[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83473-en":3,"doc-seo-83473-105":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},83473,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Rigorous Analysis of the Time-Splitting Methods for the Semiclassical Dirac Equation","Rigorous error analysis is developed for mass-preserving time-splitting schemes used to solve the semiclassical Dirac equation. The scaled Planck constant ε induces rapid oscillations in space and time when 0\u003Cε≪1, with wavelengths of order O(ε). The study derives precise dependence of approximation errors on the time step τ, the spatial mesh size h, and ε, giving temporal rates O(-τ/ε2) for S1 and O(-τ2/ε3) for S2, and spatial rate O(hm/εm) for both. Bounds for total probability density ρ and current density J are provided, together with comparisons to FDTD methods and verification via numerical experiments.","arXiv :2607 .00335v1 [math .NA] 1 Jul 2026  \nNoname manuscript No.  \n(will be inserted by the editor)  \nRigorous analysis of the time-splitting methods for thesemiclassical Dirac equation  \nHe Wang · Jia Yin  \nAbstract We provide rigorous error analysis of the mass-preserving time-splitting methods for solving the semiclassical Dirac equation. The scaled Planck constant ϵ in the equation gives rise to rapid oscillations in both space and time when 0 \u003C ϵ ≪ 1 with wavelengths of order O (ϵ) . Rigorous error estimates reveal the precise dependence of the approximation errors on the time step τ, the spatial mesh size h, and the parameter ϵ. Specifically, the temporal error scales as O 􀀀τ/ϵ2 􀀁 for the first-order splitting S1 and as O 􀀀τ2 /ϵ3 􀀁 for the second-order splitting S2 , while the spatial error scales as O (hm /ϵm ) for both methods, where m is related to the regularity of the solution. In addition, we obtain error bounds for key physical observables, including the total probability density ρ and the current density J. Compared with finite difference time domain (FDTD) methods, timesplitting approaches exhibit spectral accuracy in space and retain a relatively low computational cost. Furthermore, we demonstrate that higher accuracy can be achieved by employing the fourth-order compact time-splitting (S4c ) method. Numerical experiments are conducted to verify the reliability of the error estimates.  \nKeywords the semiclassical Dirac equation · time-splitting methods · spectral methods · ϵ-scalability  \n1 Introduction  \nThe Dirac equation, formulated by Paul Dirac in 1928, describes the relativistic quantum dynamics of spin-1/2 particles. The equation not only accounts for the fine structure of the hydrogen spectrum, but also successfully predicted the existence of antimatter, thus having a profound impact on particle physics [14] . In condensed matter systems, the massless Dirac equation governs the low-energy electronic behavior of graphene, explaining its unusual linear dispersion and the associated quantum Hall effect [11,19] . Furthermore, the Dirac equation provides the fundamental framework for understanding the topological surface states and  \nSchool of Mathematical Sciences, Fudan University, Shanghai 200433, China  \nE-mail: [hwang24@m.fudan.edu.cn](hwang24@m.fudan.edu.cn) (He Wang), [jiayin@fudan.edu.cn](jiayin@fudan.edu.cn) (Jia Yin)  \nassociated dissipationless edge currents in topological insulators and topological semimetals [16,23] . In the field of quantum simulation, it has been employed to model physical phenomena in curved spacetime [8] . In strong-field physics, the time-dependent Dirac equation is essential for simulating highly relativistic electron dynamics under ultra-intense laser fields (e.g., attosecond lasers), including highly non-linear quantum electrodynamical processes such as electron-positron pair production [13] . Among its various regimes, the Dirac equation in the semiclassical limit is of particular interest due to its distinctive physical and mathematical features, as it describes the transition from quantum to classical dynamics. According to the framework established in [3,7], the d-dimensional (d = 1 , 2 , 3) semiclassical Dirac equation admits the following formulation:  \niϵ∂tΨ = 􀀲􀀴 −iϵ 1 αj∂j + β + V (t, x)I4 − 1 Aj (t, x)α j 􀀳􀀵 Ψ, (1.1)  \nwhere i = √−1 is the imaginary unit, x = (x1 , ··· , xd) represents the spatial coordinate, t is time, I4 denotes the 4 × 4 identity matrix and the 4 × 4 matrices αj (j = 1, ··· , d), β are defined as follows:  \nαj = 􀀔0σj σj0􀀕 , j = 1 , ··· , d, β = 􀀔I20 −0I2􀀕 . (1.2)  \nIn the above expression, I2 denotes the 2 × 2 identity matrix and σ j (j = 1, ··· , d) represent the Pauli matrices which are given as  \nσ1 = 􀀔0 11 0􀀕 , σ2 = 􀀔0 −i 0i􀀕 , σ3 = 􀀔10 1􀀕 . (1.3)  \nMoreover, in the equation (1 . 1), Ψ ∈ C4 represents the four-component spinor wave function, and V, Aj ∈ R (j = 1, ··· , d) represent the electric and magnetic potentials, respectively. The","cbCaipc14qLkZ76I","https://ap.wps.com/l/cbCaipc14qLkZ76I","pdf",1501884,3,1,22,"English","en",105,"# Introduction\n## Semiclassical Dirac equation and oscillatory regime\n## Limitations of direct discretization\n## Motivation for ϵ-scalable time-splitting\n# Time-splitting error analysis","[{\"question\":\"Why do time-splitting schemes need rigorous error analysis for the semiclassical Dirac equation?\",\"answer\":\"Because the small scaled Planck constant ε (0\\u003cε≪1) creates rapid space-time oscillations that make accuracy and computational efficiency highly sensitive to discretization parameters.\"},{\"question\":\"How do the temporal errors scale for the first- and second-order splittings S1 and S2?\",\"answer\":\"The temporal error scales as O(-τ/ε2) for the first-order splitting S1 and as O(-τ2/ε3) for the second-order splitting S2.\"},{\"question\":\"What observables are controlled by the derived error bounds?\",\"answer\":\"The analysis provides error bounds for key physical observables, including the total probability density ρ and the current density J.\"}]",1784188230,55,{"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},"rigorous-analysis-of-the-time-splitting-methods-for-the-semiclassical-dirac-equation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/rigorous-analysis-of-the-time-splitting-methods-for-the-semiclassical-dirac-equation/83473/",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],{"name":72,"@type":73,"acceptedAnswer":74},"Why do time-splitting schemes need rigorous error analysis for the semiclassical Dirac equation?","Question",{"text":75,"@type":76},"Because the small scaled Planck constant ε (0\u003Cε≪1) creates rapid space-time oscillations that make accuracy and computational efficiency highly sensitive to discretization parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the temporal errors scale for the first- and second-order splittings S1 and S2?",{"text":80,"@type":76},"The temporal error scales as O(-τ/ε2) for the first-order splitting S1 and as O(-τ2/ε3) for the second-order splitting S2.",{"name":82,"@type":73,"acceptedAnswer":83},"What observables are controlled by the derived error bounds?",{"text":84,"@type":76},"The analysis provides error bounds for key physical observables, including the total probability density ρ and the current density 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