[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86232-en":3,"doc-seo-86232-105":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},86232,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Convolution Quadrature Based on a Truncated Trapezoidal Rule","Convolution quadrature (CQ) is built on the truncated trapezoidal rule, an A-stable second-order multistep family indexed by an integer J ≥ 2. Closed-form optimal coefficients are derived to minimize the principal error constant under A-stability, together with an explicit formula for that constant. As J grows, the constant decreases monotonically toward the Dahlquist optimum 1/12, remaining strictly below the BDF2 constant for every J ≥ 2. When used in CQ, analyticity reduces regularity and perturbation demands, and experiments confirm robustness under symbol perturbations and improved accuracy versus BDF2.","arXiv :2607 . 11422v1 [math .NA] 13 Jul 2026  \nConvolution quadrature based on a truncated trapezoidal rule  \nMatteo Ferrari∗   \nAbstract  \nWe study the truncated trapezoidal rule, a family of A-stable second order multistep methods parametrized by an integer J ≥ 2, a compromise between BDF2 and the trapezoidal rule. We obtain a closed-form expression for the coefficients that minimize the principal error constant under the A-stability constraint and we derive an explicit formula for the corresponding principal error constant. The latter decreases to the optimal Dahlquist value 1/12 as J increases, and is strictly smaller than the BDF2 constant for every J ≥ 2. We apply the truncated trapezoidal rule within the convolution quadrature framework. Its analyticity in a neighbourhood of the closed unit disk yields milder regularity and perturbation requirements than those of the trapezoidal rule, while the error constant can be made arbitrarily close to the optimal one by increasing J. Numerical experiments show that convolution quadrature based on the truncated trapezoidal rule remains stable under symbol perturbations, where the trapezoidal rule fails, while achieving a smaller error constant than BDF2 .  \n1 Introduction  \nConvolution Quadrature (CQ), introduced by Lubich in the seminal papers [15, 16], is a technique for the numerical evaluation of one-sided convolutions  \nK (∂t)g (t) = Z0 t κ (t − τ)g (τ)dτ, 0 ≤ t ≤ T,  \nand for the numerical solution of convolution equations K (∂ t)g = ϕ . Both the analysis and the implementation rely on the Laplace transform K of the kernel κ, rather than on κ itself. CQ has become a standard tool in the time-domain treatment of integral equations arising from wave propagation, parabolic problems, viscoelasticity, fractional diffusion, and related areas; we refer to [7] for a comprehensive overview.  \nThe key idea behind CQ is to discretize convolutions by means of an underlying ODE solver, whose generating function δ(ζ) enters the formulation through a contour integral representation of the convolution weights. The choice of the multistep method [15, 16, 17, 18] or Runge-Kutta method [19, 5, 6, 3] determines the accuracy and stability of the resulting scheme.  \nIn this work, we focus on CQ based on multistep methods. The stability of the discrete convolution requires the underlying multistep method to be A-stable, i.e. , Reδ(ei θ ) ≥ 0 for all θ ∈ [0 , 2π], which is restricted by Dahlquist’s second barrier [8] to convergence order at most two. Among A-stable second-order methods, common examples are BDF2 and the trapezoidal rule. The trapezoidal rule is non-dissipative (Reδ(ei θ ) = 0) and achieves the optimal principal error constant 1/12, but its generating function δTR (ζ) = 2(1−ζ)/(1+ζ) has a pole at ζ = −1. This singularity leads to strict regularity requirements on the data for the application of CQ [1, 10, 4] and increases the sensitivity to perturbations of the symbol K [7, §2.9] . BDF2, on the other hand, has an entire generating function and therefore requires milder regularity and perturbation requirements. However, it is dissipative with principal error constant 1/3.  \n∗ Dipartimento di Matematica, Universit`a di Pavia, Pavia, Italy ([m.ferrari@unipv.it](m.ferrari@unipv.it))  \nTo overcome these drawbacks, [2, §6] introduced the Truncated Trapezoidal Rule of order J (TTRJ ) for integer J ≥ 2: a family of multistep methods which is a compromise between BDF2 and the trapezoidal rule, where δTR is replaced by a polynomial of degree J + 1 in ζ − 1 with coefficients chosen to combine the small error constant of the trapezoidal rule with the analyticity of BDF2 . Optimal coefficients in [2] were computed numerically for J = 4 . Our main result is an explicit closed-form formula for the optimal coefficients for any J ≥ 2, under the criterion of minimizing the principal error constant of the method subject to A-stability. Asa consequence, we obtain the explicit dependence of the principal","cbCaijpyIzqnBLWk","https://ap.wps.com/l/cbCaijpyIzqnBLWk","pdf",1367301,2,1,19,"English","en",105,"# Introduction\n## Convolution Quadrature based on multistep methods\n### One-sided convolution\n# Truncated trapezoidal rule and main results\n## Theorem 3.1\n# Proof of Theorem 3.1\n# Numerical experiments","[{\"question\":\"What is the truncated trapezoidal rule in this work?\",\"answer\":\"It is a family of A-stable second-order multistep methods indexed by an integer J ≥ 2, designed as a compromise between BDF2 and the classical trapezoidal rule.\"},{\"question\":\"How is the optimal principal error constant determined?\",\"answer\":\"Optimal coefficients are obtained in closed form by minimizing the principal error constant subject to the A-stability constraint.\"},{\"question\":\"Why does truncated-trapezoidal-rule-based CQ behave better than the trapezoidal rule?\",\"answer\":\"CQ built on this method requires milder regularity and perturbation assumptions due to analyticity near the closed unit disk, and numerical experiments show stability under symbol perturbations where the trapezoidal rule fails.\"}]",1784209681,48,{"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},"convolution-quadrature-based-on-a-truncated-trapezoidal-rule","",{"@graph":36,"@context":85},[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/convolution-quadrature-based-on-a-truncated-trapezoidal-rule/86232/",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","2026-07-25","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 is the truncated trapezoidal rule in this work?","Question",{"text":75,"@type":76},"It is a family of A-stable second-order multistep methods indexed by an integer J ≥ 2, designed as a compromise between BDF2 and the classical trapezoidal rule.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the optimal principal error constant determined?",{"text":80,"@type":76},"Optimal coefficients are obtained in closed form by minimizing the principal error constant subject to the A-stability constraint.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does truncated-trapezoidal-rule-based CQ behave better than the trapezoidal rule?",{"text":84,"@type":76},"CQ built on this method requires milder regularity and perturbation assumptions due to analyticity near the closed unit disk, and numerical experiments show stability under symbol perturbations where the trapezoidal rule 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