Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis

Bellotti, Thomas; Gouarin, Loïc; Graille, Benjamin; Massot, Marc

doi:10.1016/j.jcp.2022.111670

Cited by 6 publications

(13 citation statements)

References 57 publications

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“…Moreover, we present a concise numerical study on the effect of different strategies for the treatment of the collision operator on the reliability of the adaptive method. This study confirms the relevance of the local approach, introduced in the previous works [3,4], in which the collision phase is performed on the leaves. This yields accuracy levels comparable to those of more computationally expensive strategies, which are briefly introduced in this work.…”

Section: Analysis Of the Lbm-mr Schemes Via The Equivalent Equationssupporting

confidence: 88%

“…Remark that, by (2.1), the time-step is constant across all the levels and is dictated by the finest resolution allowed for the hybrid grid, which corresponds to the resolution of reference scheme on the uniform finest mesh. This approach complies with [18,19,20] and with our previous works [3,4]. Other choices are possible to build lattice Boltzmann methods on non-uniform grids: the predominant one consists in taking a different time-step for each level of resolution [16,21,29].…”

Section: Spatial Discretizationmentioning

confidence: 79%

“…This calls for the adoption of a spatial discretization with different levels of refinement. In this Section, we summarize our adaptive approach presented in [3,4] to transform a method acting on a uniform mesh at the finest available scale into another working on a spatially adaptive grid spanning many spatial scales. The adaptive grid is dynamically updated at each time step using a regularity evaluation through the multiresolution analysis, ensuring error control.…”

Section: Adaptive Space Discretization and Adaptive Lattice Boltzmann...mentioning

confidence: 99%

“…Consider, for the sake of a simpler presentation, a bounded domain Ω = [0, 1] d , typically for d = 1, 2, 3. Following our previous contributions [3,4], we can build a hybrid partition of such a domain formed by cells at analysis of the LBM-MR schemes via the equivalent equations…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…contributions [3,4], we have introduced and tested a novel way of implementing lattice Boltzmann methods to be deployed on spatially adapted grids generated by multiresolution analysis (called LBM-MR method). The collision phase (due to its inherent local nature) was performed only on the leaves of the corresponding adaptive graded tree where the data are directly available by using the equilibria of the reference lattice Boltzmann scheme.…”

Section: Introductionmentioning

confidence: 99%

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

Bellotti¹,

Gouarin²,

Graille³

et al. 2022

The SMAI Journal of Computational Mathematics

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Multiresolution 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 previously introduced this strategy 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 target equations describing the physics. 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.

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Section: Analysis Of the Lbm-mr Schemes Via The Equivalent Equationssupporting

confidence: 88%

Section: Spatial Discretizationmentioning

confidence: 79%

Section: Adaptive Space Discretization and Adaptive Lattice Boltzmann...mentioning

confidence: 99%

Section: Spatial Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Bellotti¹,

Gouarin²,

Graille³

et al. 2022

The SMAI Journal of Computational Mathematics

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Finite Difference formulation of any lattice Boltzmann scheme

2022

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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. This structure, despite the well-known advantages from a computational standpoint, 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 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 allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequentlyused 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. Their relevance is verified by means of numerical illustrations.

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Direct simulations of external flow and noise radiation using the generalized interpolation-supplemented cascaded lattice Boltzmann method

Song,

Zhang,

Zhao

et al. 2024

Computers & Fluids

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Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis

Cited by 6 publications

References 57 publications

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

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

Finite Difference formulation of any lattice Boltzmann scheme

Direct simulations of external flow and noise radiation using the generalized interpolation-supplemented cascaded lattice Boltzmann method

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