[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83373-en":3,"doc-seo-83373-105":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},83373,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Splitting methods for nonlinear Schrödinger equation without order reduction","A high-order integration technique for the nonlinear Schrödinger equation with time-dependent Dirichlet boundary conditions is developed using Yoshida splittings built on the Strang method. The linear stiff component is advanced with a rational-like midpoint rule that computes the needed boundary values from the original data, avoiding any differentiation. While real-coefficient Yoshida splittings cannot exceed second order for parabolic stability, the modified Strang scheme still yields local order 3 and global order 2 without differentiating data, including for reaction-diffusion-type settings.","arXiv :2607 .08387v1 [math .NA] 9 Jul 2026  \nSplitting methods for nonlinear Schr¨odinger equation  \nwithout order reduction  \nC. Arranz-Sim ´on ∗ and B. Cano † IMUVA, Departamento de Matem´atica Aplicada, Facultad de Ciencias, Universidad de Valladolid,  \nPaseo de Bel´en 7, 47011 Valladolid,  \nSpain  \nAbstract  \nA technique is provided in this paper to integrate nonlinear Schr¨odinger equation with time-dependent Dirichlet boundary conditions with high-order Yoshida splittings which are based on Strang method. For that, a modification of Strang method is required in which the linear and stiff part of the equation is integrated with a rational-like version of midpoint rule for which the required boundary values can be calculated without resorting to any differentiation of data. Although Yoshida splitting (with real coefficients) cannot be applied to parabolic problems to obtain order higher than two because of stability, the modified Strang method is also applicable to such type of problems and local order 3 and global order 2 are also obtained without differentiation of data.  \n1 Introduction  \nIt is well-known the phenomenom of order reduction which splitting methods show, even when integrating linear problems with homogeneous boundary conditions and solving each part in an exact way (see [16] for the analysis of Lie-Trotter and Strang method.) Because of that, several techniques have been developed in the literature to avoid it. Up to our knowledge, the best result in this direction is the technique described in [6] for some semilinear wave problems with time-dependent boundary values, for which arbitrary high order can be observed. On the other hand, for multidimensional problems with commuting operators there are also techniques for splitting integrators which manage to improve accuracy reasonably [2, 3, 17] . However, there are other problems which are not of that type, like reaction-diffusion problems or nonlinear Schr¨odinger equation, for which local order 3 and global order 2 is the maximum  \n∗ Email: [carlos.arranz@uva.es](carlos.arranz@uva.es)  \n†Corresponding author. Email: [bcano@uva.es](bcano@uva.es)  \nwhich has been achieved in the literature when the splitting separates the linear and nonlinear part [4, 5, 12, 14, 15] . What is more, in order to achieve local order 3, numerical differentiation is required, which is well-known to be unstable for very small stepsizes. However, local and global order 2 can be obtained with a summation-by-parts argument without any numerical differentiation of data.  \nOn the other hand, a technique has been recently described in the literature to avoid order reduction of initial boundary value problems with rational-like methods [7] . The main advantage of this strategy is that it does not require either numerical nor analytical differentiation of data, but of course boundary values are required and the higher the accuracy that wants to be achieved, the higher the number of necessary boundary evaluations. The key of order reduction with splitting methods is the determination of the boundary values of the intermediate problems and the main idea of this paper is that the rational-like method which is based on the midpoint rule just requires the boundary values at the beginning and end of each step in order that no order reduction is observed. Those boundary values are easily calculable in terms of the data of the original problem and, in such a way, we get local order 3 with the so-called modified Strang method without using any differentiation of data.  \nWe focus on NLS equation in this paper because, in such a case, the nonlinear part can be exactly solved (which simplifies the description of the technique and the analysis) but, more importantly, because splitting methods do not show a stability order barrier with this equation. (We notice that in parabolic problems that order barrier is two [10, 21] when considering splitting methods with real coefficients.) In such a way, we man","cbCaigKcbS5huvRk","https://ap.wps.com/l/cbCaigKcbS5huvRk","pdf",389749,1,21,"English","en",105,"# Introduction\n# Preliminaries\n# Rational-like midpoint rule for the stiff part\n# Analysis of the modified Strang method\n# Arbitrary order via Yoshida splitting\n# Numerical experiments\n# Appendix: abstract framework","[{\"question\":\"What problem does the paper address for splitting methods applied to the nonlinear Schrödinger equation?\",\"answer\":\"It targets order reduction that typically occurs in splitting integrators, aiming to maintain higher accuracy without using differentiated data for boundary values.\"},{\"question\":\"How does the method avoid numerical differentiation of data?\",\"answer\":\"It replaces the stiff part integration with a rational-like midpoint rule where intermediate boundary values are computed directly from the original problem’s boundary data at the start and end of each step.\"},{\"question\":\"What accuracy does the modified Strang method achieve?\",\"answer\":\"It provides local order 3 and global order 2, including cases where real-coefficient Yoshida splittings are limited by stability considerations.\"}]",1784187072,53,{"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},"splitting-methods-for-nonlinear-schrodinger-equation-without-order-reduction","",{"@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/splitting-methods-for-nonlinear-schrodinger-equation-without-order-reduction/83373/",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 problem does the paper address for splitting methods applied to the nonlinear Schrödinger equation?","Question",{"text":75,"@type":76},"It targets order reduction that typically occurs in splitting integrators, aiming to maintain higher accuracy without using differentiated data for boundary values.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method avoid numerical differentiation of data?",{"text":80,"@type":76},"It replaces the stiff part integration with a rational-like midpoint rule where intermediate boundary values are computed directly from the original problem’s boundary data at the start and end of each step.",{"name":82,"@type":73,"acceptedAnswer":83},"What accuracy does the modified Strang method achieve?",{"text":84,"@type":76},"It provides local order 3 and global order 2, including cases where real-coefficient Yoshida splittings are limited by stability 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