{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:30:08Z","timestamp":1776796208318,"version":"3.51.2"},"reference-count":36,"publisher":"American Mathematical Society (AMS)","issue":"283","license":[{"start":{"date-parts":[[2014,3,19]],"date-time":"2014-03-19T00:00:00Z","timestamp":1395187200000},"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                    We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump L\u00e9vy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kru\u017ekov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this\n                    <italic>non-monotone<\/italic>\n                    method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -error estimate, and provide numerical experiments for the fractional Burgers\u2019 equation.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02690-7","type":"journal-article","created":{"date-parts":[[2013,3,19]],"date-time":"2013-03-19T10:04:09Z","timestamp":1363687449000},"page":"1489-1514","source":"Crossref","is-referenced-by-count":8,"title":["On the spectral vanishing viscosity method for periodic fractional conservation laws"],"prefix":"10.1090","volume":"82","author":[{"given":"Simone","family":"Cifani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Espen","family":"Jakobsen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,3,19]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/s00028-006-0253-z","article-title":"Entropy formulation for fractal conservation laws","volume":"7","author":"Alibaud, Natha\u00ebl","year":"2007","journal-title":"J. Evol. Equ.","ISSN":"https:\/\/id.crossref.org\/issn\/1424-3199","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1142\/S0219891607001227","article-title":"Occurrence and non-appearance of shocks in fractal Burgers equations","volume":"4","author":"Alibaud, Natha\u00ebl","year":"2007","journal-title":"J. Hyperbolic Differ. Equ.","ISSN":"https:\/\/id.crossref.org\/issn\/0219-8916","issn-type":"print"},{"key":"3","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809781","volume-title":"L\\'{e}vy processes and stochastic calculus","volume":"116","author":"Applebaum, David","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521738651","edition":"2"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/9789812816641","volume-title":"Spectral methods and their applications","author":"Guo, Ben-Yu","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/9810233337"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1006\/jdeq.1998.3458","article-title":"Fractal Burgers equations","volume":"148","author":"Biler, Piotr","year":"1998","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"6","series-title":"Scientific Computation","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral methods","author":"Canuto, C.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540307259"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1016\/j.anihpc.2009.11.008","article-title":"Regularity of solutions for the critical \ud835\udc41-dimensional Burgers\u2019 equation","volume":"27","author":"Chan, Chi Hin","year":"2010","journal-title":"Ann. Inst. H. Poincar\\'{e} C Anal. Non Lin\\'{e}aire","ISSN":"https:\/\/id.crossref.org\/issn\/0294-1449","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"847","DOI":"10.3934\/dcds.2010.27.847","article-title":"Eventual regularization of the slightly supercritical fractional Burgers equation","volume":"27","author":"Chan, Chi Hin","year":"2010","journal-title":"Discrete Contin. Dyn. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1078-0947","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1512\/iumj.2009.58.3505","article-title":"Finite time singularities and global well-posedness for fractal Burgers equations","volume":"58","author":"Dong, Hongjie","year":"2009","journal-title":"Indiana Univ. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2518","issn-type":"print"},{"issue":"204","key":"10","doi-asserted-by":"publisher","first-page":"629","DOI":"10.2307\/2153244","article-title":"Spectral viscosity approximations to multidimensional scalar conservation laws","volume":"61","author":"Chen, Gui Qiang","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"11","unstructured":"S. Cifani. On nonlinear fractional convection-diffusion equations. PhD Thesis 2011:282, NTNU, 2011."},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/j.anihpc.2011.02.006","article-title":"Entropy solution theory for fractional degenerate convection-diffusion equations","volume":"28","author":"Cifani, Simone","year":"2011","journal-title":"Ann. Inst. H. Poincar\\'{e} C Anal. Non Lin\\'{e}aire","ISSN":"https:\/\/id.crossref.org\/issn\/0294-1449","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"1090","DOI":"10.1093\/imanum\/drq006","article-title":"The discontinuous Galerkin method for fractal conservation laws","volume":"31","author":"Cifani, Simone","year":"2011","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"P. Clavin. Instabilities and nonlinear patterns of overdriven detonations in gases. Nonlinear PDE\u2019s in Condensed Matter and Reactive Flows. Kluwer, 49\u201397, 2002.","DOI":"10.1007\/978-94-010-0307-0_3"},{"key":"15","series-title":"Chapman \\& Hall\/CRC Financial Mathematics Series","isbn-type":"print","volume-title":"Financial modelling with jump processes","author":"Cont, Rama","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/1584884134"},{"issue":"269","key":"16","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1090\/S0025-5718-09-02293-5","article-title":"A numerical method for fractal conservation laws","volume":"79","author":"Droniou, J\u00e9r\u00f4me","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1007\/s00028-003-0503-1","article-title":"Global solution and smoothing effect for a non-local regularization of a hyperbolic equation","volume":"3","author":"Droniou, J.","year":"2003","journal-title":"J. Evol. Equ.","ISSN":"https:\/\/id.crossref.org\/issn\/1424-3199","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/s00205-006-0429-2","article-title":"Fractal first-order partial differential equations","volume":"182","author":"Droniou, J\u00e9r\u00f4me","year":"2006","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"19","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/019","volume-title":"Partial differential equations","volume":"19","author":"Evans, Lawrence C.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821849743","edition":"2"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/s00211-003-0502-9","article-title":"Numerical approximation of entropy solutions for hyperbolic integro-differential equations","volume":"97","author":"Dedner, Andreas","year":"2004","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"21","series-title":"Chapman \\& Hall\/CRC Research Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1201\/9781420035797","volume-title":"Second order elliptic integro-differential problems","volume":"430","author":"Garroni, Maria Giovanna","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/158488200X"},{"key":"22","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-56139-9","volume-title":"Front tracking for hyperbolic conservation laws","volume":"152","author":"Holden, Helge","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/3540432892"},{"key":"23","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139168489","volume-title":"Methods of mathematical physics","author":"Jeffreys, Harold","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0521664020"},{"key":"24","first-page":"No. 116, 23","article-title":"Stability of entropy solutions for L\u00e9vy mixed hyperbolic-parabolic equations","author":"Karlsen, Kenneth H.","year":"2011","journal-title":"Electron. J. Differential Equations"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"211","DOI":"10.4310\/DPDE.2008.v5.n3.a2","article-title":"Blow up and regularity for fractal Burgers equation","volume":"5","author":"Kiselev, Alexander","year":"2008","journal-title":"Dyn. Partial Differ. Equ.","ISSN":"https:\/\/id.crossref.org\/issn\/1548-159X","issn-type":"print"},{"key":"26","doi-asserted-by":"crossref","unstructured":"S. N. Kru\u017ekov. First order quasi-linear equations in several independent variables. Math. USSR Sbornik, 10(2):217\u2013243, 1970.","DOI":"10.1070\/SM1970v010n02ABEH002156"},{"key":"27","doi-asserted-by":"crossref","unstructured":"N. N. Kuznetsov. Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR. Comput. Math. Phys. 16 (1976), 105\u2013119.","DOI":"10.1016\/0041-5553(76)90046-X"},{"key":"28","doi-asserted-by":"crossref","unstructured":"P. Loya. Dirichlet and Fresnel integrals via iterated integration. Mathematics Magazine 78 (2005), no. 1, 63\u201367.","DOI":"10.1080\/0025570X.2005.11953303"},{"issue":"4","key":"29","doi-asserted-by":"publisher","first-page":"854","DOI":"10.1137\/0726047","article-title":"Analysis of the spectral vanishing viscosity method for periodic conservation laws","volume":"26","author":"Maday, Yvon","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"30","series-title":"Applied Mathematics and Mathematical Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-6824-1","volume-title":"Weak and measure-valued solutions to evolutionary PDEs","volume":"13","author":"M\u00e1lek, J.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/041257750X"},{"key":"31","unstructured":"K. Sato. L\u00e9vy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, 1999."},{"issue":"5","key":"32","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1137\/0727066","article-title":"The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws","volume":"27","author":"Schochet, S.","year":"1990","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"33","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1137\/0726003","article-title":"Convergence of spectral methods for nonlinear conservation laws","volume":"26","author":"Tadmor, Eitan","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"201","key":"34","doi-asserted-by":"publisher","first-page":"245","DOI":"10.2307\/2153164","article-title":"Total variation and error estimates for spectral viscosity approximations","volume":"60","author":"Tadmor, Eitan","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"35","series-title":"Applied Mathematical Sciences","isbn-type":"print","volume-title":"Partial differential equations. III","volume":"117","author":"Taylor, Michael E.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0387946527"},{"key":"36","isbn-type":"print","first-page":"241","article-title":"L\u00e9vy processes in the physical sciences","author":"Woyczy\u0144ski, Wojbor A.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/081764167X"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2013-82-283\/S0025-5718-2013-02690-7\/S0025-5718-2013-02690-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-283\/S0025-5718-2013-02690-7\/S0025-5718-2013-02690-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:34:30Z","timestamp":1776792870000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2013-82-283\/S0025-5718-2013-02690-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,19]]},"references-count":36,"journal-issue":{"issue":"283","published-print":{"date-parts":[[2013,7]]}},"alternative-id":["S0025-5718-2013-02690-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02690-7","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":[[2013,3,19]]}}}