{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T12:13:02Z","timestamp":1776859982916,"version":"3.51.2"},"reference-count":4,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,2,23]],"date-time":"2001-02-23T00:00:00Z","timestamp":982886400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider the numerical solution of the stochastic partial differential equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"partial-differential u slash partial-differential t equals partial-differential squared u slash partial-differential x squared plus sigma left-parenthesis u right-parenthesis ModifyingAbove upper W With dot left-parenthesis x comma t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u2202\n                                \n                              <\/mml:mi>\n                              <mml:mi>u<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u2202\n                                \n                              <\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msup>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u2202\n                                  \n                                <\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">\n                                \u2202\n                                \n                              <\/mml:mi>\n                              <mml:msup>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>W<\/mml:mi>\n                                <mml:mo>\n                                  \u02d9\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\partial u}\/{\\partial t}={\\partial ^2u}\/{\\partial x^2}+\\sigma (u)\\dot {W}(x,t)<\/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=\"ModifyingAbove upper W With dot\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mover>\n                              <mml:mi>W<\/mml:mi>\n                              <mml:mo>\n                                \u02d9\n                                \n                              <\/mml:mo>\n                            <\/mml:mover>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\dot {W}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is space-time white noise, using finite differences. For this equation Gy\u00f6ngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ModifyingAbove upper W With dot\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mover>\n                              <mml:mi>W<\/mml:mi>\n                              <mml:mo>\n                                \u02d9\n                                \n                              <\/mml:mo>\n                            <\/mml:mover>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\dot {W}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma left-parenthesis u right-parenthesis equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma (u)=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma left-parenthesis u right-parenthesis equals u\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma (u)=u<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) we show that no such improvements are possible.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01224-2","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"121-134","source":"Crossref","is-referenced-by-count":112,"title":["Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations"],"prefix":"10.1090","volume":"70","author":[{"given":"A.","family":"Davie","sequence":"first","affiliation":[]},{"given":"J.","family":"Gaines","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,2,23]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1017\/CBO9780511526213.005","article-title":"Numerical experiments with S(P)DE\u2019s","author":"Gaines, J. G.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521483190"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"1132","DOI":"10.1137\/S0036139992235706","article-title":"Random generation of stochastic area integrals","volume":"54","author":"Gaines, J. G.","year":"1994","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"I. Gy\u00f6ngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II, Potential Anal. 11 (1999), 1\u201337.","DOI":"10.1023\/A:1008699504438"},{"key":"4","series-title":"North-Holland Mathematical Library","isbn-type":"print","volume-title":"Stochastic differential equations and diffusion processes","volume":"24","author":"Ikeda, Nobuyuki","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0444873783","edition":"2"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01224-2\/S0025-5718-00-01224-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01224-2\/S0025-5718-00-01224-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:30:19Z","timestamp":1776724219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01224-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,2,23]]},"references-count":4,"journal-issue":{"issue":"233","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["S0025-5718-00-01224-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01224-2","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":[[2000,2,23]]}}}