[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83098-en":3,"doc-seo-83098-105":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},83098,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Sparse Space-Time Spectral Methods Can Time-Step by Peel and Pass","Global space-time spectral methods achieve high spectral accuracy in time but normally require storing the entire space-time history on a single tensor-product domain. This work shows that with an endpoint-benign Legendre or Chebyshev-T time basis, where basis polynomials equal one at the right endpoint, the final time slice of a space-time block is recovered exactly via an endpoint summation of stored time coefficients. Interpreting this as a Jacobi endpoint identity yields derivative formulas, enabling a sparse block-by-block advance that stores only one block, with models for cost, error propagation, and memory.","arXiv :2607 .06449v1 [math .NA] 7 Jul 2026  \nSparse space-time spectral methods can time-step  \nby peel and pass  \nTimon S. Gutleb 1*  \n1 School of Computer Science, University of Leeds, Leeds, UK.  \nCorresponding author(s). E-mail(s): [T.S.Gutleb@leeds.ac.uk](T.S.Gutleb@leeds.ac.uk) ;  \nAbstract  \nGlobal space-time spectral methods give spectral accuracy in time but typically require the whole space-time history to be resolved and stored on a single tensorproduct domain T × Ω . We record that in an endpoint-benign Legendre or Chebyshev-T time basis, whose polynomials all equal one at the right endpoint, the final time slice of a space-time block is recovered exactly by summing the stored coefficients along the time index. This peel-and-pass step is a special case of a Jacobi endpoint identity, which also gives derivative formulae for higher-order equations. Writing such higher-order equations as first-order systems preserves the benign value-passing structure. The result is a sparse space-time spectral element method that advances block by block, stores only one block, and needs far fewer time coefficients per solve for long-time problems. We prove the identities, give resident-memory, solve-cost and error-propagation models, and demonstrate the method on (1+1)D heat, wave and Klein–Gordon equations, and on (2+1)D fractional heat on the disk with weighted Zernike polynomials in space.  \nKeywords: space-time spectral methods, spectral elements, orthogonal polynomials, ultraspherical spectral method, time stepping  \nMSC Classification: 65M70 , 65M12 , 65M60 , 41A10  \n1 Introduction  \nSpectral methods approximate the solution of a differential equation by a truncated expansion in a global basis, converging rapidly for sufficiently well-behaved solutions. The ultraspherical spectral method of Olver and Townsend [1] represents the solution  \n1  \nin a Gegenbauer or, more generally, Jacobi basis and expresses differentiation and multiplication as sparse, banded operators between these bases, in contrast to the dense and increasingly ill-conditioned matrices of classical collocation. This pairing of spectral accuracy with banded, well-conditioned linear algebra [2] has since been carried to multivariate and non-trivial geometries by constructing sparse-operator bases of orthogonal polynomials adapted to triangles [3], disks and disk slices [4, 5], spherical caps [6], algebraic curves [7–9], annuli and cylinders [10, 11], and generalised nonclassical orthogonal polynomials [12, 13] . Decomposing a complicated spatial domain into sparse spectral elements of this kind is likewise established [3, 10 , 11 , 14] .  \nWhen the differential equation is time-dependent, the dominant strategy is the method of lines: discretise space and march the resulting system of ordinary differential equations with a time stepper [15, 16] . An appealing alternative is to place time on the same footing as space, expanding the solution in a global polynomial basis in time as well and solving the whole space-time problem at once. Earlier spectral-in-time constructions include time-spectral methods for hyperbolic and parabolic equations [17–19], as well as single- and multi-interval Legendre methods in time [20] . The authors in [21] use the generalised weighted residual method with Chebyshev temporal subdomains to improve sparsity and memory use. More recent space-time spectral methods include Legendre and Chebyshev collocation schemes for time-dependent PDEs [22–24], a Stokes formulation and thesis treatment by Kaur and Lui [25, 26], and linear and nonlinear examples by Kaur, Lui, Nataj and Wilegoda Liyanage [27] . Related spectral-in-time constructions have also been used for fractional problems [28] . The Achilles heel of global space-time spectral methods, and the reason they are seldom used for long-time or higher-dimensional solves, is that treating time as another coordinate raises the dimension of the discretised problem by one. The unknowns, operators ","cbCaieRfykDNAxDR","https://ap.wps.com/l/cbCaieRfykDNAxDR","pdf",1797400,1,24,"English","en",105,"# Abstract\n# Introduction\n# Sparse Space-Time Blocks\n## Orthogonal Polynomial Bases","[{\"question\":\"What key limitation of global space-time spectral methods does this paper address?\",\"answer\":\"It addresses the need to resolve and store the full space-time history on a single tensor-product domain, which increases memory and computational cost for long-time or higher-dimensional problems.\"},{\"question\":\"How does the “peel-and-pass” idea recover the next block’s initial condition?\",\"answer\":\"Using an endpoint-benign Legendre or Chebyshev-T time basis, the final time slice of a space-time block is recovered exactly by summing stored coefficients along the time index, then passing it as the next block’s initial data.\"},{\"question\":\"What equations and settings are used to demonstrate the method?\",\"answer\":\"The method is demonstrated on (1+1)D heat, wave, and Klein–Gordon equations, and on (2+1)D fractional heat on a disk using weighted Zernike polynomials in space.\"}]",1784185225,60,{"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},"sparse-space-time-spectral-methods-can-time-step-by-peel-and-pass","",{"@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/sparse-space-time-spectral-methods-can-time-step-by-peel-and-pass/83098/",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 key limitation of global space-time spectral methods does this paper address?","Question",{"text":75,"@type":76},"It addresses the need to resolve and store the full space-time history on a single tensor-product domain, which increases memory and computational cost for long-time or higher-dimensional problems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the “peel-and-pass” idea recover the next block’s initial condition?",{"text":80,"@type":76},"Using an endpoint-benign Legendre or Chebyshev-T time basis, the final time slice of a space-time block is recovered exactly by summing stored coefficients along the time index, then passing it as the next block’s initial data.",{"name":82,"@type":73,"acceptedAnswer":83},"What equations and settings are used to demonstrate the method?",{"text":84,"@type":76},"The method is demonstrated on (1+1)D heat, wave, and Klein–Gordon equations, and on (2+1)D fractional heat on a disk using weighted Zernike polynomials in 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