{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:44:16Z","timestamp":1709203456046},"reference-count":17,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007,10,1]]},"abstract":"<jats:title>On the Numerical Approximation of First-Order Hamilton-Jacobi Equations<\/jats:title><jats:p>Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.<\/jats:p>","DOI":"10.2478\/v10006-007-0033-0","type":"journal-article","created":{"date-parts":[[2007,10,12]],"date-time":"2007-10-12T14:22:52Z","timestamp":1192198972000},"page":"403-412","source":"Crossref","is-referenced-by-count":0,"title":["On the Numerical Approximation of First-Order Hamilton-Jacobi Equations"],"prefix":"10.61822","volume":"17","author":[{"given":"R\u00e9mi","family":"Abgrall","sequence":"first","affiliation":[]},{"given":"Vincent","family":"Perrier","sequence":"additional","affiliation":[]}],"member":"37438","reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1002\/(SICI)1097-0312(199612)49:12<1339::AID-CPA5>3.0.CO;2-B","article-title":"Numerical Discretization of First Order Hamilton-Jacobi Equations on Triangular Meshes","volume":"12","author":"R Abgrall","year":"1996","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"2","first-page":"2233","article-title":"Numerical discretization of boundary conditions for first order Hamilton Jacobi equations","volume":"6","author":"R Abgrall","year":"2004","journal-title":"SIAM Journal on Numerical Analyis"},{"key":"3","unstructured":"Abgrall R. (2007): <i>Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations.<\/i> (in revision)."},{"key":"4","unstructured":"Abgrall R. and Perrier V. (2007): <i>Error estimates for Hamilton-Jacobi equations with boundary conditions.<\/i> (in preparation)."},{"key":"5","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1023\/A:1007633810484","article-title":"High order numerical discretization for Hamilton-Jacobi equations on triangular meshes","volume":"2","author":"S. Augoula","year":"2000","journal-title":"Journal of Scientific Computing"},{"key":"6","first-page":"1373","article-title":"On Hopf's formula for solutions of first order Hamilton-Jacobi equations. Nonlinear Analysis Theory:","volume":"11","author":"M. Bardi","year":"1984","journal-title":"Methods and Applications"},{"key":"7","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1137\/0522022","article-title":"The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations","volume":"2","author":"M. Bardi","year":"1991","journal-title":"SIAM Journal on Mathematical Analysis"},{"key":"8","author":"G Barles","year":"1994","journal-title":"Solutions de viscosit\u00e9 des \u00e9quations de Hamilton-Jacobi"},{"key":"9","doi-asserted-by":"crossref","first-page":"271","DOI":"10.3233\/ASY-1991-4305","article-title":"Convergence of approximation schemes for fully nonlinear second order equations","volume":"3","author":"G. Barles","year":"1991","journal-title":"Asymptotic Analysis"},{"key":"10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0025-5718-1984-0744921-8","article-title":"Two approximations of solutions of Hamilton-Jacobi equations","volume":"167","author":"M. Crandall","year":"1984","journal-title":"Mathematics of Computation"},{"key":"11","doi-asserted-by":"crossref","first-page":"329","DOI":"10.4171\/IFB\/103","article-title":"Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities","volume":"3","author":"K. Deckelnick","year":"2004","journal-title":"Interfaces and Boundary"},{"key":"12","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1137\/S1064827598337282","article-title":"A discontinuous Galerkin finite element method for Hamilton Jacobi equations","volume":"2","author":"C. Hu","year":"1999","journal-title":"SIAM Journal on Scientific Computing"},{"key":"13","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1016\/j.aml.2004.10.009","article-title":"Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations","volume":"11","author":"F. Li","year":"2005","journal-title":"Applied Mathematics Letters"},{"key":"14","author":"P.-L. Lions","year":"1982","journal-title":"Generalized Solutions of Hamilton-Jacobi Equations"},{"key":"15","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1137\/0728049","article-title":"High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations","volume":"4","author":"S. Osher","year":"1991","journal-title":"SIAM Journal on Numerical Analysis"},{"key":"16","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.jcp.2004.10.003","article-title":"Hermite WENO schemes for Hamilton-Jacobi equations","volume":"1","author":"J. Qiu","year":"2005","journal-title":"Journal of Computational Physics"},{"key":"17","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1137\/S1064827501396798","article-title":"High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes","volume":"3","author":"Y. Zhang","year":"2003","journal-title":"SIAM Journal on Scientific Computing"}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/17\/3\/article-p403.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/amcs.2007.17.issue-3\/v10006-007-0033-0\/v10006-007-0033-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:26:24Z","timestamp":1709202384000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10006-007-0033-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,10,1]]},"references-count":17,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/v10006-007-0033-0","relation":{},"ISSN":["1641-876X"],"issn-type":[{"value":"1641-876X","type":"print"}],"subject":[],"published":{"date-parts":[[2007,10,1]]}}}