{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T10:41:30Z","timestamp":1790764890962,"version":"4.1.0"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We introduce a family of impulselike methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast part and a slow part. Methods of this family are specified by two weight functions $\\phi$, $\\psi$; one is used to average positions and the other to mollify the force. When the fast forces are conservative and $\\phi=\\psi$, the methods here coincide with the mollified impulse methods introduced by Garc\u00eda-Archilla, Sanz-Serna, and Skeel. On the other hand, the methods here extend to nonlinear situations a well-known class of exponential integrators introduced by Hairer and Lubich for cases of linear fast forces. A convergence analysis is presented that provides insight into the role played by the processes of mollification and averaging in avoiding order reduction. A simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction.<\/jats:p>","DOI":"10.1137\/070681636","type":"journal-article","created":{"date-parts":[[2008,3,5]],"date-time":"2008-03-05T19:23:53Z","timestamp":1204745033000},"page":"1040-1059","source":"Crossref","is-referenced-by-count":37,"title":["Mollified Impulse Methods for Highly Oscillatory Differential Equations"],"prefix":"10.1137","volume":"46","author":[{"given":"J. M.","family":"Sanz-Serna","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,5]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"W. Arendt, Ch. J. K. Batty, M. Hieber, and F. Neubrander,\n                      Vector-Valued Laplace Transforms and Cauchy Problems\n                      , Birkh\u00e4user, Basel, 2001.","DOI":"10.1007\/978-3-0348-5075-9"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-005-7121-z"},{"key":"R3","unstructured":"K. Dekker and J. G. Verwer,\n                      Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations\n                      , North-Holland, Amsterdam, 1984."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S1076-5670(02)80042-8"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596313851"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/39\/19\/S10"},{"key":"R7","unstructured":"H. Grubm\u00fcller,\n                      Dynamiksimulation sehr gro\u00dfer Makromolek\u00fcle auf einem Parallelrechner\n                      , Master's thesis, Physik-Dept. der Tech. Univ. M\u00fcnchen, Munich, 1989."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1080\/08927029108022142"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999353594"},{"key":"R10","unstructured":"E. Hairer, Ch. Lubich, and G. Wanner,\n                      Geometric Numerical Integration\n                      , 2nd ed., Springer, Berlin, 2006."},{"key":"R11","unstructured":"E. Hairer, S. P. N\u00f8rsett, and G. Wanner,\n                      Solving Ordinary Differential Equations\n                      I\n                      , Nonstiff Problems\n                      , 2nd. ed., Springer, Berlin, 1993."},{"key":"R12","doi-asserted-by":"crossref","unstructured":"E. Hairer and G. Wanner,\n                      Solving Ordinary Differential Equations\n                      II\n                      , Stiff and Differential-Algebraic Problems\n                      , 2nd ed., Springer, Berlin, 1996.","DOI":"10.1007\/978-3-642-05221-7_1"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/24.3.365"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"B. Leimkuhler and S. Reich,\n                      Simulating Hamiltonian Dynamics\n                      , Cambridge University Press, Cambridge, 2005.","DOI":"10.1017\/CBO9780511614118"},{"key":"R15","unstructured":"J. M. Sanz-Serna,\n                      Stability and convergence in numerical analysis\n                      I:\n                      Linear problems, a simple, comprehensive account\n                      , in Nonlinear Differential Equations, J. K. Hale and P. Mart\u00ednez-Amores, eds., Pitman, Boston, 1985, pp. 64\u2013113."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"J. M. Sanz-Serna and M. P. 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