{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:35:33Z","timestamp":1776800133878,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"297","license":[{"start":{"date-parts":[[2016,6,1]],"date-time":"2016-06-01T00:00:00Z","timestamp":1464739200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we study superconvergence properties of the local discontinuous Galerkin method for one-dimensional linear parabolic equations when alternating fluxes are used. We prove, for any polynomial degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , that the numerical fluxes converge at a rate of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 k plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2k+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (or\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 k plus 1 slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2k+1\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) for all mesh nodes and the domain average under some suitable initial discretization. We further prove a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th superconvergence rate for the derivative approximation and a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k+2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th superconvergence rate for the function value approximation at the Radau points. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp.\n                  <\/p>","DOI":"10.1090\/mcom\/2975","type":"journal-article","created":{"date-parts":[[2015,6,1]],"date-time":"2015-06-01T10:13:20Z","timestamp":1433153600000},"page":"63-84","source":"Crossref","is-referenced-by-count":35,"title":["Superconvergence of Local Discontinuous Galerkin methods for one-dimensional linear parabolic equations"],"prefix":"10.1090","volume":"85","author":[{"given":"Waixiang","family":"Cao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,6,1]]},"reference":[{"issue":"25-28","key":"1","doi-asserted-by":"publisher","first-page":"3331","DOI":"10.1016\/j.cma.2005.06.017","article-title":"Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem","volume":"195","author":"Adjerid, Slimane","year":"2006","journal-title":"Comput. 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