{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T07:06:47Z","timestamp":1776755207652,"version":"3.51.2"},"reference-count":25,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["21-71-20024"],"award-info":[{"award-number":["21-71-20024"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present two methods for the implicit integration of nonlinear stiff systems.\nDirect application of the Newton method to backward Euler discretization of such systems may diverge.\nWe observe that the solution is recovered by smoothing out certain eigenvalues in the Jacobian matrix.\nTo this end, we introduce a solution-dependent matrix-weighted combination of backward and forward Euler methods.\nThe weight is tuned on each Newton iteration to reproduce the solution with an exponential integrator, whereby a weight function for smoothing eigenvalues is obtained.\nWe apply the proposed techniques, namely quasi-Newton backward Euler and matrix-weighted Euler, to several stiff systems, including Lotka\u2013Volterra, Van der Pol\u2019s, and a blood coagulation cascade.<\/jats:p>","DOI":"10.1515\/cmam-2022-0083","type":"journal-article","created":{"date-parts":[[2022,9,13]],"date-time":"2022-09-13T17:18:27Z","timestamp":1663089507000},"page":"83-92","source":"Crossref","is-referenced-by-count":5,"title":["Two Methods for the Implicit Integration of Stiff Reaction Systems"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0424-6695","authenticated-orcid":false,"given":"Ivan D.","family":"Butakov","sequence":"first","affiliation":[{"name":"Moscow Institute of Physics and Technology , Institutskiy per., 9 , Dolgoprudny , Moscow Region, 141700 , Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6548-0496","authenticated-orcid":false,"given":"Kirill M.","family":"Terekhov","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology , Institutskiy per., 9 , Dolgoprudny , Moscow Region, 141700; and Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkin str., 8, Moscow, 119333 , Russia"}]}],"member":"374","published-online":{"date-parts":[[2022,9,14]]},"reference":[{"key":"2023033112515590222_j_cmam-2022-0083_ref_001","doi-asserted-by":"crossref","unstructured":"R. Alexander,\nThe modified Newton method in the solution of stiff ordinary differential equations,\nMath. Comp. 57 (1991), no. 196, 673\u2013701.","DOI":"10.1090\/S0025-5718-1991-1094939-7"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_002","doi-asserted-by":"crossref","unstructured":"D. G. Anderson,\nIterative procedures for nonlinear integral equations,\nJ. Assoc. Comput. Mach. 12 (1965), 547\u2013560.","DOI":"10.1145\/321296.321305"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_003","doi-asserted-by":"crossref","unstructured":"L. Armijo,\nMinimization of functions having Lipschitz continuous first partial derivatives,\nPacific J. Math. 16 (1966), 1\u20133.","DOI":"10.2140\/pjm.1966.16.1"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_004","doi-asserted-by":"crossref","unstructured":"W. Auzinger and R. Frank,\nAsymptotic error expansions for stiff equations: An analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case,\nNumer. Math. 56 (1989), no. 5, 469\u2013499.","DOI":"10.1007\/BF01396649"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_005","doi-asserted-by":"crossref","unstructured":"W. Auzinger, R. Frank and G. Kirlinger,\nA note on convergence concepts for stiff problems,\nComputing 44 (1990), no. 3, 197\u2013208.","DOI":"10.1007\/BF02262216"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_006","doi-asserted-by":"crossref","unstructured":"W. Auzinger, R. Frank and G. Kirlinger,\nModern convergence theory for stiff initial value problems,\nJ. Comput. Appl. Math. 45 (1993), no. 1\u20132, 5\u201316.","DOI":"10.1016\/0377-0427(93)90260-I"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_007","doi-asserted-by":"crossref","unstructured":"A. Bouchnita, K. Terekhov, P. Nony, Y. Vassilevski and V. Volpert,\nA mathematical model to quantify the effects of platelet count, shear rate, and injury size on the initiation of blood coagulation under venous flow conditions,\nPloS one 15 (2020), no. 7, Article ID e0235392.","DOI":"10.1371\/journal.pone.0235392"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_008","doi-asserted-by":"crossref","unstructured":"P. N. Brown, A. C. Hindmarsh and H. F. Walker,\nExperiments with quasi-Newton methods in solving stiff ODE systems,\nSIAM J. Sci. Statist. Comput. 6 (1985), no. 2, 297\u2013313.","DOI":"10.1137\/0906022"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_009","doi-asserted-by":"crossref","unstructured":"P. N. Brown and Y. Saad,\nConvergence theory of nonlinear Newton\u2013Krylov algorithms,\nSIAM J. Optim. 4 (1994), no. 2, 297\u2013330.","DOI":"10.1137\/0804017"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_010","doi-asserted-by":"crossref","unstructured":"P. R. Brune, M. G. Knepley, B. F. Smith and X. Tu,\nComposing scalable nonlinear algebraic solvers,\nSIAM Rev. 57 (2015), no. 4, 535\u2013565.","DOI":"10.1137\/130936725"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_011","doi-asserted-by":"crossref","unstructured":"G. Dahlquist,\nA special stability problem for linear multistep methods,\nNordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27\u201343.","DOI":"10.1007\/BF01963532"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_012","unstructured":"G. Dahlquist,\nOn stability and error analysis for stiff non-linear problems part i,\nTechnical report CM-P00069396, 1975."},{"key":"2023033112515590222_j_cmam-2022-0083_ref_013","unstructured":"G. Guennebaud, B. Jacob,\nEigen v3, http:\/\/eigen.tuxfamily.org, 2010."},{"key":"2023033112515590222_j_cmam-2022-0083_ref_014","unstructured":"L. V. Kantorovi\u010d,\nOn Newton\u2019s method,\nTrudy Mat. Inst. Steklov. 28 (1949), 104\u2013144."},{"key":"2023033112515590222_j_cmam-2022-0083_ref_015","doi-asserted-by":"crossref","unstructured":"M. Y. Liu, L. Zhang and C. F. Zhang,\nStudy on banded implicit Runge\u2013Kutta methods for solving stiff differential equations,\nMath. Probl. Eng. 2019 (2019), Article ID 4850872.","DOI":"10.1155\/2019\/4850872"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_016","unstructured":"A. J. Lotka,\nElements of Physical Biology,\nWilliams & Wilkins, Baltimore, 1925."},{"key":"2023033112515590222_j_cmam-2022-0083_ref_017","doi-asserted-by":"crossref","unstructured":"P. K. Moore and L. R. Petzold,\nA stepsize control strategy for stiff systems of ordinary differential equations,\nAppl. Numer. Math. 15 (1994), no. 4, 449\u2013463.","DOI":"10.1016\/0168-9274(94)00042-5"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_018","unstructured":"Y. E. Nesterov,\nA method for solving the convex programming problem with convergence rate \n                  \n                     \n                        \n                           O\n                           \u2062\n                           \n                              (\n                              \n                                 1\n                                 \/\n                                 \n                                    k\n                                    2\n                                 \n                              \n                              )\n                           \n                        \n                     \n                     \n                     O(1\/k^{2})\n                  \n               ,\nDokl. Akad. Nauk SSSR 269 (1983), no. 3, 543\u2013547."},{"key":"2023033112515590222_j_cmam-2022-0083_ref_019","doi-asserted-by":"crossref","unstructured":"J. M. Ortega and W. C. Rheinboldt,\nIterative Solution of Nonlinear Equations in Several Variables,\nClassics Appl. Math. 30,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719468"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_020","doi-asserted-by":"crossref","unstructured":"S. Schlenkrich, A. Walther and A. Griewank,\nApplication of AD-based quasi-Newton methods to stiff ODEs,\nAutomatic Differentiation: Applications, Theory, and Implementations,\nLect. Notes Comput. Sci. Eng. 50,\nSpringer, Berlin (2006), 89\u201398.","DOI":"10.1007\/3-540-28438-9_8"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_021","doi-asserted-by":"crossref","unstructured":"F. Shen, C. J. Kastrup, Y. Liu and R. F. Ismagilov,\nThreshold response of initiation of blood coagulation by tissue factor in patterned microfluidic capillaries is controlled by shear rate,\nArteriosclerosis Thrombosis Vascular Biol. 28 (2008), no. 11, 2035\u20132041.","DOI":"10.1161\/ATVBAHA.108.173930"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_022","doi-asserted-by":"crossref","unstructured":"D. C. Sorensen,\nNewton\u2019s method with a model trust region modification,\nSIAM J. Numer. Anal. 19 (1982), no. 2, 409\u2013426.","DOI":"10.1137\/0719026"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_023","doi-asserted-by":"crossref","unstructured":"M. Takesaki,\nTheory of Operator Algebras. I. Operator Algebras and Non-commutative Geometry 5,\nEncyclopaedia Math. Sci. 124,\nSpringer, Berlin, 2002.","DOI":"10.1007\/978-3-662-10453-8"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_024","doi-asserted-by":"crossref","unstructured":"Y. Vassilevski, K. Terekhov, K. Nikitin and I. Kapyrin,\nParallel Finite Volume Computation on General Meshes,\nSpringer, Cham, 2020.","DOI":"10.1007\/978-3-030-47232-0"},{"key":"2023033112515590222_j_cmam-2022-0083_ref_025","doi-asserted-by":"crossref","unstructured":"P. Wolfe,\nConvergence conditions for ascent methods,\nSIAM Rev. 11 (1969), 226\u2013235.","DOI":"10.1137\/1011036"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0083\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0083\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:17:29Z","timestamp":1680283049000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0083\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,14]]},"references-count":25,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,10,6]]},"published-print":{"date-parts":[[2023,1,1]]}},"alternative-id":["10.1515\/cmam-2022-0083"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0083","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,14]]}}}