Method and Analysis I
2:00pm - 2:20pm
"A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme".
1LMSSC, CNAM Paris, France; 2LMO, Université Paris-Saclay, France
We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution towards the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair be-
ing given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation.
2:20pm - 2:40pm
A comparative study of 3D Cumulant and Central Moments lattice Boltzmann schemes with interpolated boundary conditions for the simulation of thermal flows in high Prandtl number regime
1Warsaw University of Technology, Poland; 2School of Mechanical and Mining Engineering, The University of Queensland, St Lucia, Australia
Thermal flows characterized by high Prandtl number are numerically challenging as the transfer of momentum and heat occurs at different time scales. To account for very low thermal conductivity and obey the Courant–Friedrichs–Lewy condition, the numerical diffusion of the scheme has to be reduced. As a consequence, the numerical artefacts are dominated by the dispersion errors commonly known as wiggles. In this study, we explore possible remedies for these issues in the framework of lattice Boltzmann method by means of applying novel collision kernels, lattices with large number of discrete velocities, namely D3Q27, and a second-order boundary conditions.
For the first time, the cumulant-based collision operator, is utilised to simulate both the hydrodynamic and the thermal field. Alternatively, the advected field is computed using the central moments’ collision operator. Different relaxation strategies have been examined to account for additional degrees of freedom introduced by a higher order lattice.
To validate the proposed kernels for a pure advection-diffusion problem, the numerical simulations are compared against analytical solution of a Gaussian hill. The structure of the numerical dispersion is shown by simulating advection and diffusion of a square indicator function. Next, the influence of the interpolated boundary conditions on the quality of the results is measured in the case of the heat conduction between two concentric cylinders. Finally, a study of steady forced heat convection from a confined cylinder is performed and compared against a Finite Element Method solution.
It has been found, that the relaxation scheme of the advected field must be adjusted to profit from lattice with a larger number of discrete velocities, like D3Q27. Obtained results show clearly that it is not sufficient to assume that only the first-order central moments/cumulants contribute to solving the macroscopic advection-diffusion equation. In the case of central moments, the beneficial effect of the two relaxation time approach is presented .
 Gruszczyński, G., and Ł. Łaniewski-Wołłk. "A comparative study of 3D Cumulant and Central Moments lattice Boltzmann schemes with interpolated boundary conditions for the simulation of thermal flows in high Prandtl number regime." arXiv preprint arXiv:2203.01316 (2022).
2:40pm - 3:00pm
Finite Difference formulation of lattice Boltzmann schemes: consequences on consistency and stability
1CMAP, Ecole polytechnique, France; 2Institut de Mathématique d'Orsay, Université Paris-Saclay, France
Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. Despite the well-known advantages from a computational standpoint, this structure is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any “classical” lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme, which can be studied without explicitly constructing this scheme, allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Our formalism could lead to a better understanding of other important topics, such as boundary conditions and initial conditions.
3:00pm - 3:20pm
Extended comparison between lattice Boltzmann and Navier-Stokes solvers for unsteady aerodynamic and aeroacoustic computations
1DAAA, ONERA, Université Paris Saclay, F-92322 Châtillon - France; 2Laboratoire DynFluid, F-75013 Paris - France; 3Conservatoire National des Arts et Métiers, F-75003 Paris - France
Computational Fluid Dynamics (CFD) has become an important tool in aerospace sciences enabling both researchers and engineers to get more insight into complex fluid phenomena. The increasing computational power and the growing need of high-fidelity methods has lead to the development of Large Eddy Simulations (LES) tools among which structured finite-type Navier-Stokes (NS) methods and lattice Boltzmann methods (LBM) are the most promising ones to achieve industrial level computations . Consequently, one question which naturally arises is: Which method is the most competitive, in terms of accuracy and computational cost, on canonical aerodynamic and aeroacoustic applications ?
Previous work on the comparison of the LBM with traditional NS methods focused on different topics such as convergence order , achievable error  and runtimes . However, there still is a lack of fair one-to-one comparisons. Indeed, runtime-based results were obtained with two different solvers developed independently and having different levels of optimisation. In addition, the numerical properties of the lattice Boltzmann method are highly dependent on the collision operator  such that the conclusions of  have to be tempered.
This work aims at rigourously comparing a lattice Boltzmann solver with an LES-type finite-volume Navier-Stokes solver. The comparison takes place in ONERA's Cassiopée/Fast CFD environment implementing high-performance flow solvers relying on the same code architecture and optimisation layers. To do so, an extended von Neumann analysis of both lattice Boltzmann and Navier-Stokes schemes is proposed. The study is completed by numerical test cases to highlight the capabilities of each method. The implementation and computational times are also discussed. Finally, some trends about the performance of each methods are outlined.
 R. Löhner, “Towards overcoming the LES crisis,” Int. J. Comut. Fluid Dyn., vol. 33, no. 3, pp. 87–97, Mar. 2019.
 D. R. Noble, J. G. Georgiadis, and R. O. Buckius, “Comparison of accuracy and performance for lattice Boltzmann and finite difference simulations of steady viscous flow,” Int. J. Numer. Methods Fluids, vol. 23, no. 1, pp. 1–18, 1996.
 S. Marié, D. Ricot, and P. Sagaut, “Comparison between lattice Boltzmann method and Navier-Stokes high order schemes for computational aeroacoustics,” J. Comput. Phys., vol. 228, no. 4, pp. 1056–1070, Mar. 2009.
 K.-R. Wichmann, M. Kronbichler, R. Löhner, and W. A. Wall, “A runtime based comparison of highly tuned lattice Boltzmann and finite difference solvers,” Int. J. High Perform. Comput. Appl., pp. 370–390, Apr. 2021.
 G. Wissocq, C. Coreixas, and J.-F. Boussuge, “Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods,” Phys. Rev. E, vol. 102, no. 5, p. 053305, Nov. 2020.