[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82101-en":3,"doc-seo-82101-105":29,"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":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},82101,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","A Tensor-Train Discontinuous Galerkin Method for the Vlasov Maxwell System","Presents a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov–Maxwell system that couples modal DG discretization with low-rank tensor representations of the phase-space solution and discrete operators. The tensor-product structure enables quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form. Evaluations on standard 1D2V benchmarks (streaming Weibel, weak Landau damping, two-stream instability) match full-grid DG accuracy and conservation while substantially lowering memory and runtime. Compression exceeds 10^4 for weakly nonlinear cases and remains effective for strong nonlinearity despite reduced compressibility from filamentation, demonstrating reduced-cost deterministic kinetic plasma simulation.","arXiv :2607 .08936v1 [math .NA] 9 Jul 2026  \nA Tensor-Train Discontinuous Galerkin Method for the Vlasov-Maxwell System  \nRujeko Chinomonaa,∗, Dibyendu Adaka,d , William J. Barhama , Duc P. Truonga , Nathan V. Robertsb , Kim Ø . Rasmussena , Boian S. Alexandrova,c  \na Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Sandia National Laboratories, Albuquerque, NM, USA  \ncSLIC.AI, Santa Fe, NM, USA  \nd Indian Institute of Technology-Kharagpur, WB 721302, India  \nAbstract  \nWe present a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov–Maxwell system that combines a modal DG discretization with low-rank tensor representations of the phasespace solution and discrete operators. The formulation exploits the tensor-product structure of the DG discretization to perform quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form.  \nThe method is evaluated on several standard 1D2V Vlasov–Maxwell benchmark problems, including the streaming Weibel instability, weak Landau damping, and two-stream instability problems. Across these problems, the TT formulation reproduces the accuracy and conservation behavior of the underlying full-grid DG discretization while substantially reducing memory usage and runtime. For weakly nonlinear problems, compression ratios exceeding 104 are obtained together with significant speedups relative to the full-grid solver. For the strongly nonlinear twostream instability problem, the TT formulation remains effective despite reduced compressibility caused by fine-scale phase-space filamentation.  \nThese results demonstrate that tensor-train representations provide an effective approach for reducing the computational cost of deterministic DG-based kinetic plasma simulations while retaining the favorable numerical properties of the underlying discretization.  \n1. Introduction  \nThe Vlasov–Maxwell (VM) system is a fundamental kinetic model for collisionless plasmas describing the coupled evolution of a particle distribution function in phase-space and electromagnetic fields. In the VM system, the Vlasov equation governs the evolution of the particle distribution function under electromagnetic forces, while Maxwell’s equations determine the electric and magnetic fields generated by the charged particles. These equations arise in applications including magnetic confinement fusion [1], laser–plasma interaction [2], beam physics [3], and space and astrophysical plasmas [4] .  \n∗ Corresponding author. Email: [crujeko@lanl.gov](crujeko@lanl.gov)  \nUnlike fluid or magnetohydrodynamic models [5, 6], the VM system resolves velocity-space effects such as Landau damping, particle trapping, filamentation, and anisotropy. However, this kinetic description comes with substantial computational cost. Deterministic numerical solution of the VM system requires discretizing a phase-space of dimension d = dx + dv . In the present work, we focus on the 1D2V setting, where dx = 1 and dv = 2, giving a three-dimensional phase-space. Even in this reduced setting, storage and computational cost scale as O (N3 ) on a uniform grid with N points per dimension, and the problem rapidly becomes more expensive under refinement. Extending to higher-dimensional VM systems, where d reaches 4 , 5, or 6, makes full-grid simulation prohibitively expensive in both memory and runtime. Developing numerical methods that retain the accuracy and conservation properties of the VM system while reducing this computational cost remains a major challenge in computational plasma physics.  \nWe consider the VM system for a single species of nonrelativistic electrons, while the ions are treated as a fixed, spatially uniform background. Using a common nondimensionalization [7], with time scaled by the inverse plasma frequency, velocity by the speed of light, and length by the electron skin depth, the dimensionless VM equations are given by  \n∂t f + v · ∇xf + (E + v ×","cbCaikLwfni12t1F","https://ap.wps.com/l/cbCaikLwfni12t1F","pdf",17396211,1,26,"English","en",105,"# 1. Introduction\n## Vlasov–Maxwell model and computational challenge\n## Dimensionless formulation and conserved quantities","[{\"question\":\"What does the TT-DG method combine for the Vlasov–Maxwell system?\",\"answer\":\"It combines a modal discontinuous Galerkin (DG) discretization with tensor-train low-rank representations of the phase-space solution and the discrete operators.\"},{\"question\":\"How does the tensor-train formulation reduce computational cost?\",\"answer\":\"It leverages the DG tensor-product structure to carry out quadrature, differentiation, upwind flux evaluation, and time integration directly in compressed form, reducing memory usage and runtime.\"},{\"question\":\"Which benchmark problems are used to evaluate the method and what is the main outcome?\",\"answer\":\"Standard 1D2V benchmarks include streaming Weibel instability, weak Landau damping, and two-stream instability; results reproduce full-grid DG accuracy and conservation while substantially reducing memory and runtime.\"}]",1784178217,66,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-tensor-train-discontinuous-galerkin-method-for-the-vlasov-maxwell-system","",{"@graph":35,"@context":85},[36,53,68],{"@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":52},"https://docshare.wps.com/document/a-tensor-train-discontinuous-galerkin-method-for-the-vlasov-maxwell-system/82101/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the TT-DG method combine for the Vlasov–Maxwell system?","Question",{"text":75,"@type":76},"It combines a modal discontinuous Galerkin (DG) discretization with tensor-train low-rank representations of the phase-space solution and the discrete operators.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the tensor-train formulation reduce computational cost?",{"text":80,"@type":76},"It leverages the DG tensor-product structure to carry out quadrature, differentiation, upwind flux evaluation, and time integration directly in compressed form, reducing memory usage and runtime.",{"name":82,"@type":73,"acceptedAnswer":83},"Which benchmark problems are used to evaluate the method and what is the main outcome?",{"text":84,"@type":76},"Standard 1D2V benchmarks include streaming Weibel instability, weak Landau damping, and two-stream instability; 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