-preserving in the low-Mach limit of the mixture model.

To address the computational cost associated with high-fidelity simulations of multiscale flows, reduced order modeling strategies are also considered. These approaches aim at constructing low-dimensional representations of the solution space while remaining consistent with the underlying full-order discretization. In this context, recent developments combining semi-implicit IMEX schemes with collocated reduced order models (cROM) are presented, with particular attention to the preservation of the temporal splitting at the reduced level. In this context, a central question is whether the asymptotic-preserving properties of the high-fidelity scheme, such as the correct behavior in singular regimes (e.g. low-Mach or low-Froude limits), can also be retained in the low-fidelity setting.