{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T22:05:56Z","timestamp":1776290756080,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T00:00:00Z","timestamp":1564963200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T00:00:00Z","timestamp":1564963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002835","name":"Chalmers University of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100002835","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2020,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie\u2013Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectrum in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie\u2013Poisson structure. Here, we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie\u2013Poisson structure. The methods are surprisingly simple and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie\u2013Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to several classical isospectral flows.<\/jats:p>","DOI":"10.1007\/s10208-019-09428-w","type":"journal-article","created":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T20:07:52Z","timestamp":1565035672000},"page":"889-921","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Lie\u2013Poisson Methods for Isospectral Flows"],"prefix":"10.1007","volume":"20","author":[{"given":"Klas","family":"Modin","sequence":"first","affiliation":[]},{"given":"Milo","family":"Viviani","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,8,5]]},"reference":[{"key":"9428_CR1","unstructured":"S.\u00a0Bates and A.\u00a0Weinstein, Lectures on the Geometry of Quantization, vol.\u00a08, AMS, Berkeley, 1997."},{"key":"9428_CR2","doi-asserted-by":"crossref","unstructured":"A.\u00a0M. Bloch and A.\u00a0Iserles, On an isospectral Lie\u2013Poisson system and its Lie algebra, Foundations of Computational Mathematics 6 (2006), 121\u2013144.","DOI":"10.1007\/s10208-005-0173-2"},{"key":"9428_CR3","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1007\/s10208-015-9257-9","volume":"16","author":"G Bogfjellmo","year":"2016","unstructured":"G.\u00a0Bogfjellmo and H.\u00a0Marthinsen, High-Order Symplectic Partitioned Lie Group Methods, Foundations of Computational Mathematics 16 (2016), 493\u2013530.","journal-title":"Foundations of Computational Mathematics"},{"key":"9428_CR4","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0024-3795(91)90021-N","volume":"146","author":"RW Brockett","year":"1991","unstructured":"R.\u00a0W. Brockett, Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems, Lin. Alg. Appl. 146 (1991), 79\u201391.","journal-title":"Lin. Alg. Appl."},{"key":"9428_CR5","first-page":"87","volume":"3","author":"MT Chu","year":"1994","unstructured":"M.\u00a0T. Chu, A list of matrix flows with applications, Fields Institute Communications 3 (1994), 87\u201397.","journal-title":"Fields Institute Communications"},{"key":"9428_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906340019","volume":"17","author":"MT Chu","year":"2008","unstructured":"M.\u00a0T. Chu, Linear algebra algorithms as dynamical systems, Acta Numer. 17 (2008), 1\u201386.","journal-title":"Acta Numer."},{"key":"9428_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0720001","volume":"20","author":"P Deift","year":"1983","unstructured":"P.\u00a0Deift, T.\u00a0Nanda, and C.\u00a0Tomei, Ordinary differential equations and the symmetric eigenvalue problem, SIAM J. Numer. Anal. 20 (1983), 1\u201322.","journal-title":"SIAM J. Numer. Anal."},{"key":"9428_CR8","doi-asserted-by":"crossref","unstructured":"H.\u00a0Flaschka, The Toda lattice. II. Existence of integrals, Physical Review B 9 (1974), 1924.","DOI":"10.1103\/PhysRevB.9.1924"},{"key":"9428_CR9","volume-title":"Geometric Numerical Integration","author":"E Hairer","year":"2006","unstructured":"E.\u00a0Hairer, C.\u00a0Lubich, and G.\u00a0Wanner, Geometric Numerical Integration, vol.\u00a031, Springer, Berlin, Heidelberg, 2006."},{"key":"9428_CR10","doi-asserted-by":"crossref","unstructured":"M.\u00a0H\u00e9non, Integrals of the Toda lattice, vol.\u00a09, 1974.","DOI":"10.1103\/PhysRevB.9.1921"},{"key":"9428_CR11","unstructured":"J.\u00a0Hoppe, Ph.D. thesis MIT Cambridge, (1982)."},{"key":"9428_CR12","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1007\/s002200050379","volume":"195","author":"J Hoppe","year":"1998","unstructured":"J.\u00a0Hoppe and S.-T. Yau, Some properties of matrix harmonics on S2, Commun. Math. Phys. 195 (1998), 66\u201377.","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"9428_CR13","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1137\/0733019","volume":"33","author":"L Jay","year":"1996","unstructured":"L.\u00a0Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal. 33(1) (1996), 368\u2013387.","journal-title":"SIAM J. Numer. Anal."},{"key":"9428_CR14","doi-asserted-by":"crossref","unstructured":"A.W. Knapp, Lie groups beyond an introduction, Birkh\u00e4user, 1996.","DOI":"10.1007\/978-1-4757-2453-0"},{"key":"9428_CR15","doi-asserted-by":"crossref","unstructured":"H.J. Landau, The inverse eigenvalue problem for real symmetric toeplitz matrices, Journal of the American Mathematical Society 7 (1994).","DOI":"10.2307\/2152790"},{"key":"9428_CR16","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1007\/BF01076037","volume":"10","author":"SV Manakov","year":"1976","unstructured":"S.V. Manakov, Note on the integration of the Euler\u2019s equations of the dynamics of a n-dimensional rigid body, Functional Anal. Appl. 10 (1976), 328\u2013329.","journal-title":"Functional Anal. Appl."},{"key":"9428_CR17","doi-asserted-by":"crossref","unstructured":"J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, vol. vol.17 of Texts in Applied Mathematics, Springer-Verlag, New York, 1999.","DOI":"10.1007\/978-0-387-21792-5"},{"issue":"6","key":"9428_CR18","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1088\/0951-7715\/27\/6\/1525","volume":"27","author":"R McLachlan","year":"2014","unstructured":"R.\u00a0McLachlan, K.\u00a0Modin, and O.\u00a0Verdier, Collective symplectic integrators, Nonlinearity 27(6) (2014), 1525\u20131542.","journal-title":"Nonlinearity"},{"key":"9428_CR19","doi-asserted-by":"publisher","first-page":"061301","DOI":"10.1103\/PhysRevE.89.061301","volume":"89","author":"R McLachlan","year":"2014","unstructured":"R.\u00a0McLachlan, K.\u00a0Modin, and O.\u00a0Verdier, Symplectic integrators for spin systems, Phys. Rev. E 89:061301 (2014).","journal-title":"Phys. Rev. E"},{"issue":"307","key":"9428_CR20","doi-asserted-by":"publisher","first-page":"2325","DOI":"10.1090\/mcom\/3153","volume":"86","author":"R McLachlan","year":"2017","unstructured":"R.\u00a0McLachlan, K.\u00a0Modin, and O.\u00a0Verdier, A minimal-variable symplectic integrator on spheres, Math. Comp. 86(307) (2017), 2325\u20132344.","journal-title":"Math. Comp."},{"issue":"2","key":"9428_CR21","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/s10208-013-9163-y","volume":"14","author":"R McLachlan","year":"2014","unstructured":"R.\u00a0McLachlan, K.\u00a0Modin, O.\u00a0Verdier, and M.\u00a0Wilkins, Geometric Generalisations of SHAKE and RATTLE, Found. Comput. Math. (FoCM) 14(2) (2014), 339\u2013370.","journal-title":"Found. Comput. Math. (FoCM)"},{"key":"9428_CR22","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1017\/S0962492902000053","volume":"11","author":"R McLachlan","year":"2002","unstructured":"R.\u00a0McLachlan and G.R.W. Quispel, Splitting methods, Acta Numerica 11 (2002), 341\u2013434.","journal-title":"Acta Numerica"},{"issue":"3","key":"9428_CR23","doi-asserted-by":"publisher","first-page":"335","DOI":"10.3934\/jgm.2017014","volume":"9","author":"K Modin","year":"2017","unstructured":"K.\u00a0Modin, Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry, J. Geom. Mech. 9(3) (2017), 335\u2013390.","journal-title":"J. Geom. Mech."},{"key":"9428_CR24","doi-asserted-by":"crossref","unstructured":"K.\u00a0Modin and M.\u00a0Viviani, A casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics, arXiv:1812.11055 (2018).","DOI":"10.1017\/jfm.2019.944"},{"key":"9428_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9290-3","volume-title":"The N-Vortex Problem, Analytical Techniques","author":"PK Newton","year":"2001","unstructured":"P.\u00a0K. Newton, The N-Vortex Problem, Analytical Techniques , vol. 145, Springer, Berlin, Heidelberg, 2001."},{"key":"9428_CR26","unstructured":"H.\u00a0Poincar\u00e9, Sur une forme nouvelle des \u00e9 quations de la m\u00e9 canique, C.R. Acad. Sci. 132 (1901), 369\u2013371."},{"key":"9428_CR27","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1007\/BF01954907","volume":"28","author":"JM Sanz-Serna","year":"1988","unstructured":"J.\u00a0M. Sanz-Serna, Runge-kutta schemes for hamiltonian systems, BIT Numerical Mathematics 28 (1988), 877\u2013883.","journal-title":"BIT Numerical Mathematics"},{"key":"9428_CR28","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/0167-2789(82)90069-0","volume":"4","author":"W Symes","year":"1982","unstructured":"W.\u00a0Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D 4 (1982), 275\u2013280.","journal-title":"Phys. D"},{"key":"9428_CR29","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1143\/PTPS.45.174","volume":"45","author":"M Toda","year":"1970","unstructured":"M.\u00a0Toda, Waves in nonlinear lattice, Progress of Theoretical Physics Supplement 45 (1970), 174\u2013200.","journal-title":"Progress of Theoretical Physics Supplement"},{"key":"9428_CR30","doi-asserted-by":"publisher","first-page":"511","DOI":"10.3934\/jgm.2013.5.511","volume":"5","author":"C Tomei","year":"2013","unstructured":"C.\u00a0Tomei, The Toda lattice, old and new, J. Geom. Mech. 5 (2013), 511\u2013530.","journal-title":"J. Geom. Mech."},{"key":"9428_CR31","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1137\/1026075","volume":"26","author":"DS Watkins","year":"1984","unstructured":"D.S. Watkins, Isospectral flows, SIAM Rev. 26 (1984), 379\u2013391.","journal-title":"SIAM Rev."},{"key":"9428_CR32","unstructured":"M.\u00a0Webb, Isospectral algorithms, Toeplitz matrices and orthogonal polynomials, Ph.D. thesis, (2017)."}],"updated-by":[{"DOI":"10.1007\/s10208-024-09661-y","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000}}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-019-09428-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-019-09428-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-019-09428-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T12:39:57Z","timestamp":1724762397000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-019-09428-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,5]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["9428"],"URL":"https:\/\/doi.org\/10.1007\/s10208-019-09428-w","relation":{"correction":[{"id-type":"doi","id":"10.1007\/s10208-024-09661-y","asserted-by":"object"}]},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,5]]},"assertion":[{"value":"8 August 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 August 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2024","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s10208-024-09661-y","URL":"https:\/\/doi.org\/10.1007\/s10208-024-09661-y","order":8,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}