[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86524-en":3,"doc-seo-86524-105":29,"detail-sidebar-cat-0-en-105":83},{"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":20,"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},86524,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Performance Evaluation of Stabilized Corrections for Mixed Precision Runge-Kutta Methods","Mixed precision Runge–Kutta methods speed up diagonally implicit Runge–Kutta schemes by performing implicit solves in low precision and computing inexpensive explicit corrections to retain accuracy for larger time steps. Explicit corrections, however, can significantly harm stability. Stabilized corrections introduce a stabilization matrix in the correction step, improving accuracy and stability, but their runtime cost is not quantified. This study numerically measures runtime performance on inviscid Burgers’ and porous medium equations across multiple precision pairings and SDIRK orders.","Performance Evaluation of Stabilized Corrections for Mixed Precision Runge–Kutta Methods  \nCsar Herrera  \nDepartment of Mathematics Purdue University West Lafayette, IN, USA [herre125@purdue.edu](herre125@purdue.edu)  \nJohn Driscoll & Sigal Gottlieb & Zachary J. Grant & Tej Sai Kakumanu  \nMathematics Department, UMass Dartmouth North Dartmouth, MA, USA [jdriscoll9@umassd.edu & sgottlieb@umassd.edu](jdriscoll9@umassd.edu & sgottlieb@umassd.edu)[ ](jdriscoll9@umassd.edu & sgottlieb@umassd.edu)[zgrant@umassd.edu & tkakumanu@umassd.edu](zgrant@umassd.edu & tkakumanu@umassd.edu)  \nAndrew Christlieb  \nDepartment of CMS&E Michigan State University East Lansing, MI, USA [christli@msu.edu](christli@msu.edu)  \narXiv :2607 . 10967v1 [math .NA] 13 Jul 2026  \nAbstract—Mixed precision Runge–Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge-–Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers’ equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.  \nIndex Terms—Mixed precision, Runge–Kutta, numerical methods  \nI. INTRODUCTION  \nThe use of mixed precision to accelerate numerical algorithms has become increasingly popular in recent years (see, e.g., [1]–[3]), driven both by the runtime benefits of low precision arithmetic and by the proliferation of hardware that natively supports multiple floating-point formats. Such algorithms aim to combine the efficiency of low precision with the accuracy of high precision.  \nIn the context of ordinary differential equations and the time-evolution of partial differential equations, Z. J. Grant proposed in [4] a perturbed Runge–Kutta framework that treats the use of a cheaper, lower-accuracy implicit solve asan additive perturbation of a Runge–Kutta method, enabling the design of mixed precision methods whose low precision errors are suppressed. The performance of these methods was subsequently evaluated in [5], where speedups ranging from roughly 2 × to 15 × were reported, and their stability was analyzed in [6] .  \nThe accuracy lost to the low precision perturbation can be recovered by appending inexpensive, explicit high precision  \ncorrections to the implicit stages [4] . Although these corrections are effective for sufficiently small time steps, it was numerically shown in [6] that they shrink the region of linear stability and can introduce instabilities for larger time steps. To address this limitation, stabilized corrections were proposed in [7] . While the analysis and numerical experiments in [7] show that these new correction approaches improve both accuracy and stability, they do not quantify the associated runtime costs. The purpose of this paper is to evaluate the computational runtimes of these stabilized corrections. When applying these stabilized corrections to mixed-precision DIRK methods of second, third, and fourth order, our numerical experiments show speedups ","cbCaikE0FsVgw4lc","https://ap.wps.com/l/cbCaikE0FsVgw4lc","pdf",1532086,7,1,"English","en",105,"# Introduction\n# Mixed Precision DIRK Methods & Corrections\n## Mixed Precision DIRK Methods\n## Explicit Corrections\n# Performance Evaluation","[{\"question\":\"Which precision pairings and method orders are compared?\",\"answer\":\"Experiments compare half, single, double, and quadruple precision pairings for SDIRK methods of orders two through four, including speedups achieved when combining half precision with double or quadruple precision.\"}]",1784212384,18,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"performance-evaluation-of-stabilized-corrections-for-mixed-precision-runge-kutta-methods","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":21},"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/performance-evaluation-of-stabilized-corrections-for-mixed-precision-runge-kutta-methods/86524/",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","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Which precision pairings and method orders are compared?","Question",{"text":75,"@type":76},"Experiments compare half, single, double, and quadruple precision pairings for SDIRK methods of orders two through four, including speedups achieved when combining half precision with double or quadruple precision.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,111,114,119,122,126],{"id":21,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":45,"category_name":108,"show_sort_weight":109,"slug":110},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]