{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T06:36:16Z","timestamp":1773815776985,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T00:00:00Z","timestamp":1712275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["19-71-30012"],"award-info":[{"award-number":["19-71-30012"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg\u2013de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme.<\/jats:p>","DOI":"10.3390\/computation12040076","type":"journal-article","created":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T08:27:52Z","timestamp":1712305672000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Why Stable Finite-Difference Schemes Can Converge to Different Solutions: Analysis for the Generalized Hopf Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7353-106X","authenticated-orcid":false,"given":"Vladimir A.","family":"Shargatov","sequence":"first","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"given":"Anna P.","family":"Chugainova","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"given":"Georgy V.","family":"Kolomiytsev","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"given":"Irik I.","family":"Nasyrov","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-2362-5174","authenticated-orcid":false,"given":"Anastasia M.","family":"Tomasheva","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"given":"Sergey V.","family":"Gorkunov","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]},{"given":"Polina I.","family":"Kozhurina","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,5]]},"reference":[{"key":"ref_1","unstructured":"Jeltsch, R., and Fey, M. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Proceedings of the Hyperbolic Problems: Theory, Numerics, Applications."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1134\/S0018151X12040165","article-title":"Mechanism of the appearance of the cellular structure of a shock wave in the region of its ambiguous representation","volume":"50","author":"Likhachev","year":"2012","journal-title":"High Temp."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1017\/S0263034607000687","article-title":"Multi-phase equation of state for aluminum","volume":"25","author":"Lomonosov","year":"2007","journal-title":"Laser Part. Beams"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101905","DOI":"10.1063\/1.2894197","article-title":"Theoretical investigation of shock wave stability in metals","volume":"92","author":"Lomonosov","year":"2008","journal-title":"Appl. Phys. Lett."},{"key":"ref_5","first-page":"87","article-title":"Some problems in the theory of quasilinear equations","volume":"14","year":"1959","journal-title":"Usp. Mat. Nauk."},{"key":"ref_6","first-page":"165","article-title":"Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation","volume":"14","author":"Oleinik","year":"1959","journal-title":"Uspekhi Mat. Nauk."},{"key":"ref_7","unstructured":"Goritskii, A.Y., Kruzhkov, S.N., and Chechkin, G.A. (1999). First Order Partial. Differ. Equations, Department of Mechanics and Mathematics, MSU."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1017\/S0308210500013111","article-title":"Undercompressive Shocks and Riemann Problems for Scalar Conservation Laws with Nonconvex Fluxes","volume":"129","author":"Hayes","year":"1999","journal-title":"R. Soc. Edinb.-Proc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1006\/jdeq.2000.4009","article-title":"Diffusive\u2013Dispersive Traveling Waves and Kinetic Relations: Part I: Nonconvex Hyperbolic Conservation Laws","volume":"178","author":"Bedjaoui","year":"2002","journal-title":"J. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1017\/S0308210500003504","article-title":"Diffusive\u2013Dispersive travelling waves and kinetic relations V. Singular diffusion and nonlinear dispersion","volume":"134","author":"Bedjaoui","year":"2004","journal-title":"Proc. R. Soc. Edinb. Sect. Math."},{"key":"ref_11","first-page":"1349","article-title":"The possible effect of oscillations in a discontinuity structure on the set of admissible discontinuities","volume":"275","author":"Kulikovskii","year":"1984","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"ref_12","first-page":"285","article-title":"Strong discontinuities in flows of continua and their structure","volume":"182","author":"Kulikovskii","year":"1990","journal-title":"Proc. Steklov Inst. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1002\/cpa.3160100406","article-title":"Hyperbolic systems of conservation laws","volume":"10","author":"Lax","year":"1957","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1090\/qam\/1866551","article-title":"A nonconvex scalar conservation law with a trilinear flux","volume":"59","author":"Hayes","year":"2001","journal-title":"Q. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1134\/S0040577918100094","article-title":"Reflection and Refraction of Solitons by the KdV\u2013Burgers Equation in Nonhomogeneous Dissipative Media","volume":"197","author":"Samokhin","year":"2018","journal-title":"Theor. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"101723","DOI":"10.1016\/j.difgeo.2021.101723","article-title":"The KdV soliton crosses a dissipative and dispersive border","volume":"75","author":"Samokhin","year":"2021","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-020-00659-3","article-title":"Traveling waves and undercompressive shocks in solutions of the generalized Korteweg\u2013de Vries\u2013Burgers equation with a time-dependent dissipation coefficient distribution","volume":"135","author":"Chugainova","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113654","DOI":"10.1016\/j.cam.2021.113654","article-title":"Stability analysis of traveling wave solutions of a generalized Korteweg\u2013de Vries\u2013Burgers equation with variable dissipation parameter","volume":"397","author":"Shargatov","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1134\/S0081543823040211","article-title":"Structures of Classical and Special Discontinuities for the Generalized Korteweg\u2013de Vries\u2013Burgers Equation in the Case of a Flux Function with Four Inflection Points","volume":"322","author":"Shargatov","year":"2023","journal-title":"Proc. Steklov Inst. Math."},{"key":"ref_20","unstructured":"Lyapidevskii, V.Y., and Teshukov, V. (2000). Mathematical Models of Propagation of Long Waves in an Inhomogeneous Fluid, Publishing House of the Siberian Branch of the Russian Academy of Sciences."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0167-2789(93)90197-9","article-title":"Oscillatory instability of traveling waves for a KdV-Burgers equation","volume":"67","author":"Pego","year":"1993","journal-title":"Phys. Nonlinear Phenom."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.cnsns.2018.06.008","article-title":"Analytical description of the structure of special discontinuities described by a generalized KdV-Burgers equation","volume":"66","author":"Chugainova","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/15M1015650","article-title":"Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws","volume":"59","author":"El","year":"2017","journal-title":"Siam Rev."},{"key":"ref_24","first-page":"1007","article-title":"On the decay of an arbitrary initial discontinuity in an elastic medium","volume":"52","author":"Kulikovskii","year":"1988","journal-title":"Prikl. Mat. Mekh."},{"key":"ref_25","first-page":"47","article-title":"Eigenvalues, and instabilities of solitary waves","volume":"340","author":"Pego","year":"1992","journal-title":"Philos. Trans. R. Soc. London. Ser. Phys. Eng. Sci."},{"key":"ref_26","first-page":"199","article-title":"Undercompressive shocks for nonstrictly hyperbolic conservation laws","volume":"3","author":"Shearer","year":"1990","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1017\/S0308210500003577","article-title":"Non-classical Riemann solvers with nucleation","volume":"134","author":"LeFloch","year":"2004","journal-title":"Proc. R. Soc. Edinb. Sect. Math."},{"key":"ref_28","unstructured":"Isaacson, E., and Keller, H. (1994). Analysis of Numerical Methods, Dover Books on Mathematics, Dover Publications."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1002\/cpa.3160090206","article-title":"Survey of the stability of linear finite difference equations","volume":"9","author":"Lax","year":"1956","journal-title":"Commun. Pure Appl. Math."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/4\/76\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:23:47Z","timestamp":1760106227000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/4\/76"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,5]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["computation12040076"],"URL":"https:\/\/doi.org\/10.3390\/computation12040076","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,5]]}}}