pylbm

pylbm is an all-in-one package for numerical simulations using Lattice Boltzmann solvers.This package gives all the tools to describe your lattice Boltzmann scheme in 1D, 2D and 3D problems.We choose the D'Humières formalism to describe the problem. You can have complex geometry with a set of simple shape like circle, sphere, ...pylbm performs the numerical scheme using Cython, NumPy or Loo.py from the scheme and the domain given by the user. Pythran and Numba wiil be available soon. pylbm has MPI support with mpi4py

Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic- Woodbury-GenEO) are constructed algebraically. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non- assembled form is known by construction. The extra cost is a second (and to this day rather expensive) coarse solve in the preconditioner. One of the AWG preconditioners has already been presented in the short preprint [38]. This article is the first full presentation of the larger family of AWG preconditioners. It includes proofs of the spectral bounds as well as numerical illustrations

GenEO

The open source software GenEO, written in Python, includes two new families of preconditioners for symmetric positive definite linear systems. 1) First, the AWG preconditioners (for Algebraic-Woodbury-GenEO) have the feature of being algebraic \cite{zbMATH07846109,10.1007/978-3-030-95025-5_81}. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non-assembled form is known by construction. The extra cost is a second coarse solve in the preconditioner. 2) Second, the framework for Krylov subspace methods with adaptive multipreconditioning is implemented. Multipreconditiioning is a technique that allows to apply more than one preconditioner at each step. Domain decomposition is a natural application. Since a multipreconditioned iteration is more expensive than a classical iteration, it is advantageous to multiprecondition only when necessary. To this end, an adapativity scheme was proposed in \cite{zbMATH06601530} and is implemented in GenEO.GenEO uses Petsc4py and Dolfinx to solve 2D and 3D problems. Then, it is easy to compare this new family of preconditioners with those already defined in Petsc and see their impact on various problems with highly heterogeneous coefficients.@Article{zbMATH06601530, Author = {Spillane, Nicole}, Title = {An adaptive multipreconditioned conjugate gradient algorithm}, FJournal = {SIAM Journal on Scientific Computing}, Journal = {SIAM J. Sci. Comput.}, ISSN = {1064-8275}, Volume = {38}, Number = {3}, Pages = {a1896--a1918}, Year = {2016}, Language = {English}, DOI = {10.1137/15M1028534}, Keywords = {65F10,65N30,65N55}, zbMATH = {6601530}, Zbl = {1416.65087}}@Article{zbMATH07846109, Author = {Gouarin, Lo{\"{\i}}c and Spillane, Nicole}, Title = {Fully algebraic domain decomposition preconditioners with adaptive spectral bounds}, FJournal = {ETNA. Electronic Transactions on Numerical Analysis}, Journal = {ETNA, Electron. Trans. Numer. Anal.}, ISSN = {1068-9613}, Volume = {60}, Pages = {169--196}, Year = {2024}, Language = {English}, DOI = {10.1553/etna_vol60s169}, Keywords = {65F10,65N30,65N55}, URL = {etna.mcs.kent.edu/volumes/2021-2030/vol60/abstract.php?vol=60&pages=169-196}, zbMATH = {7846109}}@inproceedings{10.1007/978-3-030-95025-5_81,abstract = {The starting point for the algebraic preconditioner is to relax condition (1) by allowing symmetric, but possibly indefinite, matrices in the splitting of A.},address = {Cham},author = {Spillane, Nicole},booktitle = {Domain Decomposition Methods in Science and Engineering XXVI},editor = {Brenner, Susanne C. and Chung, Eric and Klawonn, Axel and Kwok, Felix and Xu, Jinchao and Zou, Jun},isbn = {978-3-030-95025-5},pages = {745--752},publisher = {Springer International Publishing},title = {Toward a New Fully Algebraic Preconditioner for Symmetric Positive Definite Problems},year = {2022}

High accuracy analysis of adaptive multiresolution-based lattice Boltzmann schemes via the equivalent equations

International audienceMultiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have introduced this strategy before to construct adaptive lattice Boltzmann methods with this interesting feature.Furthermore, these schemes allow for an effective memory compression of the solution when spatially localized phenomena -- such as shocks or fronts -- are involved, to rely on the original scheme without any manipulation at the finest level of grid and to reach a high level of accuracy on the solution. Nevertheless, the peculiar way of modeling the desired physical phenomena in the lattice Boltzmann schemes calls, besides the possibility of controlling the error introduced by the mesh adaptation, for a deeper and more precise understanding of how mesh adaptation alters the physics approximated by the numerical strategy. In this contribution, this issue is studied by performing the equivalent equations analysis of the adaptive method after writing the scheme under an adapted formalism. It provides an essential tool to master the perturbations introduced by the adaptive numerical strategy, which can thus be devised to preserve the desired features of the reference scheme at a high order of accuracy. The theoretical considerations are corroborated by numerical experiments in both the 1D and 2D context, showing the relevance of the analysis. In particular, we show that our numerical method outperforms traditional approaches, whether or not the solution of the reference scheme converges to the solution of the target equation. Furthermore, we discuss the influence of various collision strategies for non-linear problems, showing that they have only a marginal impact on the quality of the solution, thus further assessing the proposed strategy

Multiresolution-based mesh adaptation and error control for lattice Boltzmann methods with applications to hyperbolic conservation laws

International audienceLattice Boltzmann Methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this work, we design a class of fully adaptive LBM methods with dynamic mesh adaptation and error control relying on multiresolution analysis. This wavelet-based approach allows to adapt the mesh based on the regularity of the solution and leads to a very efficient compression of the solution without loosing its quality and with the preservation of the properties of the original LBM method on the finest grid. This yields a general approach for a large spectrum of schemes and allows precise error bounds, without the need for deep modifications on the reference scheme. An error analysis is proposed. For the purpose of assessing the approach, we conduct a series of test-cases for various schemes and scalar and systems of conservation laws, where solutions with shocks are to be found and local mesh adaptation is especially relevant. Theoretical estimates are retrieved while a reduced memory footprint is observed. It paves the way to an implementation in a multi-dimensional framework and high computational efficiency of the method for both parabolic and hyperbolic equations, which is the subject of a companion paper

Does the multiresolution lattice Boltzmann method allow to deal with waves passing through mesh jumps?

International audienceWe consider an adaptive multiresolution-based lattice Boltzmann scheme, which we have recently introduced and studied from the perspective of the error control and the theory of the equivalent equations. This numerical strategy leads to high compression rates, error control and its high accuracy has been explained on uniform and dynamically adaptive grids. However, one key issue with non-uniform meshes within the framework of lattice Boltzmann schemes is to properly handle acoustic waves passing through a level jump of the grid. It usually yields spurious effects, in particular reflected waves. In this paper, we propose a simple mono-dimensional test-case for the linear wave equation with a fixed adapted mesh characterized by a potentially large level jump. We investigate this configuration with our original strategy and prove that we can handle and control the amplitude of the reflected wave, which is of fourth order in the space step of the finest mesh. Numerical illustrations show that the proposed strategy outperforms the existing methods in the literature and allow to assess the ability of the method to handle the mesh jump properly