{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:49:45Z","timestamp":1776721785759,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 times 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2 \\times 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    systems appear, for example, in the context of resonant waves in gas dynamics equations. This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L normal i normal p Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {Lip}^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -stability) together with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W Superscript negative 1 comma 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">W^{-1,1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence\n                    <italic>rate<\/italic>\n                    estimates, both global (quantified in terms of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper W Superscript s comma p\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>W<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">W^{s,p}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norms) and local. We focus our attention on the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L normal i normal p Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {Lip}^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L normal i normal p Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {Lip}^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L normal i normal p Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">p<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {Lip}^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -stability.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00716-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"575-586","source":"Crossref","is-referenced-by-count":3,"title":["Convergence rate of approximate solutions to weakly coupled nonlinear systems"],"prefix":"10.1090","volume":"65","author":[{"given":"Haim","family":"Nessyahu","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/978-1-4613-9136-4_7","article-title":"Interacting weakly nonlinear hyperbolic and dispersive waves","author":"Hunter, John K.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975918"},{"key":"2","series-title":"Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11","volume-title":"Hyperbolic systems of conservation laws and the mathematical theory of shock waves","author":"Lax, Peter D.","year":"1973"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1002\/sapm1988793205","article-title":"A canonical system of integrodifferential equations arising in resonant nonlinear acoustics","volume":"79","author":"Majda, Andrew","year":"1988","journal-title":"Stud. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2526","issn-type":"print"},{"issue":"6","key":"4","doi-asserted-by":"publisher","first-page":"1505","DOI":"10.1137\/0729087","article-title":"The convergence rate of approximate solutions for nonlinear scalar conservation laws","volume":"29","author":"Nessyahu, Haim","year":"1992","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0731001","article-title":"The convergence rate of Godunov type schemes","volume":"31","author":"Nessyahu, Haim","year":"1994","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"H. Nessyahu and T. Tassa, \"Convergence rate of approximate solutions to conservation laws with initial rarefactions\", Siam J. on Numer. Anal., Vol. 31 (1994), pp. 628-654.","DOI":"10.1137\/0731034"},{"issue":"3","key":"7","first-page":"3","article-title":"Discontinuous solutions of non-linear differential equations","volume":"12","author":"Ole\u012dnik, O. A.","year":"1957","journal-title":"Uspehi Mat. Nauk (N.S.)"},{"key":"8","volume-title":"Maximum principles in differential equations","author":"Protter, Murray H.","year":"1967"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/BF00375117","article-title":"The regularized Chapman-Enskog expansion for scalar conservation laws","volume":"119","author":"Schochet, Steven","year":"1992","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"10","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-0152-3","volume-title":"Shock waves and reaction-diffusion equations","volume":"258","author":"Smoller, Joel","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0387907521"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1137\/0728048","article-title":"Local error estimates for discontinuous solutions of nonlinear hyperbolic equations","volume":"28","author":"Tadmor, Eitan","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"201","key":"12","doi-asserted-by":"publisher","first-page":"245","DOI":"10.2307\/2153164","article-title":"Total variation and error estimates for spectral viscosity approximations","volume":"60","author":"Tadmor, Eitan","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00716-8\/S0025-5718-96-00716-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00716-8\/S0025-5718-96-00716-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:06:51Z","timestamp":1776719211000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00716-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":12,"journal-issue":{"issue":"214","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["S0025-5718-96-00716-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00716-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}