{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T21:56:48Z","timestamp":1775599008662,"version":"3.50.1"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T00:00:00Z","timestamp":1742515200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,3,21]],"date-time":"2025-03-21T00:00:00Z","timestamp":1742515200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004063","name":"Knut and Alice Wallenberg Foundation","doi-asserted-by":"crossref","award":["KAW 2016.0498"],"award-info":[{"award-number":["KAW 2016.0498"]}],"id":[{"id":"10.13039\/501100004063","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100008757","name":"Swedish Defence Research Agency","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100008757","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider point sources in hyperbolic equations discretized by finite differences. If the source is stationary, appropriate source discretization has been shown to preserve the accuracy of the finite difference method. Moving point sources, however, pose two challenges that do not appear in the stationary case. First, the discrete source must not excite modes that propagate with the source velocity. Second, the discrete source spectrum amplitude must be independent of the source position. We derive a source discretization that meets these requirements and prove design-order convergence of the numerical solution for the one-dimensional advection equation. Numerical experiments indicate design-order convergence also for the acoustic wave equation in two dimensions. The source discretization covers on the order of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\sqrt{N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msqrt>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:msqrt>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> grid points on an <jats:italic>N<\/jats:italic>-point grid and is applicable for source trajectories that do not touch domain boundaries.\n<\/jats:p>","DOI":"10.1007\/s10915-025-02834-z","type":"journal-article","created":{"date-parts":[[2025,3,22]],"date-time":"2025-03-22T15:32:46Z","timestamp":1742657566000},"update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Approximating Moving Point Sources in Hyperbolic Partial Differential Equations"],"prefix":"10.1007","volume":"103","author":[{"given":"Ylva","family":"Ljungberg Rydin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Almquist","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,21]]},"reference":[{"key":"2834_CR1","volume-title":"Quantitative seismology","author":"K Aki","year":"2002","unstructured":"Aki, K., Richards, P.G.: Quantitative seismology, 2nd edn. University Science Books, Sausalito (2002)","edition":"2"},{"key":"2834_CR2","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.jcp.2019.01.039","volume":"386","author":"MJ Bencomo","year":"2019","unstructured":"Bencomo, M.J., Symes, W.W.: Discretization of multipole sources in a finite difference setting for wave propagation problems. J. Comput. Phys. 386, 296\u2013322 (2019). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2019.01.039","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2834_CR3","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1006\/jcph.1994.1057","volume":"111","author":"MH Carpenter","year":"1994","unstructured":"Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220\u2013236 (1994). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1006\/jcph.1994.1057","journal-title":"J. Comput. Phys."},{"key":"2834_CR4","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.compfluid.2014.02.016","volume":"95","author":"DC Del Rey Fern\u00e1ndez","year":"2014","unstructured":"Del Rey Fern\u00e1ndez, D.C., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171\u2013196 (2014). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.compfluid.2014.02.016","journal-title":"Comput. Fluids"},{"issue":"C","key":"2834_CR5","doi-asserted-by":"publisher","first-page":"176","DOI":"10.1016\/j.apnum.2014.06.005","volume":"93","author":"L Dovgilovich","year":"2015","unstructured":"Dovgilovich, L., Sofronov, I.: High-accuracy finite-difference schemes for solving elastodynamic problems in curvilinear coordinates within multiblock approach. Appl. Numer. Math 93(C), 176\u2013194 (2015). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.apnum.2014.06.005","journal-title":"Appl. Numer. Math"},{"issue":"130","key":"2834_CR6","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1090\/S0025-5718-1975-0386296-7","volume":"29","author":"B Gustafsson","year":"1975","unstructured":"Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comp. 29(130), 396\u2013406 (1975). (https:\/\/2.zoppoz.workers.dev:443\/https\/www.jstor.org\/stable\/2156498)","journal-title":"Math. Comp."},{"key":"2834_CR7","doi-asserted-by":"publisher","unstructured":"Gustafsson, B., Kreiss,H.-O., Oliger, J.: Time dependent problems and difference methods. John Wiley & Sons, Inc., (1995). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/9781118548448","DOI":"10.1002\/9781118548448"},{"key":"2834_CR8","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1121\/1.3514426","volume":"129","author":"R Makarewicz","year":"2011","unstructured":"Makarewicz, R.: Is a wind turbine a point source? (L). J. Acoust. Soc. Am. 129, 579 (2011). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1121\/1.3514426","journal-title":"J. Acoust. Soc. Am."},{"key":"2834_CR9","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1016\/j.jcp.2017.06.030","volume":"346","author":"O O\u2019Reilly","year":"2017","unstructured":"O\u2019Reilly, O., Lundquist, T., Dunham, E.M., Nordstr\u00f6m, J.: Energy stable and high-order-accurate finite difference methods on staggered grids. J. Comput. Phys. 346, 572\u2013589 (2017). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2017.06.030","journal-title":"J. Comput. Phys."},{"key":"2834_CR10","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1017\/S0962492902000077","volume":"11","author":"CS Peskin","year":"2002","unstructured":"Peskin, C.S.: The immersed boundary method. Acta Numerica 11, 479\u2013517 (2002). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1017\/S0962492902000077","journal-title":"Acta Numerica"},{"key":"2834_CR11","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1016\/j.jcp.2016.05.060","volume":"321","author":"NA Petersson","year":"2016","unstructured":"Petersson, N.A., O\u2019Reilly, O., Sj\u00f6green, B., Bydlon, S.: Discretizing singular point sources in hyperbolic wave propagation problems. J. Comput. Phys. 321, 532\u2013555 (2016). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2016.05.060","journal-title":"J. Comput. Phys."},{"issue":"5","key":"2834_CR12","doi-asserted-by":"publisher","first-page":"1074","DOI":"10.4208\/cicp.041109.120210a","volume":"8","author":"NA Petersson","year":"2010","unstructured":"Petersson, N.A., Sj\u00f6green, B.: Stable grid refinement and singular source discretization for seismic wave simulations. Commun. Comput. Phys. 8(5), 1074\u20131110 (2010). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.4208\/cicp.041109.120210a","journal-title":"Commun. Comput. Phys."},{"key":"2834_CR13","doi-asserted-by":"publisher","first-page":"1278","DOI":"10.1007\/s10915-018-0751-5","volume":"77","author":"Y Rydin","year":"2018","unstructured":"Rydin, Y., Mattsson, K., Werpers, J.: High-fidelity sound propagation in a varying 3D atmosphere. J. Sci. Comput. 77, 1278\u20131302 (2018). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/s10915-018-0751-5","journal-title":"J. Sci. Comput."},{"key":"2834_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-operators schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17\u201338 (2014). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2014.02.031","journal-title":"J. Comput. Phys."},{"issue":"2","key":"2834_CR15","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1016\/j.jcp.2004.04.011","volume":"200","author":"A-K Tornberg","year":"2004","unstructured":"Tornberg, A.-K., Engquist, B.: Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200(2), 462\u2013488 (2004). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2004.04.011","journal-title":"J. Comput. Phys."},{"key":"2834_CR16","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1002\/(SICI)1098-2426(199907)15:4<503::AID-NUM6>3.0.CO;2-Q","volume":"15","author":"J Wald\u00e9n","year":"1999","unstructured":"Wald\u00e9n, J.: On the approximation of singular source terms in differential equations. Numer. Meth. Part. D. E. 15, 503\u2013520 (1999). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/(SICI)1098-2426(199907)15:4<503::AID-NUM6>3.0.CO;2-Q","journal-title":"Numer. Meth. Part. D. E."},{"issue":"6","key":"2834_CR17","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1016\/j.jcp.2009.11.030","volume":"229","author":"S Zahedi","year":"2010","unstructured":"Zahedi, S., Tornberg, A.-K.: Delta function approximations in level set methods by distance function extension. J. Comput. Phys. 229(6), 2199\u20132219 (2010). https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2009.11.030","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02834-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/article\/10.1007\/s10915-025-02834-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02834-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T19:10:26Z","timestamp":1745694626000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/10.1007\/s10915-025-02834-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,21]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["2834"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/s10915-025-02834-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,21]]},"assertion":[{"value":"20 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interest"}},{"value":"The codes used to generate the results in the numerical experiments are available from the corresponding author upon request.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"40"}}