Analysis of Carlemann Linearization of Lattice Boltzmann

TL;DR

This paper investigates Carlemann linearization of the lattice Boltzmann collision term to facilitate quantum algorithm development, analyzing error bounds, variable growth, and impacts on streaming accuracy.

Contribution

It introduces the application of Carlemann linearization to lattice Boltzmann methods, providing analytical bounds and insights for quantum algorithm design.

Findings

Error decreases exponentially with linearization order.

Linearizing collision affects streaming exactness.

Number of variables grows with linearization order.

Abstract

We explore the Carlemann linearization of the collision term of the lattice Boltzmann formulation, as a first step towards formulating a quantum lattice Boltzmann algorithm. Specifically, we deal with the case of a single, incompressible fluid with the Bhatnagar Gross and Krook equilibrium function. Under this assumption, the error in the velocities is proportional to the square of the Mach number. Then, we showcase the Carlemann linearization technique for the system under study. We compute an upper bound to the number of variables as a function of the order of the Carlemann linearization. We study both collision and streaming steps of the lattice Boltzmann formulation under Carlemann linearization. We analytically show why linearizing the collision step sacrifices the exactness of streaming in lattice Boltzmann, while also contributing to the blow up in the number of Carlemann variables in the classical algorithm. The error arising from Carlemann linearization has been shown analytically and numerically to improve exponentially with the Carlemann linearization order. This bodes well for the development of a corresponding quantum computing algorithm based on the Lattice Boltzmann equation.