{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:58:09Z","timestamp":1776848289120,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"305","license":[{"start":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T00:00:00Z","timestamp":1497916800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11471306"],"award-info":[{"award-number":["11471306"]}],"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":["11371342"],"award-info":[{"award-number":["11371342"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {L}^{2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    stability and error estimates are proven. More precisely, we prove the sub-optimal (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus one half\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k+\\frac {1}{2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) convergence for monotone fluxes and optimal (\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                    ) convergence for an upwind flux when a piecewise\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P Superscript k\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">P^k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover, we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.\n                  <\/p>","DOI":"10.1090\/mcom\/3126","type":"journal-article","created":{"date-parts":[[2015,10,21]],"date-time":"2015-10-21T10:08:39Z","timestamp":1445422119000},"page":"1203-1232","source":"Crossref","is-referenced-by-count":44,"title":["Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension"],"prefix":"10.1090","volume":"86","author":[{"given":"Christian","family":"Klingenberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gero","family":"Schn\u00fccke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yinhua","family":"Xia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,6,20]]},"reference":[{"key":"1","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The finite element method for elliptic problems","volume":"40","author":"Ciarlet, Philippe G.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0898715148"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/978-3-662-03882-6_2","article-title":"Discontinuous Galerkin methods for convection-dominated problems","author":"Cockburn, Bernardo","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/3540658939"},{"issue":"190","key":"3","doi-asserted-by":"publisher","first-page":"545","DOI":"10.2307\/2008501","article-title":"The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. 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