{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:40:36Z","timestamp":1740123636427,"version":"3.37.3"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,2,27]],"date-time":"2021-02-27T00:00:00Z","timestamp":1614384000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,2,27]],"date-time":"2021-02-27T00:00:00Z","timestamp":1614384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2018-05084 VR"],"award-info":[{"award-number":["2018-05084 VR"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre\u2013Gauss\u2013Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.<\/jats:p>","DOI":"10.1007\/s10915-021-01434-x","type":"journal-article","created":{"date-parts":[[2021,2,27]],"date-time":"2021-02-27T08:02:25Z","timestamp":1614412945000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7972-6183","authenticated-orcid":false,"given":"Jan","family":"Nordstr\u00f6m","sequence":"first","affiliation":[]},{"given":"Andrew R.","family":"Winters","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,27]]},"reference":[{"key":"1434_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral Methods: Fundamentals in Single Domains","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)"},{"issue":"157","key":"1434_CR2","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1090\/S0025-5718-1982-0637287-3","volume":"38","author":"C Canuto","year":"1982","unstructured":"Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38(157), 67\u201386 (1982)","journal-title":"Math. Comput."},{"issue":"5","key":"1434_CR3","doi-asserted-by":"publisher","first-page":"B835","DOI":"10.1137\/130932193","volume":"36","author":"M Carpenter","year":"2014","unstructured":"Carpenter, M., Fisher, T., Nielsen, E., Frankel, S.: Entropy stable spectral collocation schemes for the Navier\u2013Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835\u2013B867 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"1434_CR4","doi-asserted-by":"publisher","first-page":"A1233","DOI":"10.1137\/120890144","volume":"35","author":"G Gassner","year":"2013","unstructured":"Gassner, G.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233\u2013A1253 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3\u20134","key":"1434_CR5","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/s00162-011-0253-7","volume":"27","author":"GJ Gassner","year":"2013","unstructured":"Gassner, G.J., Beck, A.D.: On the accuracy of high-order discretizations for underresolved turbulence simulations. Theoret. Comput. Fluid Dyn. 27(3\u20134), 221\u2013237 (2013)","journal-title":"Theoret. Comput. Fluid Dyn."},{"issue":"263","key":"1434_CR6","doi-asserted-by":"publisher","first-page":"1425","DOI":"10.1090\/S0025-5718-08-02110-8","volume":"77","author":"JS Hesthaven","year":"2008","unstructured":"Hesthaven, J.S., Kirby, R.M.: Filtering in Legendre spectral methods. Math. Comput. 77(263), 1425\u20131452 (2008)","journal-title":"Math. Comput."},{"key":"1434_CR7","doi-asserted-by":"crossref","unstructured":"Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer (2008)","DOI":"10.1007\/978-0-387-72067-8"},{"key":"1434_CR8","unstructured":"Kennedy, C.A., Carpenter, M.H.: Comparison of Several Numerical Methods for Simulation of Compressible Shear Layers, Vol. 3484. NASA, Langley Research Center (1997)"},{"key":"1434_CR9","doi-asserted-by":"crossref","unstructured":"Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Springer, Scientific Computation (2009)","DOI":"10.1007\/978-90-481-2261-5"},{"issue":"4","key":"1434_CR10","doi-asserted-by":"publisher","first-page":"A2076","DOI":"10.1137\/130928650","volume":"36","author":"DA Kopriva","year":"2014","unstructured":"Kopriva, D.A., Gassner, G.J.: An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems. SIAM J. Sci. Comput. 36(4), A2076\u2013A2099 (2014)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1434_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-019-01116-9","volume":"82","author":"T Lundquist","year":"2020","unstructured":"Lundquist, T., Nordstr\u00f6m, J.: Stable and accurate filtering procedures. J. Sci. Comput. 82(1), 1\u201321 (2020)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"1434_CR12","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s10915-016-0303-9","volume":"71","author":"J Nordstr\u00f6m","year":"2016","unstructured":"Nordstr\u00f6m, J.: A roadmap to well posed and stable problems in computational physics. J. Sci. Comput. 71(1), 365\u2013385 (2016)","journal-title":"J. Sci. Comput."},{"key":"1434_CR13","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/j.jcp.2018.03.003","volume":"364","author":"J Nordstr\u00f6m","year":"2018","unstructured":"Nordstr\u00f6m, J., Linders, V.: Well-posed and stable transmission problems. J. Comput. Phys. 364, 95\u2013110 (2018)","journal-title":"J. Comput. Phys."},{"key":"1434_CR14","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.jcp.2014.02.031","volume":"268","author":"M Sv\u00e4rd","year":"2014","unstructured":"Sv\u00e4rd, M., Nordstr\u00f6m, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17\u201338 (2014)","journal-title":"J. Comput. Phys."},{"key":"1434_CR15","doi-asserted-by":"crossref","unstructured":"Tadmor, E., Zhong, W.: Energy-preserving and stable approximations for the two-dimensional shallow water equations. In: Mathematics and Computation, a Contemporary View, pp. 67\u201394. Springer (2008)","DOI":"10.1007\/978-3-540-68850-1_4"},{"key":"1434_CR16","doi-asserted-by":"crossref","unstructured":"Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer (2009)","DOI":"10.1007\/b79761"},{"issue":"2","key":"1434_CR17","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/BF01062118","volume":"6","author":"H Vandeven","year":"1991","unstructured":"Vandeven, H.: Family of spectral filters for discontinuous problems. J. Sci. Comput. 6(2), 159\u2013192 (1991)","journal-title":"J. Sci. Comput."},{"key":"1434_CR18","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/0021-9991(80)90033-9","volume":"35","author":"JH Williamson","year":"1980","unstructured":"Williamson, J.H.: Low-storage Runge\u2013Kutta schemes. J. Comput. Phys. 35, 48\u201356 (1980)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01434-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-021-01434-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01434-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,29]],"date-time":"2021-03-29T13:03:25Z","timestamp":1617023005000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-021-01434-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,27]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["1434"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01434-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2021,2,27]]},"assertion":[{"value":"22 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 February 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with Ethical Standards"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Open access funding provided by Link\u00f6ping University.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Funding"}},{"value":"All data generated or analysed during this study are included in this published article.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Availability of data and material"}},{"value":"The code used to generate the results in this work is available upon request with Andrew Winters (andrew.ross.winters@liu.se).","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"17"}}