# A Non-Reflecting Boundary Condition for the Multi-Speed Lattice Boltzmann Method

My name is Friedemann Klaß, I am a PhD student in the Applied and Computational Mathematics (ACM) working group at the University of Wuppertal.

I am working in the field of Computational Fluid Dynamics, where I am developing boundary conditions (BC) for the multi-speed Lattice Boltzmann Method (LBM). The LBM can be viewed as a systematic approximation of the Boltzmann equation. In contrast to the famous Navier-Stokes equations, where a fluid is described in terms of macroscopic fields such as density and velocity, the LBM evolves a probability distribution function for the fluid particles. Algorithmically, so-called populations are streamed on a discrete grid along a finite number of directions and collide at the nodes [1].

Operating at this mesoscopic scale facilitates the formulation of complex physical interactions, such as complex geometries, multi-phase or multi-component flows. In addition, the algorithm is inherently highly parallelizable.

It has been found that the use of a higher order quadrature rule in the discretization of the Boltzmann equation extends the applicability of the methods [2]. However, this leads to multi-speed velocity stencils, i.e. populations will move to nodes beyond their nearest neighbours.

At a boundary node, not all post-streaming populations are known due to the lack of neighbours. Therefore, a BC must assign values to the unknown populations in such a way as to produce the desired behaviour (e.g. prescribed velocity at the boundary) at the macroscopic scale. Multi-speed velocity stencils have multiple layers of boundary nodes, making this task even more complex.

There is little literature that investigates BCs for the multi-speed case. To fill this gap, I developed a non-reflecting BC for the multi-speed LBM [3], which is designed to mimic an open boundary and not interact with the bulk dynamics of the flow. Such BCs are often placed at artificial boundaries, that result from truncating a problem’s physical domain to a smaller computational domain. I have also formulated a Dirichlet BC for the simulation of thermal flow using multi-speed velocity stencils [4].

References
[1] Krueger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., & Viggen, E. M. (2016). The Lattice Boltzmann Method: Principles and Practice. (Graduate Texts in Physics). Springer.
[2] Siebert, D. N., Hegele, L. A., & Philippi, P. C. (2008). Lattice Boltzmann equation linear stability analysis: Thermal and athermal models. Physical Review E (Vol. 77, Issue 2). American Physical Society (APS).
[3] Klass, F., Gabbana, A. & Bartel, A. (2023). A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods. Communications in Computational Physics (Vol. 33, Issue 1, pp. 101–117). Global Science Press.
[4] Klass, F., Gabbana, A., & Bartel, A. (2021). A non-equilibrium bounce-back boundary condition for thermal multispeed LBM. Journal of Computational Science (Vol. 53, p. 101364). Elsevier BV.