A low-rank algorithm for weakly compressible flow

TL;DR

This paper introduces a low-rank numerical method for weakly compressible fluid flow that efficiently solves the Boltzmann equation and can be combined with various discretization strategies, avoiding the sonic CFL constraint.

Contribution

It presents a novel low-rank splitting scheme for the Boltzmann equation that simplifies computations and integrates with spectral and semi-Lagrangian methods.

Findings

Efficiently captures relevant flow dynamics with low-rank approximation.

Avoids sonic CFL condition in numerical schemes.

Compatible with multiple discretization techniques.

Abstract

In this paper, we propose a numerical method for solving weakly compressible fluid flow based on a dynamical low-rank projector splitting. The low-rank splitting scheme is applied to the Boltzmann equation with BGK collision term, which results in a set of constant coefficient advection equations. This procedure is numerically efficient as a small rank is sufficient to obtain the relevant dynamics (described by the Navier--Stokes equations). The resulting method can be combined with a range of different discretization strategies; in particular, it is possible to implement spectral and semi-Lagrangian methods, which allows us to design numerical schemes that are not encumbered by the sonic CFL condition.