[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82977-en":3,"doc-seo-82977-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82977,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Automated Derivation of Lattice Boltzmann Schemes for Systems of Conservation Laws","Lattice Boltzmann Methods used to simulate multiphysics conservation-law systems typically require labor-intensive, hand-derived schemes per target PDE, which limits model retargeting. The presented framework generalizes a flux-in-first-moment construction into a single automated derivation for conservation-form systems covering hyperbolic, parabolic, and mixed types. Fluxes are mapped to first-order discrete moments, while spatial gradients are handled via advection-relaxation cascades. Validation with Method of Manufactured Solutions confirms near-second-order convergence in double precision and single-precision behavior via equilibrium-shifted formulation, and OpenLB GPU kernels achieve up to 96% of peak single-precision memory-bandwidth efficiency.","arXiv :2607 .05668v 1 [ cs .MS] 6 Jul 2026  \nAUTOMATED DERIVATION OF LATTICE BOLTZMANN SCHEMES FOR SYSTEMS OF CONSERVATION LAWS  \nADRIAN KUMMERLÄNDER∗ ,† , FEDOR BUKREEV∗ ,‡ , AND MATHIAS J. KRAUSE∗ ,†,‡  \nAbstract. The simulation of multiphysics phenomena with Lattice Boltzmann Methods (LBM) traditionally requires a specialized scheme hand-derived for each targeted Partial Differential Equation (PDE), making the retargeting of physical models a labor-intensive bottleneck. To resolve this, we recognize the flux-in-first-moment construction of a recently proposed class of LBM schemes as a discrete-kinetic relaxation approximation of conservation laws, and generalize their case-by-case, handderived construction into a single automated derivation for conservation-form systems of hyperbolic, parabolic, and mixed type. This decouples the quadrature lattice from physical transport, and we exercise the approach across twelve transport-equation systems, including compressible Navier–Stokes– Fourier flow, magnetohydrodynamics, nonlinear elasticity, and electromagnetics. Nonlinear fluxes map directly onto the first-order discrete moments, while spatial gradients are tracked point-wise via advection-relaxation cascades, replacing finite-volume flux reconstruction with local kinetic updates. We encapsulate the approach in an automated PDE2LBM symbolic compiler, driven by a coordinatefree Domain-Specific Language (DSL) that transforms abstract PDEs into LBMs. Validation across all systems using a Method of Manufactured Solutions (MMS) confirms convergence at or near second order in double precision, and the reference-and equilibrium-shifted formulation retains convergence in single precision. Targeting the platform-transparent framework OpenLB, the generated GPU kernels approach the memory-bandwidth roofline, reaching up to 96% of peak in single precision. Unlike existing LBM code generators, which require the discrete scheme as input, this framework derives the scheme from the declared PDE itself: the equilibrium, gradient-tracking cascade, and unit scaling all follow from the conservation law alone.  \nKey words. Lattice Boltzmann methods, automatic code generation, symbolic computation, method of manufactured solutions, high-performance computing  \n1. Introduction. Traditional frameworks for approximating continuous systems of conservation laws on unstructured meshes, most prominently finite volume and finite element methods, evaluate spatial derivatives and interfacial flux tensors by gathering extended, irregularly addressed node neighborhoods. While robust and geometrically flexible, these indirect, non-local memory access patterns map poorly onto coalesced, throughput-oriented memory systems on contemporary High-Performance Computing (HPC) architectures. As modern HPC nodes commonly lean on thousands of concurrent execution threads within heterogeneous GPU accelerators, the performance of Partial Differential Equation (PDE) solvers is increasingly bound by memory bandwidth and synchronization limits rather than raw floating-point throughput.  \nThe classical Lattice Boltzmann Method (LBM) [34] circumvents this bottleneck by considering the continuous problem at the mesoscopic level. By tracking particle distribution functions across a discrete velocity space, the algorithm decomposes into a perfectly parallel collision step and a neighborhood-local streaming step along discrete characteristics. This locality yields excellent scalability on heterogeneous supercomputers [4, 22, 35, 37, 54] . However, standard scalar LBM formulations are structurally rigid. Constructing a scheme for a new target is fundamentally a momentmatching problem: the discrete velocity moments of the equilibrium distribution must reproduce, order by order, the macroscopic densities and fluxes of the target equations. This matching is classically realized through high-order Hermite expansions of the  \n∗ The three authors contributed equally to this work. All author","cbCais7Sf6y4JuLC","https://ap.wps.com/l/cbCais7Sf6y4JuLC","pdf",1295776,2,1,60,"English","en",105,"# Introduction\n## LBM background and limitations\n## Flux-in-first-moment kinetic schemes\n## From conservation laws to automated derivation","[{\"question\":\"Why is retargeting Lattice Boltzmann Method (LBM) models traditionally labor-intensive?\",\"answer\":\"Standard LBM workflows require a specialized, hand-derived scheme for each target PDE, making model retargeting a bottleneck.\"},{\"question\":\"What is the core idea behind the automated derivation approach?\",\"answer\":\"The method generalizes flux-in-first-moment LBM constructions by treating them as a discrete-kinetic relaxation approximation of conservation laws, enabling one unified derivation procedure.\"},{\"question\":\"How are spatial gradients and nonlinear fluxes represented in the generated schemes?\",\"answer\":\"Nonlinear fluxes map directly onto first-order discrete moments, while spatial gradients are tracked point-wise using advection-relaxation cascades implemented as local kinetic 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is retargeting Lattice Boltzmann Method (LBM) models traditionally labor-intensive?","Question",{"text":75,"@type":76},"Standard LBM workflows require a specialized, hand-derived scheme for each target PDE, making model retargeting a bottleneck.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the core idea behind the automated derivation approach?",{"text":80,"@type":76},"The method generalizes flux-in-first-moment LBM constructions by treating them as a discrete-kinetic relaxation approximation of conservation laws, enabling one unified derivation procedure.",{"name":82,"@type":73,"acceptedAnswer":83},"How are spatial gradients and nonlinear fluxes represented in the generated schemes?",{"text":84,"@type":76},"Nonlinear fluxes map directly onto first-order discrete moments, while spatial gradients are tracked point-wise using advection-relaxation cascades implemented as local kinetic 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