[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83143-en":3,"doc-seo-83143-105":30,"detail-sidebar-cat-0-en-105":95},{"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},83143,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","On the order of Runge Kutta methods reusing last stage","Explicit Runge–Kutta (RK) schemes for initial value problems in differential equations are studied under a cost-reduction strategy: the last function evaluation from one step is reused as the first evaluation in the next step. This reuse lowers the per-step workload from s to s−1 function evaluations after the first step, but generally degrades the convergence order. The paper derives coefficient conditions for an s-stage RK method of order p to retain global accuracy order p when applying this reuse, including the highest-order families (p=s) and supporting numerical experiments.","arXiv :2607 .06788v1 [math .NA] 7 Jul 2026  \nOn the order of Runge Kutta methods reusing last stage ⋆  \nM. Calvoa , J.I. Montijanoa,∗, L. Rándeza  \na Departamento Matemática Aplicada, Universidad de Zaragoza, 50009-Zaragoza, Spain.  \nAbstract  \nIn this paper we consider explicit Runge–Kutta (RK) methods for the numerical solution of Initial Value Problems (IVPs) in differential equations in which the last function evaluation in a step is reused, substituting the first evaluation for the next step. This requirement implies that, except for the first step, the computational cost of an s–stage explicit RK method reduces from sto (s−1) function evaluations per step. It will be seen that, in general, if a RK method has order p, when introducing this reused stage the new method has in general order \u003C p.  \nIn this context we study the conditions on the coefficients of an explicit RK method with order p such that the global error of the method reusing the last stage has the same order p of accuracy. A detailed study on the conditions that the coefficients of an s stage method with order p must satisfy for the method reusing the last stage has also order p. In particular, for the families of s–stage explicit RK methods with the highest order p = s we identify those methods reusing the last stage which retain the order of of the original method. The results of some numerical experiments are presented to confirm the theoretical results.  \nKeywords:  \n2000 MSC: 65L05, 65L07  \n1. Introduction  \nRunge Kutta (RK) methods for the numerical solution of IVPs in systems of differential equations  \ndyd(tt) = f (t, y(t)), y(0) = y0 ∈ Rd , t ∈ [0, T] (1)  \nare among the most popular algorithms for the numerical solution of (1) . In particular explicit s–stage RK methods advance the numerical solution of (1)  \n⋆ This work was supported by Ministerio de Ciencia, Innovación y Universidades, Agencia estatal de Investigación, Spain, under the project PID2022–141385NB–I00 .  \n∗ Corresponding author  \nEmail addresses: calvo@unizar .es (M. Calvo), monti@unizar .es (J.I. Montijano), [randez@unizar.es](randez@unizar.es) (L. Rández)  \nfrom (tn = nh, yn) to (tn+1 = tn + h, yn+1) with the formulas  \nyn+1 = yn + h [b1 fn,1 + b2 fn,2 + ... + bn fn,s], (2)  \nwhere  \nfn,1 = f (tn , yn)  \nfn,2 = f (tn + c2 h, yn + h a21 fn,1)  \n... ... (3) fn,s = f (tn + csh, yn + has,1fn,1 + ... + has,s−1fn,s−1) .  \nIt is common to collect the available parameters of (2),(3) , bj , cj , ajk in the form of the Butcher array [3, 11]  \nc 1 = 0 c2  \n.  \n.  \n.  \n.  \n..  \ncs  \nwith the condition  \n[b](b1 b2 ... bs)[1](b1 b2 ... bs)[ b](b1 b2 ... bs)[2](b1 b2 ... bs)[ ... b](b1 b2 ... bs)[s](b1 b2 ... bs)  \ni−1  \nci = X aij i = 1 , .., sj=1  \nA  \nbT  \n(4)  \nUsually the free parameters of (4) are selected in such a way that the method achieves the highest accuracy possible for all IVP (1) with f (t, y) sufficiently smooth. Denoting by y(t;tn , yn) the local solution of differential problem y′(t) = f (t, y(t)) that passes by the point (tn , yn), that is, y (tn ) = yn , the accuracy is measured by the local error, defined as the error after one step ∥yn+1 − y(tn + h;tn , yn)∥ . Then the method has order p [11] if  \n∥yn+1 − y(tn + h;tn , yn)∥ = O (hp+1) , h → 0 (5)  \nfor all IVP (1) . This condition implies that the global error of (1) behaves as ∥y(T) − yN ∥ = O (hp ) with Nh = T. In this setting it is well known that for s ≤ 4 it is possible to get order p = s whereas for s > 4 the maximum order p \u003C s.  \nClearly the computational cost of an explicit RK method is associated to the number of function evaluations at each step i.e. the number of stages. With the aim to reduce this cost we examine here explicit RK methods with cs = 1 in which the first evaluation of the n-th step, fn,1 = f (tn , yn), is substituted by the last evaluation of the (n−1)-th step, fn−1,s i.e. fn,1 = f (tn−1 +cs h, yn−1 + hPsj11 as,j fn−1,j) . These methods will be called explicit RK methods reusing last stage, and we","cbCaidoFLjvzHKwW","https://ap.wps.com/l/cbCaidoFLjvzHKwW","pdf",480334,2,1,21,"English","en",105,"# Introduction\n## Reusing the last stage strategy\n## Order and error concepts\n## Two-approximation one-step formulation\n## Butcher series analysis","[{\"question\":\"What does “reusing the last stage” mean in the Runge–Kutta method studied here?\",\"answer\":\"The method replaces the first function evaluation of a step with the last function evaluation computed in the previous step, so the next step starts from the reused stage value rather than recomputing it.\"},{\"question\":\"How does reusing the last stage affect computational cost per step?\",\"answer\":\"After the first step, an explicit s-stage Runge–Kutta method reduces from requiring s function evaluations per step to requiring s−1 function evaluations per step due to the reuse.\"},{\"question\":\"Why might the convergence order drop when reusing the last stage?\",\"answer\":\"Although the original RK method may have order p, the reused-stage modification changes the effective one-step formulation. In general, the resulting approximation has accuracy order q \\u003c p, leading to overall order min(q+1, p).\"},{\"question\":\"What conditions are derived to ensure the reused-last-stage method keeps global order p?\",\"answer\":\"The paper establishes coefficient constraints for an s-stage explicit RK method of order p such that the global error of the reused-last-stage variant maintains order p, with special attention to families where p=s.\"}]",1784185580,53,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"on-the-order-of-runge-kutta-methods-reusing-last-stage","",{"@graph":36,"@context":89},[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/on-the-order-of-runge-kutta-methods-reusing-last-stage/83143/",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","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What does “reusing the last stage” mean in the Runge–Kutta method studied here?","Question",{"text":75,"@type":76},"The method replaces the first function evaluation of a step with the last function evaluation computed in the previous step, so the next step starts from the reused stage value rather than recomputing it.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does reusing the last stage affect computational cost per step?",{"text":80,"@type":76},"After the first step, an explicit s-stage Runge–Kutta method reduces from requiring s function evaluations per step to requiring s−1 function evaluations per step due to the reuse.",{"name":82,"@type":73,"acceptedAnswer":83},"Why might the convergence order drop when reusing the last stage?",{"text":84,"@type":76},"Although the original RK method may have order p, the reused-stage modification changes the effective one-step formulation. In general, the resulting approximation has accuracy order q \u003C p, leading to overall order min(q+1, p).",{"name":86,"@type":73,"acceptedAnswer":87},"What conditions are derived to ensure the reused-last-stage method keeps global order p?",{"text":88,"@type":76},"The paper establishes coefficient constraints for an s-stage explicit RK method of order p such that the global error of the reused-last-stage variant maintains order p, with special attention to families where p=s.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]