On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation

 L. Pareschi, T. Rey (11/11/2020 preprint ArXiv: 2011.05811)

Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.

High order pressure-based semi-implicit IMEX schemes for the 3D Navier-Stokes equations at all Mach numbers

Walter Boscheri, Lorenzo Pareschi (4/8/2020 preprint arXiv:2008.01789)

This article aims at developing a high order pressure-based solver for the solution of the 3D compressible Navier-Stokes system at all Mach numbers. We propose a cell-centered discretization of the governing equations that splits the fluxes into a fast and a slow scale part, that are treated implicitly and explicitly, respectively. A novel semi-implicit discretization is proposed for the kinetic energy as well as the enthalpy fluxes in the energy equation, hence avoiding any need of iterative solvers. The implicit discretization yields an elliptic equation on the pressure that can be solved for both ideal gas and general equation of state (EOS).

Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

Giulia Bertaglia, Lorenzo Pareschi (8/7/2020, ESAIM: Mathematical Modelling and Numerical Analysis to appear,  arXiv:2007.04019)

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit.

Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model

Giulia Bertaglia, Valerio Caleffi, Lorenzo Pareschi, Alessandro Valiani (3/7/2020 preprint arXiv:2007.01907)

This work aims at identifying and quantifying uncertainties related to elastic and viscoelastic parameters, which characterize the arterial wall behavior, in one-dimensional modeling of the human arterial hemodynamics. The chosen uncertain parameters are modeled as random Gaussian-distributed variables, making stochastic the system of governing equations. The proposed methodology is initially discussed for a model equation, presenting a thorough convergence study which confirms the spectral accuracy of the stochastic collocation method and the second-order of accuracy of the IMEX finite volume scheme chosen to solve the mathematical model.