{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,21]],"date-time":"2026-06-21T05:38:35Z","timestamp":1782020315295,"version":"3.54.5"},"reference-count":51,"publisher":"American Mathematical Society (AMS)","issue":"356","license":[{"start":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T00:00:00Z","timestamp":1761868800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2208391"],"award-info":[{"award-number":["2208391"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1929284"],"award-info":[{"award-number":["1929284"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The Runge\u2013Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the\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                    approximation in the final stage, even if we drop the\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                    th-order polynomial modes and use the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P Superscript k minus 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">P^{k-1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed by Chen, Sun, and Xing [arXiv preprint, arXiv:2402.15150, 2024]. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability type and accuracy order of the original method.\n                  <\/p>","DOI":"10.1090\/mcom\/4037","type":"journal-article","created":{"date-parts":[[2024,10,9]],"date-time":"2024-10-09T09:38:06Z","timestamp":1728466686000},"page":"2795-2838","source":"Crossref","is-referenced-by-count":3,"title":["Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge\u2013Kutta discontinuous Galerkin method"],"prefix":"10.1090","volume":"94","author":[{"given":"Zheng","family":"Sun","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2024,10,31]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1741","DOI":"10.1137\/21M1435495","article-title":"\ud835\udc3f\u00b2 error estimate to smooth solutions of high order Runge-Kutta discontinuous Galerkin method for scalar nonlinear conservation laws with and without sonic points","volume":"60","author":"Ai, Jingqi","year":"2022","journal-title":"SIAM J. 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