Beatrice Battisti

CNRS – Université Savoie Mont Blanc – 🇫🇷 Chambéry (France)

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Implicit-explicit collocated Reduced Order Model schemes for multiscale systems of PDEs

Many physical phenomena are governed by nonlinear systems of partial differential equations, whose solutions span a wide range of spatial and temporal scales, often over large computational domains. Implicit-Explicit (IMEX) time integration schemes address the temporal stiffness of such problems by separating fast and slow dynamics, thus relaxing the severe time step constraint imposed by fully explicit methods, and, therefore, improving the overall computational efficiency. Collocation-based reduced order models (cROMs), on the other hand, drastically reduce computational cost by confining high-fidelity (HF) evaluations to a small set of collocation cells distributed across the spatial mesh, while recovering the full solution via projection onto a reduced basis. In this work, we introduce a novel unified IMEX cROM framework, enabling efficient and accurate simulation of stiff multiscale parametric problems, at significantly reduced computational cost. The proposed scheme is validated on a range of test cases of increasing complexity, for both in-sample and out-of-sample parameter values. For the one-dimensional advection-diffusion equation, we consider HF accuracy up to second order in space and time, and extensions to nonlinear fluxes and variable viscosity coefficients. For the one-dimensional shallow water equations, the Asymptotic Preserving (AP) property of the IMEX scheme is verified, for the first time, at the reduced level in the low-Froude limit. Finally, the scheme is tested on a dam-break problem featuring a strong discontinuity, discretized on a two-dimensional unstructured mesh.