@Article{bouchut08entropy,
  author   = {Bouchut, Fran{\c{c}}ois and Morales de Luna, Tom{\'a}s},
  title    = {{A}n entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
  journal  = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},
  year     = {2008},
  abstract = {We consider the system of partial differential equations governing
	the one-dimensional flow of two superposed immiscible layers of shallow
	water. The difficulty in this system comes from the coupling terms
	involving some derivatives of the unknowns that make the system nonconservative,
	and eventually nonhyperbolic. Due to these terms, a numerical scheme
	obtained by performing an arbitrary scheme to each layer, and using
	time-splitting or other similar techniques leads to instabilities
	in general. Here we use entropy inequalities in order to control
	the stability. We introduce a stable well-balanced time-splitting
	scheme for the two-layer shallow water system that satisfies a fully
	discrete entropy inequality. In contrast with Roe type solvers, it
	does not need the computation of eigenvalues, which is not simple
	for the two-layer shallow water system. The solver has the property
	to keep the water heights nonnegative, and to be able to treat vanishing
	values.},
  keywords = {Two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity},
  url      = {http://dx.doi.org/10.1051/m2an:2008019},
}

{"_id":"hHgxmWFibcm9fYyGp","bibbaseid":"bouchut-moralesdeluna-anentropysatisfyingschemefortwolayershallowwaterequationswithuncoupledtreatment-2008","title":"An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment","author_short":["Bouchut, F.","Morales de Luna, T."],"year":2008,"bibtype":"article","biburl":"http://www.uma.es/media/files/edanyabib.bib","bibdata":{"bibtype":"article","type":"article","author":[{"propositions":[],"lastnames":["Bouchut"],"firstnames":["François"],"suffixes":[]},{"propositions":["Morales","de"],"lastnames":["Luna"],"firstnames":["Tomás"],"suffixes":[]}],"title":"An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment","journal":"ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)","year":"2008","abstract":"We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.","keywords":"Two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity","url":"http://dx.doi.org/10.1051/m2an:2008019","bibtex":"@Article{bouchut08entropy,\n author = {Bouchut, Fran{\\c{c}}ois and Morales de Luna, Tom{\\'a}s},\n title = {{A}n entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},\n journal = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)},\n year = {2008},\n abstract = {We consider the system of partial differential equations governing\n\tthe one-dimensional flow of two superposed immiscible layers of shallow\n\twater. The difficulty in this system comes from the coupling terms\n\tinvolving some derivatives of the unknowns that make the system nonconservative,\n\tand eventually nonhyperbolic. Due to these terms, a numerical scheme\n\tobtained by performing an arbitrary scheme to each layer, and using\n\ttime-splitting or other similar techniques leads to instabilities\n\tin general. Here we use entropy inequalities in order to control\n\tthe stability. We introduce a stable well-balanced time-splitting\n\tscheme for the two-layer shallow water system that satisfies a fully\n\tdiscrete entropy inequality. In contrast with Roe type solvers, it\n\tdoes not need the computation of eigenvalues, which is not simple\n\tfor the two-layer shallow water system. The solver has the property\n\tto keep the water heights nonnegative, and to be able to treat vanishing\n\tvalues.},\n keywords = {Two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity},\n url = {http://dx.doi.org/10.1051/m2an:2008019},\n}\n\n","author_short":["Bouchut, F.","Morales de Luna, T."],"key":"bouchut08entropy","id":"bouchut08entropy","bibbaseid":"bouchut-moralesdeluna-anentropysatisfyingschemefortwolayershallowwaterequationswithuncoupledtreatment-2008","role":"author","urls":{"Paper":"http://dx.doi.org/10.1051/m2an:2008019"},"keyword":["Two-layer shallow water","nonconservative system","complex eigenvalues","time-splitting","entropy inequality","well-balanced scheme","nonnegativity"],"metadata":{"authorlinks":{"morales de luna, t":"https://bibbase.org/show?bib=http%3A%2F%2Fwww.uco.es%2F%7Ema1molut%2Fmorales.bib&theme=simple"}}},"search_terms":["entropy","satisfying","scheme","two","layer","shallow","water","equations","uncoupled","treatment","bouchut","morales de luna"],"keywords":["two-layer shallow water","nonconservative system","complex eigenvalues","time-splitting","entropy inequality","well-balanced scheme","nonnegativity"],"authorIDs":["9aGyuQQGPpD8hSE8h"],"dataSources":["yN9P3cwTmyLBqLYsP","3JTHNdbCwFwtr9YyG","hpo4NBzrFaStcKS6z","NWWpJWWcpEiCgSNQB","pgrvhZxwMgj4EDTEA","pLaL5rYzWMf7bTeYm"]}