{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T23:30:06Z","timestamp":1776209406868,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["PRIN 2017"],"award-info":[{"award-number":["PRIN 2017"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \u201cFrancesco Severi\u201d","doi-asserted-by":"publisher","award":["GNCS 2019"],"award-info":[{"award-number":["GNCS 2019"]}],"id":[{"id":"10.13039\/100009112","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":[[2021,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.<\/jats:p>","DOI":"10.1007\/s10915-021-01518-8","type":"journal-article","created":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T16:03:04Z","timestamp":1622908984000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems"],"prefix":"10.1007","volume":"88","author":[{"given":"Luca","family":"Bonaventura","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elisa","family":"Calzola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elisabetta","family":"Carlini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2270-4629","authenticated-orcid":false,"given":"Roberto","family":"Ferretti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,5]]},"reference":[{"key":"1518_CR1","doi-asserted-by":"crossref","unstructured":"Bokanowski, O., Simarmata, G.: Semi-Lagrangian discontinuous Galerkin schemes for some first-and second-order partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 50:1699\u20131730, 2016","DOI":"10.1051\/m2an\/2016004"},{"key":"1518_CR2","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1006\/jcph.1999.6414","volume":"158","author":"L Bonaventura","year":"2000","unstructured":"Bonaventura, L.: A semi-implicit, semi-Lagrangian scheme using the height coordinate for a nonhydrostatic and fully elastic model of atmospheric flows. J. Computat. Phys. 158, 186\u2013213 (2000)","journal-title":"J. Computat. Phys."},{"key":"1518_CR3","doi-asserted-by":"crossref","unstructured":"Bonaventura, L., Calzola, E., Carlini, E., Ferretti, R.: A fully semi-Lagrangian method for the Navier-Stokes equations in primitive variables. In Proceedings of the: Finite Elements in Flows Conference, p. 2020. Springer Verlag, Rome (2017)","DOI":"10.1007\/978-3-030-30705-9_6"},{"key":"1518_CR4","doi-asserted-by":"publisher","first-page":"A2458","DOI":"10.1137\/140969713","volume":"36","author":"L Bonaventura","year":"2014","unstructured":"Bonaventura, L., Ferretti, R.: Semi-Lagrangian methods for parabolic problems in divergence form. SIAM J. Sci. Comput. 36, A2458\u2013A2477 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"1518_CR5","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.amc.2017.11.030","volume":"323","author":"L Bonaventura","year":"2018","unstructured":"Bonaventura, L., Ferretti, R., Rocchi, L.: A fully semi-Lagrangian discretization for the 2D Navier-Stokes equations in the vorticity-streamfunction formulation. Appl. Mathemat. Comput. 323, 132\u2013144 (2018)","journal-title":"Appl. Mathemat. Comput."},{"key":"1518_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23831-4","volume-title":"Earth System Modelling 2: Algorithms","author":"L Bonaventura","year":"2012","unstructured":"Bonaventura, L., Redler, R., Budich, R.: Earth System Modelling 2: Algorithms. Code Infrastructure and Optimisation. Springer Verlag, New York (2012)"},{"key":"1518_CR7","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1007\/s10915-016-0267-9","volume":"70","author":"L Bonaventura","year":"2017","unstructured":"Bonaventura, L., Della Rocca, A.: Unconditionally strong stability preserving extensions of the TR-BDF2 method. J. Sci. Comput. 70, 859\u2013895 (2017)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"1518_CR8","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1093\/amrx\/abv001","volume":"2015","author":"C-E Br\u00e9hier","year":"2015","unstructured":"Br\u00e9hier, C.-E., Faou, E.: Analysis of the Monte-Carlo error in a hybrid semi-Lagrangian scheme. Appl. Mathemat. Res. Expre. 2015(2), 167\u2013203 (2015)","journal-title":"Appl. Mathemat. Res. Expre."},{"key":"1518_CR9","unstructured":"Cacace, S., Ferretti, R.: Efficient implementation of characteristic-based schemes on unstructured triangular grids. Preprint arXiv:2104.09189, submitted"},{"key":"1518_CR10","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1051\/m2an\/1995290100971","volume":"29","author":"M Camilli","year":"1995","unstructured":"Camilli, M., Falcone, M.: An approximation scheme for the optimal control of diffusion processes. Mathemat. Modell. Numer. Anal. 29, 97\u2013122 (1995)","journal-title":"Mathemat. Modell. Numer. Anal."},{"key":"1518_CR11","doi-asserted-by":"crossref","unstructured":"Canuto, C., Hussaini, M.Y., Quarteron, A., Zang, T.A.: Spectral methods: Fundamentals in single domains. Springer Science & Business Media, 2006","DOI":"10.1007\/978-3-540-30726-6"},{"key":"1518_CR12","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/0021-9991(90)90091-E","volume":"86","author":"V Casulli","year":"1990","unstructured":"Casulli, V.: Semi-implicit finite difference methods for the two dimensional shallow water equations. J. Computat. Phys. 86, 56\u201374 (1990)","journal-title":"J. Computat. Phys."},{"issue":"4","key":"1518_CR13","first-page":"99","volume":"27","author":"V Casulli","year":"1994","unstructured":"Casulli, V., Cattani, E.: Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow. Comput. Mathemat. Appl. 27(4), 99\u2013112 (1994)","journal-title":"Comput. Mathemat. Appl."},{"key":"1518_CR14","doi-asserted-by":"publisher","first-page":"2003","DOI":"10.1175\/1520-0493(1988)116<2003:ATTLSL>2.0.CO;2","volume":"116","author":"J Cot\u00e9","year":"1988","unstructured":"Cot\u00e9, J., Staniforth, A.: A two time level semi-Lagrangian semi-implicit scheme for spectral models. Monthly Weather Rev. 116, 2003\u20132012 (1988)","journal-title":"Monthly Weather Rev."},{"key":"1518_CR15","doi-asserted-by":"publisher","first-page":"1433","DOI":"10.1090\/S0025-5718-2012-02632-9","volume":"82","author":"K Debrabant","year":"2013","unstructured":"Debrabant, K., Jakobsen, E.: Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Mathemat. Computat. 82, 1433\u20131462 (2013)","journal-title":"Mathemat. Computat."},{"key":"1518_CR16","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1016\/j.procs.2012.04.106","volume":"9","author":"C Erath","year":"2012","unstructured":"Erath, C., Lauritzen, P.H., Garcia, J.H., Tufo, H.M.: Integrating a scalable and efficient semi-Lagrangian multi-tracer transport scheme in HOMME. Procedia Comput. Sci. 9, 994\u20131003 (2012)","journal-title":"Procedia Comput. Sci."},{"key":"1518_CR17","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1515\/caim-2016-0023","volume":"7","author":"C Erath","year":"2016","unstructured":"Erath, C., Taylor, M.A., Nair, R.D.: Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core. Commun. Appl. Indus. Mathemat. 7, 74\u201398 (2016)","journal-title":"Commun. Appl. Indus. Mathemat."},{"key":"1518_CR18","doi-asserted-by":"crossref","unstructured":"Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM (2013)","DOI":"10.1137\/1.9781611973051"},{"key":"1518_CR19","first-page":"679","volume":"7","author":"X Feng","year":"2013","unstructured":"Feng, X., Song, H., Tang, T., Yang, J.: Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inv. Prob. Imag. 7, 679\u2013695 (2013)","journal-title":"Inv. Prob. Imag."},{"key":"1518_CR20","doi-asserted-by":"publisher","first-page":"445","DOI":"10.4208\/cicp.070709.011209a","volume":"8","author":"R Ferretti","year":"2010","unstructured":"Ferretti, R.: A technique for high-order treatment of diffusion terms in semi-Lagrangian schemes. Commun. Computat. Phys. 8, 445\u2013470 (2010)","journal-title":"Commun. Computat. Phys."},{"key":"1518_CR21","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s00211-012-0505-5","volume":"124","author":"R Ferretti","year":"2013","unstructured":"Ferretti, R.: On the relationship between semi-Lagrangian and Lagrange-Galerkin schemes. Numerische Mathematik 124, 31\u201356 (2013)","journal-title":"Numerische Mathematik"},{"key":"1518_CR22","unstructured":"Ferretti, R., Mehrenberger, M.: Stability of semi-Lagrangian schemes of arbitrary odd degree under constant and variable advection speed. Mathematics of Computation, to appear"},{"key":"1518_CR23","doi-asserted-by":"crossref","unstructured":"Ferretti, R., Perrone, G.: On the stability of semi-Lagrangian advection schemes under finite element interpolations. In Applied And Industrial Mathematics In Italy II, pages 339\u2013350. World Scientific, 2007","DOI":"10.1142\/9789812709394_0030"},{"key":"1518_CR24","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.jcp.2014.01.033","volume":"264","author":"F Garcia","year":"2014","unstructured":"Garcia, F., Bonaventura, L., Net, M., S\u00e1nchez, J.: Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells. J. Computat. Phys. 264, 41\u201354 (2014)","journal-title":"J. Computat. Phys."},{"key":"1518_CR25","doi-asserted-by":"crossref","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica, pages 209\u2013286, 2010","DOI":"10.1017\/S0962492910000048"},{"key":"1518_CR26","doi-asserted-by":"crossref","unstructured":"Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Applications of Mathematics (New York), vSpringer-Verlag, Berlin (1992)","DOI":"10.1007\/978-3-662-12616-5"},{"key":"1518_CR27","unstructured":"Kushner, H., Dupuis, P.G.: Numerical methods for stochastic control problems in continuous time, volume\u00a024. Springer Science & Business Media, 2013"},{"key":"1518_CR28","volume-title":"Numerical methods for ordinary differential systems","author":"JD Lambert","year":"1991","unstructured":"Lambert, J.D.: Numerical methods for ordinary differential systems. Wiley, Springer (1991)"},{"key":"1518_CR29","doi-asserted-by":"crossref","unstructured":"Leveque, R.J. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge","DOI":"10.1017\/CBO9780511791253"},{"key":"1518_CR30","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1002\/num.10020","volume":"18","author":"GN Milstein","year":"2002","unstructured":"Milstein, G.N.: The probability approach to numerical solution of nonlinear parabolic equations. Numer. Methods Partial Diff. Equat. 18, 490\u2013522 (2002)","journal-title":"Numer. Methods Partial Diff. Equat."},{"key":"1518_CR31","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1090\/S0025-5718-99-01134-5","volume":"69","author":"GN Milstein","year":"2000","unstructured":"Milstein, G.N., Tretyakov, M.V.: Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations. Mathemat. Computat. 69, 237\u2013567 (2000)","journal-title":"Mathemat. Computat."},{"key":"1518_CR32","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1093\/imanum\/21.4.887","volume":"21","author":"GN Milstein","year":"2001","unstructured":"Milstein, G.N., Tretyakov, M.V.: Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach. IMA J. Numer. Anal. 21, 887\u2013917 (2001)","journal-title":"IMA J. Numer. Anal."},{"key":"1518_CR33","unstructured":"Milstein, G.N., Tretyakov, M.V.: Stochastic numerics for mathematical physics. Springer Science & Business Media, 2013"},{"key":"1518_CR34","doi-asserted-by":"crossref","unstructured":"Picarelli, A., Reisinger, C.: Probabilistic error analysis for some approximation schemes to optimal control problems. Systems & Control Letters 137, (2020)","DOI":"10.1016\/j.sysconle.2019.104619"},{"key":"1518_CR35","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1080\/07055900.1984.9649200","volume":"22","author":"J Pudykiewicz","year":"1984","unstructured":"Pudykiewicz, J., Staniforth, A.: Some properties and comparative performance of the semi-Lagrangian method of Robert in the solution of the advection diffusion equation. Atmosphere-Ocean 22, 283\u2013304 (1984)","journal-title":"Atmosphere-Ocean"},{"key":"1518_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical approximation of partial differential equations","author":"A Quarteroni","year":"1994","unstructured":"Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer Verlag, Newyork (1994)"},{"key":"1518_CR37","doi-asserted-by":"publisher","first-page":"319","DOI":"10.2151\/jmsj1965.60.1_319","volume":"60","author":"A Robert","year":"1982","unstructured":"Robert, A.: A semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations. J. Meteorol. Soc. Japan 60, 319\u2013325 (1982)","journal-title":"J. Meteorol. Soc. Japan"},{"key":"1518_CR38","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/j.jcp.2004.10.013","volume":"204","author":"G Rosatti","year":"2005","unstructured":"Rosatti, G., Cesari, D., Bonaventura, L.: Semi-implicit, semi-Lagrangian environmental modelling on Cartesian grids with cut cells. J. Computat. Phys. 204, 353\u2013377 (2005)","journal-title":"J. Computat. Phys."},{"key":"1518_CR39","doi-asserted-by":"publisher","first-page":"2206","DOI":"10.1175\/1520-0493(1991)119<2206:SLISFA>2.0.CO;2","volume":"119","author":"A Staniforth","year":"1991","unstructured":"Staniforth, A., Cot\u00e9, J.: Semi-Lagrangian integration schemes for atmospheric models-a review. Monthly Weather Rev. 119, 2206\u20132223 (1991)","journal-title":"Monthly Weather Rev."},{"key":"1518_CR40","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1002\/qj.49712757107","volume":"127","author":"C Temperton","year":"2001","unstructured":"Temperton, C., Hortal, M., Simmons, A.: A two-time-level semi-Lagrangian global spectral model. Quart. J. Royal Meteorol. Soc. 127, 111\u2013127 (2001)","journal-title":"Quart. J. Royal Meteorol. Soc."},{"key":"1518_CR41","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1002\/qj.49711347714","volume":"113","author":"C Temperton","year":"1987","unstructured":"Temperton, C., Staniforth, A.: An efficient two-time-level semi-Lagrangian semi-implicit integration scheme. Quart .J. Royal Meteorol. Soc. 113, 1025\u20131039 (1987)","journal-title":"Quart .J. Royal Meteorol. Soc."},{"key":"1518_CR42","doi-asserted-by":"publisher","first-page":"2582","DOI":"10.1002\/qj.2544","volume":"141","author":"G Tumolo","year":"2015","unstructured":"Tumolo, G., Bonaventura, L.: A semi-implicit, semi-Lagrangian, DG framework for adaptive numerical weather prediction. Quart. J. Royal Meteorol. Soc. 141, 2582\u20132601 (2015)","journal-title":"Quart. J. Royal Meteorol. Soc."},{"key":"1518_CR43","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.jcp.2012.06.006","volume":"232","author":"G Tumolo","year":"2013","unstructured":"Tumolo, G., Bonaventura, L., Restelli, M.: A semi-implicit, semi-Lagrangian, $$p-$$adaptive discontinuous Galerkin method for the shallow water equations. J. Computat. Phys. 232, 46\u201367 (2013)","journal-title":"J. Computat. Phys."},{"key":"1518_CR44","unstructured":"Wedi, N.P., Bauer, P., Diamantakis, M., Hamrud, M., Malardel, S., Mogensen, K., Mozdzynski, G., Smolarkiewicz, P.K.: The modelling infrastructure of the Integrated Forecasting System: Recent advances and future challenges. Technical Report 760, ECMWF, 2015"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01518-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01518-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01518-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,23]],"date-time":"2021-06-23T07:32:20Z","timestamp":1624433540000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01518-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,5]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["1518"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01518-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,5]]},"assertion":[{"value":"8 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Data and codes produced with this work will be made available upon reasonable request.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Data Availability"}}],"article-number":"23"}}