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CLT is an explicit finite difference code, whereas M3D-C1 is an implicit finite element code. Despite fundamentally different implementations, results from both codes agree closely for linear and nonlinear growth rates, mode structures, nonlinear saturation levels, Poincaré plots, and scaling laws, confirming accuracy and reliability of the nonlinear simulations.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/linear-and-nonlinear-benchmarks-between-the-clt-code-and-the-m3d-c1-code-for-the-21-resistive-tearing-mode-and-the-11-resistive-kink-mode-benchmark-study/268684/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/linear-and-nonlinear-benchmarks-between-the-clt-code-and-the-m3d-c1-code-for-the-21-resistive-tearing-mode-and-the-11-resistive-kink-mode-benchmark-study/268684.png","ImageObject",300,407,{"name":92,"@type":93},"supergirl","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Which instabilities and mode numbers are benchmarked between CLT and M3D-C1?","Question",{"text":113,"@type":114},"The benchmarks cover the 2/1 resistive tearing mode and the 1/1 resistive kink mode, including both linear and nonlinear regimes.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What are the main differences between the CLT and M3D-C1 codes?",{"text":118,"@type":114},"CLT is an explicit finite-difference 3D nonlinear MHD code for toroidal geometry, while M3D-C1 is an implicit high-order finite-element 3D code solving time-dependent two-fluid MHD equations.",{"name":120,"@type":111,"acceptedAnswer":121},"What aspects of the results match between the two codes despite different implementations?",{"text":122,"@type":114},"They report close agreement in linear and nonlinear growth rates, mode structures, nonlinear saturation levels, Poincaré plots, and scaling laws.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},268684,1789434712,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":46,"preview_page_count":4},962088121634,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Computer Physics Communications 269 (2021) 108134  \nContents lists available at ScienceDirect  \nComputer Physics Communications  \n[www.elsevier.com/locate/cpc](www.elsevier.com/locate/cpc)  \nLinear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode ✩  \nW. Zhang a , S.C. Jardin b , Z.W. Ma a,∗ , A. Kleinerb , H.W. Zhang a  \na Institute for Fusion Theory and Simulation, Department of Physics, Zhejiang University, Hangzhou 310027, China b Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08543, USA  \n\n| a r t i c l e i n f o |  | a b s t r a c t |\n| --- | --- | --- |\n| Article history:\u003Cbr>Received 28 February 2020\u003Cbr>Received in revised form 11 July 2021 Accepted 5 August 2021\u003Cbr>Available online 11 August 2021 |  | The linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode are presented. CLT is an explicit finite difference code, while M3D-C1 is an implicit finite element code. Although the implementations of CLT and M3D-C1 are totally different, we find that the simulation results of the resistive-kink mode and the m/n = 2/1 tearing mode from M3D-C1 and CLT are almost the same, including the linear and nonlinear growth rates, the mode structures, the nonlinear saturation levels, the Poincare plots, and the scaling laws. This confirms that the nonlinear results for the 1/1 resistive-kink mode and 2/1 tearing mode are accurate and reliable.\u003Cbr>© 2021 Elsevier B.V. All rights reserved. |\n| Keywords:\u003Cbr>Benchmark\u003Cbr>Finite element\u003Cbr>Finite difference\u003Cbr>Tearing mode\u003Cbr>Resistive kink mode\u003Cbr>Magnetohydrodynamic instabilities |  |  |\n\n1. Introduction  \nTearing modes are a common phenomenon in tokamaks. The tearing modes are the primary cause of the degradation of tokamak performance [1] and even disruptions [2]. Sawteeth, which not only flatten the plasma temperature but also may trigger neoclassical tearing modes in nearby resonant surfaces [3,4], are believed to be related to the nonlinear evolution of the resistive-kink mode [5,6]. It is worthwhile to investigate these instabilities to increase our understanding to achieve high-performance operation in future fusion reactors [7,8].  \nThe resistive tearing mode was first studied by Furth et al., who found that the linear growth rate asymptotically scales like γ ∼ S −3/5 , where S is the Lundquist number [9]. The first analytical theory of the resistive-kink mode was given by Coppi et al.  \n[6], who found that the linear growth rate asymptotically scales like γ ∼ S −1/3 . Not only linear but also nonlinear theoretical studies have been reported in the past decades [10–12]. It should be noted that significant simplifications of the physical model and the  \n✩  \n*  \nThe review of this paper was arranged by Prof. David W. Walker. Corresponding author.  \n[E-mail addresses:](E-mail addresses: wzhang9008@163.com)[ wzhang9008@163.com](E-mail addresses: wzhang9008@163.com) (W. Zhang), [jardin@pppl.gov](jardin@pppl.gov)  \n(S.C. Jardin), [zwma@zju.edu.cn](zwma@zju.edu.cn) (Z.W. Ma), [akleiner@pppl.gov](akleiner@pppl.gov) (A. Kleiner),  \n[changhw@zju.edu.cn](changhw@zju.edu.cn) (H.W. Zhang).  \ngeometry are applied in these analytical studies. Simulation studies have been widely performed [13, 14] to confirm that these assumptions are reasonable and gain more insight into those instabilities. Since the implementations of different codes are different [15–19], and the nonlinear evolution from different simulation codes sometimes is significantly different, it is difficult to say which simulations are accurate and reliable.  \nOne way to verify the simulation results is to benchmark between different codes. This method can be useful, but it is still a challenge since the nonlinear simulation results can be sensitive to many details of the formulation, the equilibrium, the boundary conditions, and the initial conditions. The M3D-C1","cbCaichKAcKQmA6y","https://ap.wps.com/l/cbCaichKAcKQmA6y","pdf",3497739,"English","# Introduction\n## Background and motivation for tokamak instabilities\n## Linear theories and scaling laws\n## Need for code-to-code validation via benchmarking\n## Description of CLT and M3D-C1 codes","[{\"question\":\"Which instabilities and mode numbers are benchmarked between CLT and M3D-C1?\",\"answer\":\"The benchmarks cover the 2/1 resistive tearing mode and the 1/1 resistive kink mode, including both linear and nonlinear regimes.\"},{\"question\":\"What are the main differences between the CLT and M3D-C1 codes?\",\"answer\":\"CLT is an explicit finite-difference 3D nonlinear MHD code for toroidal geometry, while M3D-C1 is an implicit high-order finite-element 3D code solving time-dependent two-fluid MHD equations.\"},{\"question\":\"What aspects of the results match between the two codes despite different implementations?\",\"answer\":\"They report close agreement in linear and nonlinear growth rates, mode structures, nonlinear saturation levels, Poincaré plots, and scaling laws.\"}]","Linear and nonlinear benchmarks between the CLT code and the M3D-C1 code for the 2/1 resistive tearing mode and the 1/1 resistive kink mode - Benchmark study | PDF"]