{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:37:54Z","timestamp":1776724674365,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A Cauchy problem for a one\u2013dimensional diffusion\u2013reaction equation is solved on a grid by a random walk method, in which the diffusion part is solved by random walk of particles, and the (nonlinear) reaction part is solved via Euler\u2019s polygonal arc method. Unlike in the literature, we do not assume monotonicity for the initial condition. It is proved that the algorithm converges and the rate of convergence is of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the spatial mesh length.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00917-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"593-602","source":"Crossref","is-referenced-by-count":4,"title":["Convergence of a random walk method for a partial differential equation"],"prefix":"10.1090","volume":"67","author":[{"given":"Weidong","family":"Lu","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1017\/S0022112073002016","article-title":"Numerical study of slightly viscous flow","volume":"57","author":"Chorin, Alexandre Joel","year":"1973","journal-title":"J. Fluid Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1120","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"A. J. Chorin, Vortex sheet approximation of boundary layer, J. Comp. Phys, 27(1978), 428\u2013442.","DOI":"10.1016\/0021-9991(78)90019-0"},{"key":"3","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-0082-3","volume-title":"A mathematical introduction to fluid mechanics","author":"Chorin, A. J.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0387904069"},{"key":"4","volume-title":"An introduction to probability theory and its applications. Vol. II","author":"Feller, William","year":"1971","edition":"2"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1002\/cpa.3160400204","article-title":"Convergence of the random vortex method","volume":"40","author":"Goodman, Jonathan","year":"1987","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1137\/0902007","article-title":"Convergence of random methods for a reaction-diffusion equation","volume":"2","author":"Hald, Ole H.","year":"1981","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"1373","DOI":"10.1137\/0907091","article-title":"Convergence of a random method with creation of vorticity","volume":"7","author":"Hald, Ole H.","year":"1986","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"8","unstructured":"D. G. Long, Convergence of the random vortex method in one and two dimensions, Ph.D. Thesis, Univ. of California, Berkeley, 1986."},{"key":"9","series-title":"Interscience Tracts in Pure and Applied Mathematics, No. 4","volume-title":"Difference methods for initial-value problems","author":"Richtmyer, Robert D.","year":"1967","edition":"2"},{"issue":"186","key":"10","doi-asserted-by":"publisher","first-page":"615","DOI":"10.2307\/2008485","article-title":"Convergence of a random particle method to solutions of the Kolmogorov equation \ud835\udc62_{\ud835\udc61}=\ud835\udf08\ud835\udc62\u2093\u2093+\ud835\udc62(1-\ud835\udc62)","volume":"52","author":"Puckett, Elbridge Gerry","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/0021-9991(85)90154-8","article-title":"Accuracy of the random vortex method for a problem with nonsmooth initial conditions","volume":"58","author":"Roberts, Stephen","year":"1985","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"186","key":"12","doi-asserted-by":"publisher","first-page":"647","DOI":"10.2307\/2008486","article-title":"Convergence of a random walk method for the Burgers equation","volume":"52","author":"Roberts, Stephen","year":"1989","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\/1998-67-222\/S0025-5718-98-00917-X\/S0025-5718-98-00917-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00917-X\/S0025-5718-98-00917-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:44:07Z","timestamp":1776721447000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-222\/S0025-5718-98-00917-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":12,"journal-issue":{"issue":"222","published-print":{"date-parts":[[1998,4]]}},"alternative-id":["S0025-5718-98-00917-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00917-x","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":[[1998]]}}}