{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:36:16Z","timestamp":1776728176410,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"235","license":[{"start":{"date-parts":[[2001,5,23]],"date-time":"2001-05-23T00:00:00Z","timestamp":990576000000},"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                    When shock speed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    times\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta t slash normal upper Delta x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta t\/\\Delta x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is rational, the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved. A sufficient condition under which the above solutions can be extended to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"negative normal infinity greater-than j greater-than normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-\\infty &gt;j&gt;\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , implying the existence of discrete shock profiles of numerical schemes, is provided. A class of monotonicity preserving schemes, including all monotonicity preserving schemes with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^{1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    numerical flux functions, the second order upwinding flux based MUSCL scheme, the second order flux based MUSCL scheme with Lax-Friedrichs\u2019 splitting, and the Godunov scheme for scalar conservation laws are found to satisfy this condition. Thus, the existence of discrete shock profiles for these schemes is established when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s normal upper Delta t slash normal upper Delta x\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\Delta t\/\\Delta x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is rational.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01254-0","type":"journal-article","created":{"date-parts":[[2002,11,6]],"date-time":"2002-11-06T14:02:22Z","timestamp":1036591342000},"page":"1043-1069","source":"Crossref","is-referenced-by-count":6,"title":["Existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws"],"prefix":"10.1090","volume":"70","author":[{"given":"Haitao","family":"Fan","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,5,23]]},"reference":[{"key":"1","unstructured":"[EY] B. 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