{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:38:03Z","timestamp":1776785883758,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,7,6]],"date-time":"2007-07-06T00:00:00Z","timestamp":1183680000000},"content-version":"am","delay-in-days":365,"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 finite difference approximations for the following linear stationary convection-diffusion equations:\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"one half sigma squared left-parenthesis x right-parenthesis u left-parenthesis x right-parenthesis plus b left-parenthesis x right-parenthesis u prime left-parenthesis x right-parenthesis minus u left-parenthesis x right-parenthesis equals minus f left-parenthesis x right-parenthesis comma x element-of double-struck upper R comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mfrac>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mfrac>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{1\\over 2}\\sigma ^2(x)u(x) + b(x)u\u2019(x) - u(x) =-f(x),\\quad x\\in \\mathbb {R},<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c3\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is allowed to be\n                    <italic>degenerate<\/italic>\n                    . We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is\n                    <italic>sharp<\/italic>\n                    . To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01876-x","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1755-1778","source":"Crossref","is-referenced-by-count":6,"title":["Rate of convergence of finite difference approximations for degenerate ordinary differential equations"],"prefix":"10.1090","volume":"75","author":[{"given":"Jianfeng","family":"Zhang","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,7,6]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1051\/m2an:2002002","article-title":"On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations","volume":"36","author":"Barles, Guy","year":"2002","journal-title":"M2AN Math. 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