{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:42:26Z","timestamp":1767919346978,"version":"3.49.0"},"reference-count":24,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801396"],"award-info":[{"award-number":["11801396"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11802193"],"award-info":[{"award-number":["11802193"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20170374"],"award-info":[{"award-number":["BK20170374"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Local discontinuous Galerkin method is considered\nfor time-dependent singularly perturbed semilinear problems with boundary layer.\nThe method is equipped with a general numerical flux\nincluding two kinds of independent parameters.\nBy virtue of the weighted estimates and suitably designed global projections,\nwe establish optimal <jats:inline-formula id=\"j_cmam-2019-0185_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo stretchy=\"false\">(<\/m:mo>\n                              <m:mrow>\n                                 <m:mi>k<\/m:mi>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                              <m:mo stretchy=\"false\">)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2019-0185_eq_0386.png\"\/>\n                        <jats:tex-math>{(k+1)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-th error estimate in a local region\nat a distance of <jats:inline-formula id=\"j_cmam-2019-0185_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi mathvariant=\"script\">\ud835\udcaa<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>h<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mi>log<\/m:mi>\n                                       <m:mo>\u2061<\/m:mo>\n                                       <m:mrow>\n                                          <m:mo stretchy=\"false\">(<\/m:mo>\n                                          <m:mfrac>\n                                             <m:mn>1<\/m:mn>\n                                             <m:mi>h<\/m:mi>\n                                          <\/m:mfrac>\n                                          <m:mo stretchy=\"false\">)<\/m:mo>\n                                       <\/m:mrow>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\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-2019-0185_eq_0550.png\"\/>\n                        <jats:tex-math>{\\mathcal{O}(h\\log(\\frac{1}{h}))}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> from domain boundary.\nHere <jats:italic>k<\/jats:italic> is the degree of piecewise polynomials in the discontinuous finite element space\nand <jats:italic>h<\/jats:italic> is the maximum mesh size.\nBoth semi-discrete LDG method\nand fully discrete LDG method with a third-order explicit Runge\u2013Kutta\ntime-marching are considered.\nNumerical experiments support our theoretical results.<\/jats:p>","DOI":"10.1515\/cmam-2019-0185","type":"journal-article","created":{"date-parts":[[2020,9,7]],"date-time":"2020-09-07T13:33:15Z","timestamp":1599485595000},"page":"31-52","source":"Crossref","is-referenced-by-count":9,"title":["Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems"],"prefix":"10.1515","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9500-9398","authenticated-orcid":false,"given":"Yao","family":"Cheng","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences , Suzhou University of Science and Technology , Suzhou , P. R. China"}]},{"given":"Chuanjing","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , Suzhou University of Science and Technology , Suzhou , P. R. China"}]},{"given":"Yanjie","family":"Mei","sequence":"additional","affiliation":[{"name":"International Education School , Suzhou University of Science and Technology , Suzhou , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2020,9,1]]},"reference":[{"key":"2023033111222133070_j_cmam-2019-0185_ref_001","doi-asserted-by":"crossref","unstructured":"J. L.  Bona, H.  Chen, O.  Karakashian and Y.  Xing,\nConservative, discontinuous Galerkin-methods for the generalized Korteweg\u2013de Vries equation,\nMath. Comp. 82 (2013), no. 283, 1401\u20131432.","DOI":"10.1090\/S0025-5718-2013-02661-0"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_002","doi-asserted-by":"crossref","unstructured":"P.  Castillo, B.  Cockburn, I.  Perugia and D.  Sch\u00f6tzau,\nAn a priori error analysis of the local discontinuous Galerkin method for elliptic problems,\nSIAM J. Numer. Anal. 38 (2000), no. 5, 1676\u20131706.","DOI":"10.1137\/S0036142900371003"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_003","doi-asserted-by":"crossref","unstructured":"P.  Castillo, B.  Cockburn, D.  Sch\u00f6tzau and C.  Schwab,\nOptimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems,\nMath. Comp. 71 (2002), no. 238, 455\u2013478.","DOI":"10.1090\/S0025-5718-01-01317-5"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_004","doi-asserted-by":"crossref","unstructured":"Y.  Cheng,\nOptimal error estimate of the local discontinuous Galerkin methods based on the generalized alternating numerical fluxes for nonlinear convection-diffusion equations,\nNumer. Algorithms 80 (2019), no. 4, 1329\u20131359.","DOI":"10.1007\/s11075-018-0529-8"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_005","doi-asserted-by":"crossref","unstructured":"Y.  Cheng, X.  Meng and Q.  Zhang,\nApplication of generalized Gauss\u2013Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations,\nMath. Comp. 86 (2017), no. 305, 1233\u20131267.","DOI":"10.1090\/mcom\/3141"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_006","doi-asserted-by":"crossref","unstructured":"Y.  Cheng and Q.  Zhang,\nLocal analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem,\nJ. Sci. Comput. 72 (2017), no. 2, 792\u2013819.","DOI":"10.1007\/s10915-017-0378-y"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_007","unstructured":"Y.  Cheng, Q.  Zhang and H.  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Guzm\u00e1n,\nLocal analysis of discontinuous Galerkin methods applied to singularly perturbed problems,\nJ. Numer. Math. 14 (2006), no. 1, 41\u201356.","DOI":"10.1163\/156939506776382157"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_011","doi-asserted-by":"crossref","unstructured":"N.  Heuer and M.  Karkulik,\nA robust DPG method for singularly perturbed reaction-diffusion problems,\nSIAM J. Numer. Anal. 55 (2017), no. 3, 1218\u20131242.","DOI":"10.1137\/15M1041304"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_012","doi-asserted-by":"crossref","unstructured":"C.  Johnson, U.  N\u00e4vert and J.  Pitk\u00e4ranta,\nFinite element methods for linear hyperbolic problems,\nComput. Methods Appl. Mech. Engrg. 45 (1984), no. 1\u20133, 285\u2013312.","DOI":"10.1016\/0045-7825(84)90158-0"},{"key":"2023033111222133070_j_cmam-2019-0185_ref_013","doi-asserted-by":"crossref","unstructured":"N.  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