{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T00:12:31Z","timestamp":1760573551766,"version":"build-2065373602"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"NTNU Norwegian University of Science and Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We prove that <jats:inline-formula>\n              <jats:tex-math>$$\\alpha$$<\/jats:tex-math>\n            <\/jats:inline-formula>-dissipative solutions to the Cauchy problem of the Hunter\u2013Saxton equation, where <jats:inline-formula>\n              <jats:tex-math>$$\\alpha \\in W^{1, \\infty }(\\mathbb {R}, [0, 1))$$<\/jats:tex-math>\n            <\/jats:inline-formula>, can be computed numerically with order <jats:inline-formula>\n              <jats:tex-math>$$\\mathcal {O}({\\varDelta x}^{1\/8}+{\\varDelta x}^{\\beta \/4})$$<\/jats:tex-math>\n            <\/jats:inline-formula> in <jats:inline-formula>\n              <jats:tex-math>$$L^{\\infty }(\\mathbb {R})$$<\/jats:tex-math>\n            <\/jats:inline-formula>, provided there exist constants <jats:inline-formula>\n              <jats:tex-math>$$C&gt; 0$$<\/jats:tex-math>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:tex-math>$$\\beta \\in (0, 1]$$<\/jats:tex-math>\n            <\/jats:inline-formula> such that the initial spatial derivative <jats:inline-formula>\n              <jats:tex-math>$${\\bar{u}}_{x}$$<\/jats:tex-math>\n            <\/jats:inline-formula> satisfies <jats:inline-formula>\n              <jats:tex-math>$$\\Vert {\\bar{u}}_x(\\cdot + h) - {\\bar{u}}_x(\\cdot )\\Vert _2 \\le Ch^{\\beta }$$<\/jats:tex-math>\n            <\/jats:inline-formula> for all <jats:inline-formula>\n              <jats:tex-math>$$h \\in (0, 2]$$<\/jats:tex-math>\n            <\/jats:inline-formula>. The derived convergence rate is exemplified by a number of numerical experiments.<\/jats:p>","DOI":"10.1007\/s00211-025-01482-7","type":"journal-article","created":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T07:42:34Z","timestamp":1758872554000},"page":"1643-1694","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rate of convergence for numerical $$\\alpha$$-dissipative solutions of the Hunter\u2013Saxton equation"],"prefix":"10.1007","volume":"157","author":[{"given":"Thomas","family":"Christiansen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Katrin","family":"Grunert","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,26]]},"reference":[{"issue":"3","key":"1482_CR1","doi-asserted-by":"publisher","first-page":"996","DOI":"10.1137\/050623036","volume":"37","author":"A Bressan","year":"2005","unstructured":"Bressan, A., Constantin, A.: Global solutions of the Hunter-Saxton equation. SIAM J. Math. Anal. 37(3), 996\u20131026 (2005)","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"1482_CR2","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.matpur.2010.02.005","volume":"94","author":"A Bressan","year":"2010","unstructured":"Bressan, A., Holden, H., Raynaud, X.: Lipschitz metric for the Hunter-Saxton equation. J. Math. Pures Appl. 94(1), 68\u201392 (2010)","journal-title":"J. Math. Pures Appl."},{"key":"1482_CR3","unstructured":"Christiansen, T.: On the convergence rate of a numerical method for the Hunter\u2013Saxton equation, arXiv:2409.18903"},{"key":"1482_CR4","unstructured":"Christiansen, T., Grunert, K.: A numerical view on \u03b1-dissipative solutions of the Hunter\u2013Saxton equation, arXiv:2404.11174"},{"issue":"1","key":"1482_CR5","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/s10915-024-02479-4","volume":"99","author":"T Christiansen","year":"2024","unstructured":"Christiansen, T., Grunert, K., Nordli, A., Solem, S.: A convergent numerical algorithm for \u03b1-dissipative solutions of the Hunter-Saxton equation. J. Sci. Comput. 99(1), 14 (2024)","journal-title":"J. Sci. Comput."},{"key":"1482_CR6","first-page":"51","volume-title":"Nonlinear approximation","author":"RA DeVore","year":"1998","unstructured":"DeVore, R.A.: Nonlinear approximation, vol. 7, pp. 51\u2013150. Acta Numerica. Cambridge Univ. Press, Cambridge (1998)"},{"key":"1482_CR7","unstructured":"DeVore, R.\u00a0A., Lorentz, G.\u00a0G.: Constructive approximation. Vol. 303. A series of Comprehensive Studies in Mathematics. Springer, Berlin, x+449 (1993)"},{"issue":"1","key":"1482_CR8","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1142\/S0219891611002366","volume":"8","author":"CM Dafermos","year":"2011","unstructured":"Dafermos, C.M.: Generalized characteristics and the Hunter-Saxton equation. J. Hyperbolic Differ. Equ. 8(1), 159\u2013168 (2011)","journal-title":"J. Hyperbolic Differ. Equ."},{"key":"1482_CR9","unstructured":"Folland, G.\u00a0B.: Real analysis. Modern techniques and their applications. Pure and Applied Mathematics, second edition. A Wiley-Interscience Publication, New York, (1999)"},{"issue":"2","key":"1482_CR10","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s40687-022-00314-6","volume":"9","author":"K Grunert","year":"2022","unstructured":"Grunert, K., Holden, H.: Uniqueness of conservative solutions for the Hunter-Saxton equation. Res. Math. Sci. 9(2), 19 (2022)","journal-title":"Res. Math. Sci."},{"key":"1482_CR11","doi-asserted-by":"crossref","unstructured":"Grunert, K., Holden, H., Raynaud, X.: A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system. Forum Math. Sigma, 3:Paper No. e1, 73 (2015)","DOI":"10.1017\/fms.2014.29"},{"key":"1482_CR12","doi-asserted-by":"publisher","first-page":"4209","DOI":"10.3934\/dcds.2012.32.4209","volume":"32","author":"K Grunert","year":"2012","unstructured":"Grunert, K., Holden, H., Raynaud, X.: Global conservative solutions to the Camassa-Holm equation for initial data with nonvanishing asymptotics. Discrete Contin. Dyn. Syst. 32, 4209\u20134227 (2012)","journal-title":"Discrete Contin. Dyn. Syst."},{"issue":"3","key":"1482_CR13","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1142\/S0219891618500182","volume":"15","author":"K Grunert","year":"2018","unstructured":"Grunert, K., Nordli, A.: Existence and Lipschitz stability for \u03b1-dissipative solutions of the two-component Hunter-Saxton system. J. Hyperbolic Differ. Equ. 15(3), 559\u2013597 (2018)","journal-title":"J. Hyperbolic Differ. Equ."},{"issue":"2","key":"1482_CR14","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s10543-020-00835-y","volume":"61","author":"K Grunert","year":"2021","unstructured":"Grunert, K., Nordli, A., Solem, S.: Numerical conservative solutions of the Hunter-Saxton equation. BIT 61(2), 441\u2013471 (2021)","journal-title":"BIT"},{"issue":"4","key":"1482_CR15","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1007\/s42985-024-00293-z","volume":"5","author":"K Grunert","year":"2024","unstructured":"Grunert, K., Tandy, M.: A Lipschitz metric for \u03b1-dissipative solutions to the Hunter-Saxton equation. Partial Differ. Equ. Appl. 5(4), 24 (2024)","journal-title":"Partial Differ. Equ. Appl."},{"issue":"258","key":"1482_CR16","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0025-5718-07-01919-9","volume":"76","author":"H Holden","year":"2007","unstructured":"Holden, H., Karlsen, K.H., Risebro, N.H.: Convergent difference schemes for the Hunter-Saxton equation. Math. Comp. 76(258), 699\u2013744 (2007)","journal-title":"Math. Comp."},{"issue":"3","key":"1482_CR17","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1007\/s00205-011-0403-5","volume":"201","author":"H Holden","year":"2011","unstructured":"Holden, H., Raynaud, X.: Global semigroup of conservative solutions of the nonlinear variational wave equation Arch. Ration. Mech. Anal. 201(3), 871\u2013964 (2011)","journal-title":"Ration. Mech. Anal."},{"issue":"6","key":"1482_CR18","doi-asserted-by":"publisher","first-page":"1498","DOI":"10.1137\/0151075","volume":"51","author":"JK Hunter","year":"1991","unstructured":"Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51(6), 1498\u20131521 (1991)","journal-title":"SIAM J. Appl. Math."},{"key":"1482_CR19","doi-asserted-by":"crossref","unstructured":"Leoni, G.: A first course in Sobolev spaces. Second edition Vol. 181. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, xxii+734 (2017)","DOI":"10.1090\/gsm\/181"},{"key":"1482_CR20","volume-title":"A course in real analysis","author":"JN McDonald","year":"2013","unstructured":"McDonald, J.N., Weiss, N.A.: A course in real analysis. Academic Press Inc, San Diego, CA, second edition (2013)"},{"issue":"2","key":"1482_CR21","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s13160-017-0252-1","volume":"34","author":"Y Miyatake","year":"2017","unstructured":"Miyatake, Y., Cohen, D., Furihata, D., Matsuo, T.: Geometric numerical integrators for Hunter-Saxton-like equations. Jpn. J. Ind. Appl. Math. 34(2), 441\u2013472 (2017)","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"1482_CR22","unstructured":"Nordli, A.: On the two-component Hunter\u2013Saxton system. PhD thesis, Norwegian University of Science and Technology (NTNU), ISBN:978-82-326-2365-5 (2017)"},{"key":"1482_CR23","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1080\/00029890.1969.12000249","volume":"76","author":"J Serrin","year":"1969","unstructured":"Serrin, J., Varberg, D.E.: A general chain rule for derivatives and the change of variables formula for the Lebesgue integral. Amer. Math. Monthly 76, 514\u2013520 (1969)","journal-title":"Amer. Math. Monthly"},{"issue":"2","key":"1482_CR24","doi-asserted-by":"publisher","first-page":"1249","DOI":"10.1137\/080714105","volume":"31","author":"X Yan","year":"2009","unstructured":"Yan, X., Shu, C.-W.: Local discontinuous Galerkin method for the Hunter-Saxton equation and its zero-viscosity and zero-dispersion limits. SIAM J. Sci. Comput. 31(2), 1249\u20131268 (2009)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"1482_CR25","doi-asserted-by":"publisher","first-page":"606","DOI":"10.4208\/jcm.1003-m0003","volume":"28","author":"X Yan","year":"2010","unstructured":"Yan, X., Shu, C.-W.: Dissipative numerical methods for the Hunter-Saxton equation. J. Comput. Math. 28(5), 606\u2013620 (2010)","journal-title":"J. Comput. Math."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01482-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01482-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01482-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T04:03:59Z","timestamp":1760501039000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01482-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,26]]},"references-count":25,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1482"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01482-7","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2025,9,26]]},"assertion":[{"value":"24 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}