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Multiscale Analyses for the Shallow Water Equations

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Niveau: Supérieur, Doctorat, Bac+8
Multiscale Analyses for the Shallow Water Equations Didier Bresch, Rupert Klein, and Carine Lucas Abstract This paper explores several asymptotic limit regimes for shallow water flows over multiscale topography. Depending on the length and time scales consid- ered and on the characteristic water depth and height of topography, a variety of mathematically quite different asymptotic limit systems emerges. Specifically, we recover the classical “lake equations” for balanced flow without gravity waves in the single time, single space scale limit (Greenspan, Cambridge Univ. Press, (1968)), discuss a weakly nonlinear and a strongly nonlinear multi-scale version of these wave-free equations involving short-range topography, and we re-derive the equa- tions for long-wave shallow water waves passing over short-range topography by Le Maıtre et al., JCP (2001). Didier Bresch LAMA, equipe EDPs2, Bat. Le Chablais, Campus scientifique, Universite de Savoie, 73376 Le Bourget du Lac, France e-mail: Rupert Klein Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195 Berlin, Germany e-mail: Carine Lucas MAPMO, Universite d'Orleans, UFR Sciences, Batiment de mathematiques - Route de Chartres, B.

  • single

  • lake equations

  • flow over

  • over multiscale

  • short-wave topography

  • nonlinear momentum

  • multiscale analyses

  • ?? sr tref


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MultiscaleAnalysesfortheShallowWaterEquationsDidierBresch,RupertKlein,andCarineLucasAbstractThispaperexploresseveralasymptoticlimitregimesforshallowwaterflowsovermultiscaletopography.Dependingonthelengthandtimescalesconsid-eredandonthecharacteristicwaterdepthandheightoftopography,avarietyofmathematicallyquitedifferentasymptoticlimitsystemsemerges.Specifically,werecovertheclassical“lakeequations”forbalancedflowwithoutgravitywavesinthesingletime,singlespacescalelimit(Greenspan,CambridgeUniv.Press,(1968)),discussaweaklynonlinearandastronglynonlinearmulti-scaleversionofthesewave-freeequationsinvolvingshort-rangetopography,andwere-derivetheequa-tionsforlong-waveshallowwaterwavespassingovershort-rangetopographybyLeMaıˆtreetal.,JCP(2001).DidierBreschLAMA,e´quipeEDPs2,Baˆt.LeChablais,Campusscientique,Universite´deSavoie,73376LeBourgetduLac,Francee-mail:Didier.Bresch@univ-savoie.frRupertKleinDepartmentofMathematicsandComputerScience,FreeUniversityofBerlin,Arnimallee6,14195Berlin,Germanye-mail:Rupert.Klein@math.fu-berlin.deCarineLucasMAPMO,Universite´dOrle´ans,UFRSciences,Baˆtimentdemathe´matiques-RoutedeChartres,B.P.6759-45067Orle´anscedex2,Francee-mail:carine.lucas@univ-orleans.fr1