{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T14:39:24Z","timestamp":1704897564618},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2020,8,5]],"date-time":"2020-08-05T00:00:00Z","timestamp":1596585600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem.\nThis method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution.\nIn each time step, on an interval of length <jats:italic>k<\/jats:italic>, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on <jats:inline-formula id=\"j_cmam-2020-0009_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>m<\/m:mi>\n                              <m:mo>\u2265<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0009_eq_0380.png\" \/>\n                        <jats:tex-math>{m\\geq 1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> intervals of length <jats:inline-formula id=\"j_cmam-2020-0009_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mfrac>\n                              <m:mi>k<\/m:mi>\n                              <m:mi>m<\/m:mi>\n                           <\/m:mfrac>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0009_eq_0235.png\" \/>\n                        <jats:tex-math>{\\frac{k}{m}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> for the convection part.\nWith <jats:italic>h<\/jats:italic> the mesh width in space, this results in an error bound of the form <jats:inline-formula id=\"j_cmam-2020-0009_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:msub>\n                                    <m:mi>C<\/m:mi>\n                                    <m:mn>0<\/m:mn>\n                                 <\/m:msub>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>h<\/m:mi>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                              <\/m:mrow>\n                              <m:mo>+<\/m:mo>\n                              <m:mrow>\n                                 <m:msub>\n                                    <m:mi>C<\/m:mi>\n                                    <m:mi>m<\/m:mi>\n                                 <\/m:msub>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>k<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0009_eq_0140.png\" \/>\n                        <jats:tex-math>{C_{0}h^{2}+C_{m}k}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> for appropriately smooth solutions, where <jats:inline-formula id=\"j_cmam-2020-0009_ineq_9996\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>C<\/m:mi>\n                                 <m:mi>m<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>\u2264<\/m:mo>\n                              <m:mrow>\n                                 <m:msup>\n                                    <m:mi>C<\/m:mi>\n                                    <m:mo>\u2032<\/m:mo>\n                                 <\/m:msup>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mfrac>\n                                    <m:msup>\n                                       <m:mi>C<\/m:mi>\n                                       <m:mo>\u2032\u2032<\/m:mo>\n                                    <\/m:msup>\n                                    <m:mi>m<\/m:mi>\n                                 <\/m:mfrac>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0009_eq_0143.png\" \/>\n                        <jats:tex-math>{C_{m}\\leq C^{\\prime}+\\frac{C^{\\prime\\prime}}{m}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nThis work complements the earlier study [V. Thom\u00e9e and A.\u2009S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283\u2013293] based on the second-order Strang splitting.<\/jats:p>","DOI":"10.1515\/cmam-2020-0009","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T07:18:26Z","timestamp":1597735106000},"page":"769-782","source":"Crossref","is-referenced-by-count":3,"title":["A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem"],"prefix":"10.1515","volume":"20","author":[{"given":"Amiya K.","family":"Pani","sequence":"first","affiliation":[{"name":"Department of Mathematics , IIT Bombay , Powai , Mumbai 400076 , India"}]},{"given":"Vidar","family":"Thom\u00e9e","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences , Chalmers University of Technology and University of Gothenburg, SE\u2013412 96 Gothenburg , Sweden"}]},{"given":"A.\u2009S.","family":"Vasudeva Murthy","sequence":"additional","affiliation":[{"name":"TIFR Centre for Applicable Mathematics , Yelahanka New Town , Bangalore 560 065 , India"}]}],"member":"374","published-online":{"date-parts":[[2020,8,5]]},"reference":[{"key":"2023033110450794638_j_cmam-2020-0009_ref_001","doi-asserted-by":"crossref","unstructured":"S.  Descombes,\nConvergence of a splitting method of high order for reaction-diffusion systems,\nMath. Comp. 70 (2001), no. 236, 1481\u20131501.","DOI":"10.1090\/S0025-5718-00-01277-1"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_002","doi-asserted-by":"crossref","unstructured":"E.  Faou, A.  Ostermann and K.  Schratz,\nAnalysis of exponential splitting methods for inhomogeneous parabolic equations,\nIMA J. Numer. Anal. 35 (2015), no. 1, 161\u2013178.","DOI":"10.1093\/imanum\/dru002"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_003","doi-asserted-by":"crossref","unstructured":"E.  Hansen and A.  Ostermann,\nExponential splitting for unbounded operators,\nMath. Comp. 78 (2009), no. 267, 1485\u20131496.","DOI":"10.1090\/S0025-5718-09-02213-3"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_004","doi-asserted-by":"crossref","unstructured":"E.  Hansen, A.  Ostermann and K.  Schratz,\nThe error structure of the Douglas\u2013Rachford splitting method for stiff linear problems,\nJ. Comput. Appl. Math. 303 (2016), 140\u2013145.","DOI":"10.1016\/j.cam.2016.02.037"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_005","doi-asserted-by":"crossref","unstructured":"W.  Hundsdorfer and J.  Verwer,\nNumerical Solution of Time-dependent Advection-diffusion-reaction Equations,\nSpringer Ser. Comput. Math. 33,\nSpringer, Berlin, 2003.","DOI":"10.1007\/978-3-662-09017-6"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_006","doi-asserted-by":"crossref","unstructured":"T.  Jahnke and C.  Lubich,\nError bounds for exponential operator splittings,\nBIT 40 (2000), no. 4, 735\u2013744.","DOI":"10.1023\/A:1022396519656"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_007","doi-asserted-by":"crossref","unstructured":"S.  MacNamara and G.  Strang,\nOperator splitting,\nSplitting Methods in Communication, Imaging, Science, and Engineering,\nSci. Comput.,\nSpringer, Cham (2016), 95\u2013114.","DOI":"10.1007\/978-3-319-41589-5_3"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_008","doi-asserted-by":"crossref","unstructured":"G.  Strang,\nOn the construction and comparison of difference schemes,\nSIAM J. Numer. Anal. 5 (1968), 506\u2013517.","DOI":"10.1137\/0705041"},{"key":"2023033110450794638_j_cmam-2020-0009_ref_009","doi-asserted-by":"crossref","unstructured":"V.  Thom\u00e9e and A. S.  Vasudeva Murthy,\nAn explicit-implicit splitting method for a convection-diffusion problem,\nComput. Methods Appl. Math. 19 (2019), no. 2, 283\u2013293.","DOI":"10.1515\/cmam-2018-0018"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p769.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0009\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0009\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:53:23Z","timestamp":1680267203000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0009\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,5]]},"references-count":9,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2020-0009"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2020-0009","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,5]]}}}