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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.05354">arXiv:2201.05354</a> <span> [<a href="https://arxiv.org/pdf/2201.05354">pdf</a>, <a href="https://arxiv.org/format/2201.05354">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Finite Difference formulation of any lattice Boltzmann scheme </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bellotti%2C+T">Thomas Bellotti</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.05354v1-abstract-short" style="display: inline;"> 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 t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.05354v1-abstract-full').style.display = 'inline'; document.getElementById('2201.05354v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.05354v1-abstract-full" style="display: none;"> 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 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. Their relevance is verified by means of numerical illustrations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.05354v1-abstract-full').style.display = 'none'; document.getElementById('2201.05354v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76M28; 65M06; 65M12; 15A15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.13816">arXiv:2105.13816</a> <span> [<a href="https://arxiv.org/pdf/2105.13816">pdf</a>, <a href="https://arxiv.org/format/2105.13816">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> High accuracy analysis of adaptive multiresolution-based lattice Boltzmann schemes via the equivalent equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bellotti%2C+T">Thomas Bellotti</a>, <a href="/search/math?searchtype=author&query=Gouarin%2C+L">Lo茂c Gouarin</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.13816v1-abstract-short" style="display: inline;"> 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 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… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.13816v1-abstract-full').style.display = 'inline'; document.getElementById('2105.13816v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.13816v1-abstract-full" style="display: none;"> 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 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.13816v1-abstract-full').style.display = 'none'; document.getElementById('2105.13816v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.12609">arXiv:2105.12609</a> <span> [<a href="https://arxiv.org/pdf/2105.12609">pdf</a>, <a href="https://arxiv.org/format/2105.12609">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Does the multiresolution lattice Boltzmann method allow to deal with waves passing through mesh jumps? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bellotti%2C+T">Thomas Bellotti</a>, <a href="/search/math?searchtype=author&query=Gouarin%2C+L">Lo茂c Gouarin</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.12609v2-abstract-short" style="display: inline;"> We 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 m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.12609v2-abstract-full').style.display = 'inline'; document.getElementById('2105.12609v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.12609v2-abstract-full" style="display: none;"> We 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.12609v2-abstract-full').style.display = 'none'; document.getElementById('2105.12609v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.02903">arXiv:2103.02903</a> <span> [<a href="https://arxiv.org/pdf/2103.02903">pdf</a>, <a href="https://arxiv.org/format/2103.02903">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jcp.2022.111670">10.1016/j.jcp.2022.111670 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bellotti%2C+T">Thomas Bellotti</a>, <a href="/search/math?searchtype=author&query=Gouarin%2C+L">Lo茂c Gouarin</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.02903v2-abstract-short" style="display: inline;"> Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.02903v2-abstract-full').style.display = 'inline'; document.getElementById('2103.02903v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.02903v2-abstract-full" style="display: none;"> Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in the literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equations, thus being less problem-dependent than the AMR approaches, which have a hard time guaranteeing an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the grid. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.02903v2-abstract-full').style.display = 'none'; document.getElementById('2103.02903v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.12163">arXiv:2102.12163</a> <span> [<a href="https://arxiv.org/pdf/2102.12163">pdf</a>, <a href="https://arxiv.org/format/2102.12163">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Multiresolution-based mesh adaptation and error control for lattice Boltzmann methods with applications to hyperbolic conservation laws </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gouarin%2C+L">Lo茂c Gouarin</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a>, <a href="/search/math?searchtype=author&query=Bellotti%2C+T">Thomas Bellotti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.12163v2-abstract-short" style="display: inline;"> Lattice 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 met… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.12163v2-abstract-full').style.display = 'inline'; document.getElementById('2102.12163v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.12163v2-abstract-full" style="display: none;"> Lattice 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.12163v2-abstract-full').style.display = 'none'; document.getElementById('2102.12163v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.12947">arXiv:2006.12947</a> <span> [<a href="https://arxiv.org/pdf/2006.12947">pdf</a>, <a href="https://arxiv.org/format/2006.12947">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Unexpected convergence of lattice Boltzmann schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boghosian%2C+B">Bruce Boghosian</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Lallemand%2C+P">Pierre Lallemand</a>, <a href="/search/math?searchtype=author&query=Tekitek%2C+M">Mohamed-Mahdi Tekitek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.12947v1-abstract-short" style="display: inline;"> In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time step tends to zero shows that the numerical soluti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.12947v1-abstract-full').style.display = 'inline'; document.getElementById('2006.12947v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.12947v1-abstract-full" style="display: none;"> In this work, we study numerically the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times when the time step is proportional to the space step and tends to zero. We do this by a combination of theory and numerical experiment. The classical formal analysis when all the relaxation parameters are fixed and the time step tends to zero shows that the numerical solution converges to solutions of the heat equation, with a constraint connecting the diffusivity, the space step and the coefficient of relaxation of the momentum. If the diffusivity is fixed and the space step tends to zero, the relaxation parameter for the momentum is very small, causing a discrepency between the previous analysis and the numerical results. We propose a new analysis of the method for this specific situation of evanescent relaxation, based on the dispersion equation of the lattice Boltzmann scheme. A new asymptotic partial differential equation, the damped acoustic system, is emergent as a result of this formal analysis. Complementary numerical experiments establish the convergence of the scalar D2Q9 lattice Boltzmann scheme with multiple relaxation times and acoustic scaling in this specific case of evanescent relaxation towards the numerical solution of the damped acoustic system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.12947v1-abstract-full').style.display = 'none'; document.getElementById('2006.12947v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Computers & Fluids, 2020, 172, pp.301 - 311 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.12215">arXiv:1911.12215</a> <span> [<a href="https://arxiv.org/pdf/1911.12215">pdf</a>, <a href="https://arxiv.org/format/1911.12215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A stability property for a mono-dimensional three velocities scheme with relative velocity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Rao%2C+S+V+R">S. V. Raghurama Rao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.12215v1-abstract-short" style="display: inline;"> In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.12215v1-abstract-full').style.display = 'inline'; document.getElementById('1911.12215v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.12215v1-abstract-full" style="display: none;"> In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.12215v1-abstract-full').style.display = 'none'; document.getElementById('1911.12215v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1905.12393">arXiv:1905.12393</a> <span> [<a href="https://arxiv.org/pdf/1905.12393">pdf</a>, <a href="https://arxiv.org/format/1905.12393">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Caetano%2C+F">Filipa Caetano</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1905.12393v3-abstract-short" style="display: inline;"> 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 toward 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 b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.12393v3-abstract-full').style.display = 'inline'; document.getElementById('1905.12393v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1905.12393v3-abstract-full" style="display: none;"> 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 toward 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 being 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. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.12393v3-abstract-full').style.display = 'none'; document.getElementById('1905.12393v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.10436">arXiv:1806.10436</a> <span> [<a href="https://arxiv.org/pdf/1806.10436">pdf</a>, <a href="https://arxiv.org/format/1806.10436">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Solar and Stellar Astrophysics">astro-ph.SR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Physics">physics.class-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Plasma Physics">physics.plasm-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1137/18M1194225">10.1137/18M1194225 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Numerical treatment of the nonconservative product in a multiscale fluid model for plasmas in thermal nonequilibrium: application to solar physics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wargnier%2C+Q">Quentin Wargnier</a>, <a href="/search/math?searchtype=author&query=Faure%2C+S">Sylvain Faure</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Magin%2C+T">Thierry Magin</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1806.10436v2-abstract-short" style="display: inline;"> This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutions of a nonconservative product present in the ele… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.10436v2-abstract-full').style.display = 'inline'; document.getElementById('1806.10436v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.10436v2-abstract-full" style="display: none;"> This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on travelling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution, that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527--599]. The numerical strategy is eventually assessed in the framework of a solar physics test case. The computational method is able to capture the travelling wave solutions in both the highly- and coarsely-resolved cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.10436v2-abstract-full').style.display = 'none'; document.getElementById('1806.10436v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2020, 42 (2), pp.B492-B519 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.08770">arXiv:1803.08770</a> <span> [<a href="https://arxiv.org/pdf/1803.08770">pdf</a>, <a href="https://arxiv.org/format/1803.08770">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Cellular Automata and Lattice Gases">nlin.CG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4208/cicp.OA-2016-0257">10.4208/cicp.OA-2016-0257 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boghosian%2C+B">Bruce Boghosian</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Lallemand%2C+P">Pierre Lallemand</a>, <a href="/search/math?searchtype=author&query=Tekitek%2C+M">Mohamed-Mahdi Tekitek</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1803.08770v1-abstract-short" style="display: inline;"> We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.08770v1-abstract-full').style.display = 'inline'; document.getElementById('1803.08770v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.08770v1-abstract-full" style="display: none;"> We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.08770v1-abstract-full').style.display = 'none'; document.getElementById('1803.08770v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Contribution published in Communications in Computational Physics, 2018, 23, pp.1263 - 1278 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.02369">arXiv:1802.02369</a> <span> [<a href="https://arxiv.org/pdf/1802.02369">pdf</a>, <a href="https://arxiv.org/format/1802.02369">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Physics">physics.class-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.4967541">10.1063/1.4967541 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Recovering the full Navier Stokes equations with lattice Boltzmann schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Lallemand%2C+P">Pierre Lallemand</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.02369v1-abstract-short" style="display: inline;"> We consider multi relaxation times lattice Boltzmann scheme with two particle distributions for the thermal Navier Stokes equations formulated with conservation of mass and momentum and dissipation of volumic entropy.Linear stability is taken into consideration to determine a coupling between two coefficients of dissipation.We present interesting numerical results for one-dimensional strong… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.02369v1-abstract-full').style.display = 'inline'; document.getElementById('1802.02369v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.02369v1-abstract-full" style="display: none;"> We consider multi relaxation times lattice Boltzmann scheme with two particle distributions for the thermal Navier Stokes equations formulated with conservation of mass and momentum and dissipation of volumic entropy.Linear stability is taken into consideration to determine a coupling between two coefficients of dissipation.We present interesting numerical results for one-dimensional strong nonlinear acoustic waves with shocks. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.02369v1-abstract-full').style.display = 'none'; document.getElementById('1802.02369v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> 30th International Symposium on Rarefied Gas Dynamics, Jul 2016, Victoria, BC, Canada. American Institute of Physics Proceedings, volume 1786 (040003), pp.40003 - 40003, 2016 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.02381">arXiv:1506.02381</a> <span> [<a href="https://arxiv.org/pdf/1506.02381">pdf</a>, <a href="https://arxiv.org/format/1506.02381">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Stability of a bidimensional relative velocity lattice Boltzmann scheme </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=F%C3%A9vrier%2C+T">Tony F茅vrier</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.02381v1-abstract-short" style="display: inline;"> In this contribution, we study the theoretical and numerical stability of a bidimensional relative velocity lattice Boltzmann scheme. These relative velocity schemes introduce a velocity field parameter called "relative velocity" function of space and time. They generalize the d'Humi猫res multiple relaxation times scheme and the cascaded automaton. This contribution studies the stability of a four… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.02381v1-abstract-full').style.display = 'inline'; document.getElementById('1506.02381v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.02381v1-abstract-full" style="display: none;"> In this contribution, we study the theoretical and numerical stability of a bidimensional relative velocity lattice Boltzmann scheme. These relative velocity schemes introduce a velocity field parameter called "relative velocity" function of space and time. They generalize the d'Humi猫res multiple relaxation times scheme and the cascaded automaton. This contribution studies the stability of a four velocities scheme applied to a single linear advection equation according to the value of this relative velocity. We especially compare when it is equal to 0 (multiple relaxation times scheme) or to the advection velocity ("cascaded like" scheme). The comparison is made in terms of L1 and L2 stability. The L1 stability area is fully described in terms of relaxation parameters and advection velocity for the two choices of relative velocity. These results establish that no hierarchy of these two choices exists for the L1 notion. Instead, choosing the parameter equal to the advection velocity improves the numerical L2 stability of the scheme. This choice cancels some dispersive terms and improve the numerical stability on a representative test case. We theoretically strengthen these results with a weighted L2 notion of stability. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.02381v1-abstract-full').style.display = 'none'; document.getElementById('1506.02381v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.04546">arXiv:1503.04546</a> <span> [<a href="https://arxiv.org/pdf/1503.04546">pdf</a>, <a href="https://arxiv.org/format/1503.04546">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Fevrier%2C+T">Tony Fevrier</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.04546v1-abstract-short" style="display: inline;"> This paper studies the stability properties of a two dimensional relative velocity scheme for the Navier-Stokes equations. This scheme inspired by the cascaded scheme has the particularity to relax in a frame moving with a velocity field function of space and time. Its stability is studied first in a linear context then on the non linear test case of the Kelvin-Helmholtz instability. The link wi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.04546v1-abstract-full').style.display = 'inline'; document.getElementById('1503.04546v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.04546v1-abstract-full" style="display: none;"> This paper studies the stability properties of a two dimensional relative velocity scheme for the Navier-Stokes equations. This scheme inspired by the cascaded scheme has the particularity to relax in a frame moving with a velocity field function of space and time. Its stability is studied first in a linear context then on the non linear test case of the Kelvin-Helmholtz instability. The link with the choice of the moments is put in evidence. The set of moments of the cascaded scheme improves the stability of the d'Humi猫res scheme for small viscosities. On the contrary, a relative velocity scheme with the usual set of moments deteriorates the stability. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.04546v1-abstract-full').style.display = 'none'; document.getElementById('1503.04546v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1502.02143">arXiv:1502.02143</a> <span> [<a href="https://arxiv.org/pdf/1502.02143">pdf</a>, <a href="https://arxiv.org/ps/1502.02143">ps</a>, <a href="https://arxiv.org/format/1502.02143">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Fevrier%2C+T">Tony Fevrier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1502.02143v1-abstract-short" style="display: inline;"> We study the formal precision of the relative velocity lattice Boltzmann schemes. They differ from the d'Humi猫res schemes by their relaxation phase: it occurs for a set of moments parametrized by a velocity field function of space and time. We deal with the asymptotics of the relative velocity schemes for one conservation law: the third order equivalent equation is exposed for an arbitrary number… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.02143v1-abstract-full').style.display = 'inline'; document.getElementById('1502.02143v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1502.02143v1-abstract-full" style="display: none;"> We study the formal precision of the relative velocity lattice Boltzmann schemes. They differ from the d'Humi猫res schemes by their relaxation phase: it occurs for a set of moments parametrized by a velocity field function of space and time. We deal with the asymptotics of the relative velocity schemes for one conservation law: the third order equivalent equation is exposed for an arbitrary number of dimensions and velocities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.02143v1-abstract-full').style.display = 'none'; document.getElementById('1502.02143v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1401.0399">arXiv:1401.0399</a> <span> [<a href="https://arxiv.org/pdf/1401.0399">pdf</a>, <a href="https://arxiv.org/ps/1401.0399">ps</a>, <a href="https://arxiv.org/format/1401.0399">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Cellular Automata and Lattice Gases">nlin.CG</span> </div> </div> <p class="title is-5 mathjax"> On rotational invariance of lattice Boltzmann schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Augier%2C+A">Adeline Augier</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Lallemand%2C+P">Pierre Lallemand</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1401.0399v1-abstract-short" style="display: inline;"> We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new "Berliner version" of the Taylor expansion method. The details of the implementation are p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.0399v1-abstract-full').style.display = 'inline'; document.getElementById('1401.0399v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1401.0399v1-abstract-full" style="display: none;"> We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new "Berliner version" of the Taylor expansion method. The details of the implementation are presented. These ideas are applied for the D2Q9, D2Q13, D3Q19 and D3Q27 lattice Boltzmann schemes. Some limitations associated with necessary stability conditions are also presented. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.0399v1-abstract-full').style.display = 'none'; document.getElementById('1401.0399v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 January, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1312.3297">arXiv:1312.3297</a> <span> [<a href="https://arxiv.org/pdf/1312.3297">pdf</a>, <a href="https://arxiv.org/ps/1312.3297">ps</a>, <a href="https://arxiv.org/format/1312.3297">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Lattice Boltzmann schemes with relative velocities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Fevrier%2C+T">Tony Fevrier</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1312.3297v2-abstract-short" style="display: inline;"> In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d'Humi猫res. They extend also the Geier's cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.3297v2-abstract-full').style.display = 'inline'; document.getElementById('1312.3297v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1312.3297v2-abstract-full" style="display: none;"> In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d'Humi猫res. They extend also the Geier's cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.3297v2-abstract-full').style.display = 'none'; document.getElementById('1312.3297v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2013. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.2465">arXiv:1112.2465</a> <span> [<a href="https://arxiv.org/pdf/1112.2465">pdf</a>, <a href="https://arxiv.org/ps/1112.2465">ps</a>, <a href="https://arxiv.org/format/1112.2465">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Linear Lattice Boltzmann Schemes for Acoustic: parameters choices and isotropy properties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Augier%2C+A">Adeline Augier</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.2465v1-abstract-short" style="display: inline;"> In this paper, we investigate the numerous parameters choices for linear lattice Boltzmann schemes according to the definition of the isotropic order given in \cite{ADG11}. This property---written in a general framework including all of the \ddqq schemes---can be read through a group operation. It implies some relations on the parameters of the scheme (equilibrium states and relaxation times) that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.2465v1-abstract-full').style.display = 'inline'; document.getElementById('1112.2465v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.2465v1-abstract-full" style="display: none;"> In this paper, we investigate the numerous parameters choices for linear lattice Boltzmann schemes according to the definition of the isotropic order given in \cite{ADG11}. This property---written in a general framework including all of the \ddqq schemes---can be read through a group operation. It implies some relations on the parameters of the scheme (equilibrium states and relaxation times) that give rigorous methodology to select them according to the desired order of isotropy. For acoustic applications in two spaces dimensions (namely \ddqn and \ddqt schemes) this methodology is used to propose a full description of the sets of parameters that involve isotropy of order $m$ ($m\in\{1,2,3,5\}$ for \ddqn and $m\in\{1,2\}$ for \ddqt). We then propose numerical illustrations for the \ddqn scheme. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.2465v1-abstract-full').style.display = 'none'; document.getElementById('1112.2465v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1110.0287">arXiv:1110.0287</a> <span> [<a href="https://arxiv.org/pdf/1110.0287">pdf</a>, <a href="https://arxiv.org/ps/1110.0287">ps</a>, <a href="https://arxiv.org/format/1110.0287">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Augier%2C+A">Adeline Augier</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1110.0287v1-abstract-short" style="display: inline;"> In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d'Humi茅res. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group op… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1110.0287v1-abstract-full').style.display = 'inline'; document.getElementById('1110.0287v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1110.0287v1-abstract-full" style="display: none;"> In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d'Humi茅res. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a lattice Boltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1110.0287v1-abstract-full').style.display = 'none'; document.getElementById('1110.0287v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2011. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0711.0681">arXiv:0711.0681</a> <span> [<a href="https://arxiv.org/pdf/0711.0681">pdf</a>, <a href="https://arxiv.org/ps/0711.0681">ps</a>, <a href="https://arxiv.org/format/0711.0681">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Plasma Physics">physics.plasm-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Physics">physics.class-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S021820250900353X">10.1142/S021820250900353X <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Kinetic Theory of Plasmas: Translational Energy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Graille%2C+B">Benjamin Graille</a>, <a href="/search/math?searchtype=author&query=Magin%2C+T+E">Thierry E. Magin</a>, <a href="/search/math?searchtype=author&query=Massot%2C+M">Marc Massot</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0711.0681v1-abstract-short" style="display: inline;"> In the present contribution, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and the reactive collisions. Given the strong disparity of mass between the electrons and heavy particles, su… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0711.0681v1-abstract-full').style.display = 'inline'; document.getElementById('0711.0681v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0711.0681v1-abstract-full" style="display: none;"> In the present contribution, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and the reactive collisions. Given the strong disparity of mass between the electrons and heavy particles, such as molecules, atoms, and ions, we conduct a dimensional analysis of the Boltzmann equation. We then generalize the Chapman-Enskog method, emphasizing the role of a multiscale perturbation parameter on the collisional operator, the streaming operator, and the collisional invariants of the Boltzmann equation. The system is examined at successive orders of approximation, each of which corresponding to a physical time scale. The multicomponent Navier-Stokes regime is reached for the heavy particles, which follow a hyperbolic scaling, and is coupled to first order drift-diffusion equations for the electrons, which follow a parabolic scaling. The transport coefficients exhibit an anisotropic behavior when the magnetic field is strong enough. We also give a complete description of the Kolesnikov effect, i.e., the crossed contributions to the mass and energy transport fluxes coupling the electrons and heavy particles. Finally, the first and second principles of thermodynamics are proved to be satisfied by deriving a total energy equation and an entropy equation. Moreover, the system of equations is shown to be conservative and the purely convective system hyperbolic, thus leading to a well-defined structure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0711.0681v1-abstract-full').style.display = 'none'; document.getElementById('0711.0681v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mathematical Models and Methods in Applied Sciences, Vol. 19, 04 (2009) 527-599 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 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