A Low-Rank Algorithm for Weakly Compressible Flow

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.

A Quasi-Conservative Dynamical Low-Rank Algorithm for the Vlasov Equation

“…Computing numerical solutions of high-dimensional evolutionary partial differential equations by dynamical low-rank approximation has only recently been considered for kinetic problems [15,14]. However, such algorithms have been investigated extensively in quantum mechanics; see, in particular, [34,33] for the MCTDH approach to molecular quantum dynamics in the chemical physics literature and [25,26] for a computational mathematics point of view of this approach.…”

“…In contrast to standard time-stepping methods, the projector-splitting methods have been shown to be robust to the typical presence of small singular values in the low-rank approximation [20]. The approach in [15,14] and in the present paper is based on an adaptation of the projector-splitting method of [28] to kinetic equations.…”

“…However, to the best of our knowledge, no low-rank algorithms are available that are able to conserve even linear invariants. Furthermore, it has recently been proposed to use low-rank numerical methods to solve fluid problems [14]. Also in this setting conservation of mass and momentum, a hallmark of traditional fluid solvers, is, of course, of great interest.…”

A low-rank projector-splitting integrator for the Vlasov–Maxwell equations with divergence correction

“…In this section, we explain the low-rank integrator for the Vlasov-Maxwell equations for the 3+3 dimensional case in full detail, following the ideas in the seminal paper [25] and the recent works [15,11,14]. First, we summarize the framework for the construction of the integrator.…”

“…Recently, a different approach was proposed in the context of the Vlasov-Poisson equations; see [15] and the follow-up papers [11,14]. The key idea is to approximate the dynamics of the system by constraining it to the low-rank manifold.…”