{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:23Z","timestamp":1787329283531,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We study a class of monotone numerical schemes for time-dependent Hamilton\u2013Jacobi equations with weak Dirichlet boundary conditions. We get a convergence rate of $\\frac{1}{2}$ under some usual assumptions on the data, plus an extra assumption on the Hamiltonian $H(Du,x)$ at the boundary $\\partial\\Omega$. More specifically the mapping $p\\to H(p,x)$ must satisfy a monotonicity condition for all p in a certain subset of $\\mathbf{R}^n$ given by $\\Omega$. This condition allows the use of the interior subsolution conditions at the boundary in the comparison arguments. We also prove a comparison result and Lipschitz regularity of the exact solution. As an example we construct a Godunov-type scheme that can handle the weakened boundary conditions.<\/jats:p>","DOI":"10.1137\/050632932","type":"journal-article","created":{"date-parts":[[2008,6,6]],"date-time":"2008-06-06T18:05:16Z","timestamp":1212775516000},"page":"2371-2392","source":"Crossref","is-referenced-by-count":4,"title":["Convergence Rate of Monotone Numerical Schemes for Hamilton\u2013Jacobi Equations with Weak Boundary Conditions"],"prefix":"10.1137","volume":"46","author":[{"given":"Knut","family":"Waagan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,6,6]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142998345980"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"M. Bardi and I. Capuzzo-Dolcetta,\n                      Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations\n                      , in Systems & Control: Foundations & Applications, Birkh\u00e4user, Boston, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia.","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.57262\/die\/1371586188"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1990.39.39024"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01445155"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1991-4305"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"I. Capuzzo Dolcetta and F. Leoni,\n                      On the Vanishing Viscosity Approximation of a Time Dependent Hamilton\u2013Jacobi Equation\n                      , in Recent Trends in Nonlinear Analysis, Birkh\u00e4user, Basel, 2000, pp. 59\u201375.","DOI":"10.1007\/978-3-0348-8411-2_8"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"M. G. Crandall,\n                      Viscosity Solutions: A Primer\n                      , in Viscosity Solutions and Applications, Springer, Berlin, 1997, pp. 1\u201343.","DOI":"10.1007\/BFb0094294"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0690039-8"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1984-0744921-8"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1985-0784180-6"},{"key":"R13","unstructured":"W. H. Fleming and H. M. Soner,\n                      Controlled Markov Processes and Viscosity Solutions\n                      , Appl. Math. 25, Springer-Verlag, New York, 1993."},{"key":"R14","first-page":"105","volume":"16","author":"Ishii H.","year":"1989","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. 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