[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83724-en":3,"doc-seo-83724-105":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},83724,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","A perfectly matched layer approach for the spectral split-step Padé method","The split-step-Padé (SSP) method is widely used to simulate wave propagation in domains such as radio physics, optics, and acoustics. The work develops and numerically validates a spectral counterpart of the SSP approach by combining Padé-based one-way Helmholtz propagation with a spectral discretization of the transverse operator. A central focus is the reliable inversion of the transverse operator when perfectly matched layers (PMLs) truncate the domain, achieved via Krylov subspace methods with an effective preconditioner. The spectral marching scheme is also analyzed under periodic transverse conditions and tested in two realistic scenarios from radio physics and underwater acoustics.","arXiv :2607 .03521v1 [math .NA] 3 Jul 2026  \nA perfectly matched layer approach for the spectral split-step Pad´e method  \nDaniel Walsken 1 , Matthias Ehrhardt 1 , ∗, Pavel Petrov2  \n1 Applied and Computational Mathematics, University of Wuppertal, Gaußstrasse 20, Wuppertal, 42119, Germany  \n2 Instituto de Matem´atica Pura e Aplicada, Rio de Janeiro, Brazil E-mail: [walsken@uni-wuppertal.de](walsken@uni-wuppertal.de1)[1](walsken@uni-wuppertal.de1) , [ehrhardt@uni-wuppertal.de](ehrhardt@uni-wuppertal.de1)[1](ehrhardt@uni-wuppertal.de1),∗  \n[pavel.petrov@impa.br](pavel.petrov@impa.br2)[2](pavel.petrov@impa.br2)  \nAbstract  \nThe split-step-Pad´e (SSP) method is widely used to model wave phenomena in various applications, including radio physics, optics and acoustics. In this method, the propagator of the one-way counterpart of the Helmholtz equation is computed through its Pad´e approximant and a finite-difference discretization of the transverse operator. This work develops and validates numerically a spectral counterpart of the SSP method. A key challenge in practical applications is inverting the transverse operator in the presence of perfectly matched layers (PMLs), which are commonly used to truncate the computational domain. Such inversion can be accomplished using Krylov subspace methods, which converge rapidly, provided that a suitable preconditioner is used. We also study the analytical properties of the spectral SSP marching scheme under periodicity conditions in the transverse variable. We validate the newly developed spectral SSP method numerically in two realistic test scenarios from radio physics and underwater acoustics.  \nKeywords: Split-Step Pad´e method, Perfectly Matched Layer, Krylov method, wide-angle parabolic equation, Spectral discretization  \nMSC: 68Q25, 68R10, 68U05  \nIntroduction  \nThe propagation of waves in inhomogeneous media is a fundamental problem in applied mathematics and computational physics. It has numerous  \n∗ Corresponding Author, email: [ehrhardt@uni-wuppertal.de](ehrhardt@uni-wuppertal.de)  \napplications in optics, radio physics, underwater and atmosphere acoustics, and geophysical exploration. The governing model is usually the Helmholtz equation, whose direct numerical solution in large or unbounded domains is computationally expensive due highly oscillatory nature of its solutions that imposes severe discretization step restriction. A common approach to reducing computational complexity is to use parabolic approximations of the Helmholtz equation, which were first introduced by Leontovich and Fock [29, 55] . These models transform the original boundary value problem into an initial value problem in a preferred propagation direction, thereby significantly reducing computational effort. Although such reformulation neglects back-scattered waves, it has proven to be extremely efficient in many applications, where parabolic equations have become main computational tools [12, 26, 30, 38], especially after the introduction of split-step Fourier (SSF) spectral solvers (mostly for narrow-angle parabolic equations) .  \nHowever, classical parabolic equation methods are limited in their ability to accurately capture wide-angle propagation effects. This limitation has been overcome by the development of wide-angle parabolic equations (WAPEs), which approximate the square-root operator arising from the factorization of the Helmholtz operator using rational Pad´e approximants [1, 12, 13, 26, 30, 45] . It is important that WAPEs substantially improve angular accuracy while preserving computational efficiency paraxial equations. The next major improvement of WAPEs was the invention of the split-step Pad´e (SSP) method [5, 13] that aims at approximating the propagator of the operator square root (i.e., the exponential that emerges from integration of one-way Helmholtz equation over small step), by a Pad´e series. This has proven to be much more efficient than approximating the square-root operator itsel","cbCaimh9V7jmGIeE","https://ap.wps.com/l/cbCaimh9V7jmGIeE","pdf",10587977,4,1,27,"English","en",105,"# Abstract\n# Introduction\n## Background: Helmholtz and parabolic approximations\n## Wide-angle parabolic equations and SSP methods\n## Spectral split-step ideas via FFT\n## Domain truncation and perfectly matched layers\n## Remaining challenge and motivation for the proposed approach","[{\"question\":\"What is the main goal of this work on the spectral split-step Padé method?\",\"answer\":\"Develop and numerically validate a spectral counterpart of the split-step Padé (SSP) method, addressing practical issues that arise when using perfectly matched layers (PMLs).\"},{\"question\":\"Why is inverting the transverse operator difficult when PMLs are used?\",\"answer\":\"PMLs are used to truncate the computational domain, and this changes the structure of the transverse operator; inversion must be done accurately and efficiently despite these effects.\"},{\"question\":\"How does the paper compute the required inversion efficiently?\",\"answer\":\"It uses Krylov subspace methods that converge rapidly when a suitable preconditioner is employed.\"}]",1784190004,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-perfectly-matched-layer-approach-for-the-spectral-split-step-pade-method","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/a-perfectly-matched-layer-approach-for-the-spectral-split-step-pade-method/83724/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","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 is the main goal of this work on the spectral split-step Padé method?","Question",{"text":75,"@type":76},"Develop and numerically validate a spectral counterpart of the split-step Padé (SSP) method, addressing practical issues that arise when using perfectly matched layers (PMLs).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is inverting the transverse operator difficult when PMLs are used?",{"text":80,"@type":76},"PMLs are used to truncate the computational domain, and this changes the structure of the transverse operator; inversion must be done accurately and efficiently despite these effects.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper compute the required inversion efficiently?",{"text":84,"@type":76},"It uses Krylov subspace methods that converge rapidly when a suitable preconditioner is employed.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]