Cross-platform programming model for many-core lattice Boltzmann simulations.

We present a novel, hardware-agnostic implementation strategy for lattice Boltzmann (LB) simulations, which yields massive performance on homogeneous and heterogeneous many-core platforms. Based solely on C++17 Parallel Algorithms, our approach does not rely on any language extensions, external libraries, vendor-specific code annotations, or pre-compilation steps. Thanks in particular to a recently proposed GPU back-end to C++17 Parallel Algorithms, it is shown that a single code can compile and reach state-of-the-art performance on both many-core CPU and GPU environments for the solution of a given non trivial fluid dynamics problem. The proposed strategy is tested with six different, commonly used implementation schemes to test the performance impact of memory access patterns on different platforms. Nine different LB collision models are included in the tests and exhibit good performance, demonstrating the versatility of our parallel approach. This work shows that it is less than ever necessary to draw a distinction between research and production software, as a concise and generic LB implementation yields performances comparable to those achievable in a hardware specific programming language. The results also highlight the gains of performance achieved by modern many-core CPUs and their apparent capability to narrow the gap with the traditionally massively faster GPU platforms. All code is made available to the community in form of the open-source project stlbm, which serves both as a stand-alone simulation software and as a collection of reusable patterns for the acceleration of pre-existing LB codes.

Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods.

The present work proposes a general methodology to study stability and isotropy properties of lattice Boltzmann (LB) schemes. As a first investigation, such a methodology is applied to better understand these properties in the context of regularized approaches. To this extent, linear stability analyses of two-dimensional models are proposed: the standard Bhatnagar-Gross-Krook collision model, the original precollision regularization, and the recursive regularized model, where off-equilibrium distributions are partially computed thanks to a recursive formula. A systematic identification of the physical content carried by each LB mode is done by analyzing the eigenvectors of the linear systems. Stability results are then numerically confirmed by performing simulations of shear and acoustic waves. This work allows drawing fair conclusions on the stability properties of each model. In particular, among the aforementioned models, recursive regularization turns out to be the most stable one for the D2Q9 lattice, especially in the zero-viscosity limit. Two major properties shared by every regularized model are highlighted: (1) a mode filtering property and (2) an incorrect, and broadly anisotropic, dissipation rate of the modes carrying physical waves in under-resolved conditions. The first property is the main source of increased stability, especially for the recursive regularization. It is a direct consequence of the reconstruction of off-equilibrium populations before each collision process, decreasing the rank of the system of discrete equations. The second property seems to be related to numerical errors directly induced by the equilibration of high-order moments. In such a case, this property is likely to occur with any collision model that follows such a stabilization methodology.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria.

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell-Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations.

Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. This article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant, or regularized form. In parallel with that, several bases (nonorthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post-collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. This work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of this article, general instructions are provided to help the reader with the implementation of the most complicated collision models.

Recursive regularization step for high-order lattice Boltzmann methods.

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from 10^{4} to 10^{6}, and where a thorough analysis of the case at Re=3×10^{4} is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.