Stability of lattice Boltzmann schemes for initial boundary value problems in raw formulation

TL;DR

This paper investigates the stability of one-dimensional linear lattice Boltzmann schemes for scalar hyperbolic equations directly in raw form, introducing strong stability notions and analyzing specific schemes with boundary conditions.

Contribution

It presents a novel approach to analyze lattice Boltzmann scheme stability without transforming into scalar form, focusing on strong stability for schemes with one-sided stencils.

Findings

Identified conditions for strong stability and instability.

Analyzed three representative schemes with various boundary conditions.

Supported theoretical results with numerical simulations.

Abstract

We study the stability of one-dimensional linear lattice Boltzmann schemes for scalar hyperbolic equations with respect to boundary data. Our approach is based on the original raw algorithm on several unknowns, thereby avoiding the need for a transformation into an equivalent scalar formulation-a challenging process in presence of boundaries. To address different behaviors exhibited by the numerical scheme, we introduce appropriate notions of strong stability. They account for the potential absence of a continuous extension of the stable vector bundle associated with the bulk scheme on the unit circle for certain components. Rather than developing a general theory, complicated by the fact that discrete boundaries in lattice Boltzmann schemes are inherently characteristic, we focus on strong stability-instability for methods whose characteristic equations have stencils of breadth one to the left. In this context, we study three representative schemes. These are endowed with various boundary conditions drawn from the literature, and our theoretical results are supported by numerical simulations.