[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83412-en":3,"doc-seo-83412-105":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},83412,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Some properties of high-order nonstandard multistep multistage methods","This paper introduces nonstandard versions of multistep multistage numerical schemes for ordinary differential equations. By defining nonstandard general linear methods, it establishes convergence and shows that the proposed nonstandard methods can reach the same order as standard counterparts. A key result is preservation of qualitative properties such as boundedness for all positive step sizes. The theoretical findings are supported by numerical experiments demonstrating the practical behavior of the schemes.","arXiv :2607 .08694v1 [math .NA] 9 Jul 2026  \nSome properties of high-order nonstandard multistep multistage methods  \nBalint M. Takacs 1,2,*  \n1 Department of Analysis and Operations Research, Institute of  \nMathematics, Budapest University of Technology and Economics, Muegyetem rakpart 3., Budapest, H-1111, Hungary.  \n2 HUN-REN-ELTE Numerical Analysis and Large Networks Research Group, P´azm´any P´eter s´et´any 1/C, Budapest, H-1117, Budapest.  \nAbstract  \nIn this paper, we introduce nonstandard versions of multistep multistage methods. While proving the convergence of these schemes, we also define nonstandard general linear methods. We show that the nonstandard methods can attain the same order as their standard counterparts while preserving certain qualitative properties (e.g., boundedness) for all positive step sizes. These results are also demonstrated by some numerical experiments.  \nKeywords: nonstandard finite difference, multistep multistage method, positivity  \npreservation, SSP methods  \nMSC Classification: 65L06 , 92D25 , 92D30  \n1 Introduction  \nApart from the aim to design numerical methods that approximate a given differential equation in a time-efficient manner (i.e., the order of a given method is high), oneof the key goals of numerical modeling is the construction of numerical schemes that behave in a qualitatively reasonable way, meaning that they preserve some properties of the original, continuous model. Although the application of adaptive step sizes is a leading area in this field [1–4], another approach is the use of nonstandard finite differences. In the last year alone, the latter methods were used in epidemiology [5–8],  \n*Corresponding author. E-mail: [takacs.balint.mate@ttk.bme.hu](takacs.balint.mate@ttk.bme.hu)[ ](takacs.balint.mate@ttk.bme.hu)1  \ncybersecurity [9, 10], ecology [11–13], among others [14–16] . The main advantage of these methods, originally developed by Mickens [17], is that they typically preserve certain properties of the original continuous model for all positive step sizes.  \nOne disadvantage of nonstandard methods is that in most cases, they only attain a convergence of order one. However, in recent years, there has been some work in the construction of higher-order methods. One early example was the investigation of higher-order nonstandard Runge-Kutta methods [18, 19], while more recent approaches include:  \n• the employment of Theta methods [20–22] with several applications [23–25],• the use of Richardson extrapolation [26–28],  \n• the change of the denominator function into a more general form [29–34],  \n• or a clever split of the right-hand side of the equation [35, 36],  \namong others [37–40] . One usual idea of these methods is to restrict ourselves to a set of equations in some special form, and therefore construct a method for that given type of equations.  \nAnother possibility is to start from well-known ’standard’ methods, and by changing the timestep ∆t to φ(∆t), we can construct nonstandard forms of these schemes. This transformation has been applied to Runge-Kutta [41, 42] and linear multistep methods [43, 44] . The core idea of this approach is that by rewriting the ’standard’schemes into a strong-stability preserving (SSP) form (see [45]), we can guarantee the preservation of some linear properties. Note that the ideas of SSP methods have also been applied in the nonstandard framework before, but in a different context [46, 47] .  \nThe main aim of the present paper is to generalize nonstandard Runge-Kutta and nonstandard linear multistep methods by introducing nonstandard versions of multistep multistage methods. In the process, we also define nonstandard versions of general linear methods. By changing ∆t to φ(∆t), these new, nonstandard schemes can preserve some properties (e.g., boundedness) for every possible ∆t > 0, while also attaining the same orders as the ’standard’ methods. It is worth noting that the work relies heavily on the paper of Constantinescu ","cbCais4SKmpHPkL0","https://ap.wps.com/l/cbCais4SKmpHPkL0","pdf",3192670,2,1,42,"English","en",105,"# Introduction\n## Motivation and background\n## Related approaches and SSP reformulations\n# Preliminaries\n## Initial value problem and assumptions\n## Discretization framework\n# Main results and structure\n## Convergence, order, and qualitative properties\n## Numerical experiments\n## Conclusions and extensions","[{\"question\":\"What are nonstandard multistep multistage methods, and why are they introduced?\",\"answer\":\"The work proposes nonstandard versions of multistep multistage schemes by transforming the timestep to a nonstandard function. The goal is to retain qualitative behavior while still achieving high accuracy.\"},{\"question\":\"How does the paper guarantee qualitative property preservation for nonstandard methods?\",\"answer\":\"It uses a reformulation related to strong-stability preserving (SSP) structure and shows preservation of properties such as boundedness for every positive step size.\"},{\"question\":\"What is the main convergence and accuracy claim compared to standard methods?\",\"answer\":\"The paper proves convergence and demonstrates that the nonstandard schemes can attain the same order as the corresponding standard methods, supported further by numerical experiments.\"}]",1784187401,106,{"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},"some-properties-of-high-order-nonstandard-multistep-multistage-methods","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/some-properties-of-high-order-nonstandard-multistep-multistage-methods/83412/",4,{"url":51,"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","2026-07-23","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 are nonstandard multistep multistage methods, and why are they introduced?","Question",{"text":75,"@type":76},"The work proposes nonstandard versions of multistep multistage schemes by transforming the timestep to a nonstandard function. The goal is to retain qualitative behavior while still achieving high accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper guarantee qualitative property preservation for nonstandard methods?",{"text":80,"@type":76},"It uses a reformulation related to strong-stability preserving (SSP) structure and shows preservation of properties such as boundedness for every positive step size.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main convergence and accuracy claim compared to standard methods?",{"text":84,"@type":76},"The paper proves convergence and demonstrates that the nonstandard schemes can attain the same order as the corresponding standard methods, supported further by numerical experiments.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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"]