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Updated: Dec 17, 2025

The Diffusion of Passive Tracers in Laminar Shear Flow
Published on: May 1, 2018
On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection-diffusion
Stephan Simonis1,2, Martin Frank2,3, Mathias J Krause1,2
1Lattice Boltzmann Research Group, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.
Abstract:
The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection-diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.
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