[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83283-en":3,"doc-seo-83283-105":29,"detail-sidebar-cat-0-en-105":89},{"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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83283,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Scaling WaterLily.jl With MPI And An Improved Geometric Multigrid Solver","Performance-oriented developments for WaterLily.jl introduce MPI-based distributed-memory parallelism in a pure Julia, scale-resolving incompressible flow solver that runs on CPU and GPU backends. Strong-scaling tests show a near-ideal linear time reduction, while weak-scaling efficiency stays above 85% until node memory-concurrency contention becomes dominant. Inter-node weak scalability remains above 96% up to grids of about one billion cells. Poisson performance is improved via an adaptive under-relaxed red-black Gauss–Seidel smoother with anisotropic coarsening operators.","arXiv :2607 .07687v2 [physics .comp-ph] 9 Jul 2026  \nParCFD 2026  \n37th International Conference on Parallel Computational Fluid Dynamics Nov 03-06 2026, Nantes, FRANCE  \nSCALING WATERLILY.JL WITH MPI AND AN IMPROVED GEOMETRIC MULTIGRID SOLVER  \nBernat FONT∗ , Marin LAUBER, Tzu-Yao HUANG,  \nGabriel D. WEYMOUTH  \nDelft University of Technology  \nFaculty of Mechanical Engineering  \nMekelweg 2, 2628 CD Delft, The Netherlands  \n∗ e-mail: [b. font@tudelft. nl](b. font@tudelft. nl), web page: [https://b-fg. github. io](https://b-fg. github. io)  \nKey words: distributed-memory parallelism, message-passing interface, geometric multigrid, finite volume, Julia  \nAbstract. We present recent performance-oriented developments in WaterLily.jl, a scale-resolving incompressible flow solver written in pure Julia that runs seamlessly on CPUs and GPUs of any vendor. Supported by the newly added MPI-based parallelism, strong-scalability tests display a near-ideal linear trend, and weak-scaling efficiency is kept above 85% before node memory-concurrency contention dominates parallel performance. Inter-node weak scalability is sustained above 96% with grid size up to 1 billion cells. We further benchmark improvements to the geometric multigrid Poisson solver enabled by an adaptive under-relaxed red-black Gauss–Seidel smoother together with anisotropic coarsening operators.  \n1 INTRODUCTION  \nHigh-performance computing (HPC) is a cornerstone of computational fluid dynamics, enabling an ever-growing resolution in scale-resolving simulations. Driven by hardware advances such as general-purpose processor (CPU) and accelerator-based (GPU) nodes, CFD software is rapidly evolving to harvest the increase in computational power. The trend comes at an added software-complexity cost, and CFD developers now heavily rely on HPC libraries to offload compute-expensive kernels and parallelisation.  \nIn this direction, we present performance-oriented developments and the first parallelscalability results of WaterLily. jl [1] . Written in pure Julia, WaterLily is designed to bea compact and minimalist finite-volume incompressible-flow solver that runs on any CPU or GPU computing backend. This feature is enabled by separating computing workloads (typically for loops) from the algorithmic space-time discretization. By using Julia’s metaprogramming support, the solver ultimately specialises each kernel to the available computing backend using KernelAbstractions. jl[2] .  \n2 MPI PARALLELISATION AND SCALING  \nFurther leveraging Julia’s scientific computing ecosystem, distributed-memory parallelism has been recently integrated in WaterLily via the ImplicitGlobalGrid. jl (IGG) package [3] . Backed by the MPI. jl wrapper, IGG uses a Cartesian MPI communicator  \nt / Δ t [ms]  \n106  \n105  \n104  \n103  \n102  \n101  \n100  \n1 2 4 8 16 32 64 128 256 512  \nnp  \nE = t1 / tnp  \n1.0  \n0.8  \n0.6  \n0.4  \n0.2  \n0.0  \n1 2 4 8 16 32 64 128 256 512  \nnp  \nFigure 1: Strong scaling (left) and weak scaling (right) results on the Rome CPU partition of the Snellius Dutch national supercomputer. The inset shows the inter-node weakscaling efficiency, normalised by the single-node (64-rank) time.  \nnaturally mapping to regular grids, and offering CUDA-aware or ROCm-aware MPI support for GPU-based applications. Additionally, IGG works with different grid sizes, thus enabling halo updates within geometric-multigrid (GMG) levels without added complexity from the IGG-user side.  \nStrong and weak scalability results are presented in Figure 1 . The tests are conducted with the SIMD single-thread CPU backend with single precision (FP32) and half-node occupancy (64 of the 128 cores ofthe dual-socket AMD EPYC 7H12 nodes) . The test cases comprise a Taylor–Green vortex (TGV) at Re = 1600 in a triple-periodic cubic domain, and flow past a sphere at ReD = 3700 in a (16 × 6 × 6)D domain. The MPI domain decomposition always minimizes the halo surface for a given number of processes. The strong-scalability results","cbCaiieWE5xmhfSd","https://ap.wps.com/l/cbCaiieWE5xmhfSd","pdf",306706,1,4,"English","en",105,"# Introduction\n# MPI Parallelisation and Scaling\n# Anisotropic Geometric Multigrid Solver with Adaptive Under-Relaxed Red-Black Gauss–Seidel Smoother","[{\"question\":\"What software and programming approach does the paper focus on?\",\"answer\":\"The paper focuses on WaterLily.jl, a scale-resolving incompressible flow solver written in pure Julia, designed to run seamlessly on CPUs and GPUs.\"},{\"question\":\"How does MPI parallelisation affect scaling performance?\",\"answer\":\"With MPI-based distributed-memory parallelism, strong scaling follows near-ideal linear time reduction, and weak-scaling efficiency stays above 85% before memory-concurrency contention limits performance.\"},{\"question\":\"What improvements are made to the geometric multigrid Poisson solver?\",\"answer\":\"Poisson solver improvements include an adaptive under-relaxed red-black Gauss–Seidel smoother and anisotropic coarsening operators to enhance multigrid efficiency.\"}]",1784186474,10,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":27},"scaling-waterlilyjl-with-mpi-and-an-improved-geometric-multigrid-solver","",{"@graph":35,"@context":83},[36,52,66],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":21},"https://docshare.wps.com/document/scaling-waterlilyjl-with-mpi-and-an-improved-geometric-multigrid-solver/83283/",{"url":51,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":23,"description":14,"dateModified":60,"datePublished":60,"encodingFormat":59,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":40,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":63,"interactionType":64,"userInteractionCount":4},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"What software and programming approach does the paper focus on?","Question",{"text":73,"@type":74},"The paper focuses on WaterLily.jl, a scale-resolving incompressible flow solver written in pure Julia, designed to run seamlessly on CPUs and GPUs.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"How does MPI parallelisation affect scaling performance?",{"text":78,"@type":74},"With MPI-based distributed-memory parallelism, strong scaling follows near-ideal linear time reduction, and weak-scaling efficiency stays above 85% before memory-concurrency contention limits performance.",{"name":80,"@type":71,"acceptedAnswer":81},"What improvements are made to the geometric multigrid Poisson solver?",{"text":82,"@type":74},"Poisson solver improvements include an adaptive under-relaxed red-black Gauss–Seidel smoother and anisotropic coarsening operators to enhance multigrid efficiency.","https://schema.org",{"og:url":51,"og:type":85,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":87,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":90},[91,95,99,103,108,113,118,121,126,129,132],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":100,"show_sort_weight":101,"slug":102},"Exam",70,"exam",{"id":104,"doc_module":4,"doc_module_name":45,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":45,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":28,"doc_module":4,"doc_module_name":45,"category_name":130,"show_sort_weight":28,"slug":131},"Lifestyle","lifestyle",{"id":133,"doc_module":4,"doc_module_name":45,"category_name":134,"show_sort_weight":104,"slug":135},19,"General","general"]