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Mixed finite element methods for the shallow water equations including current and silt sedimentation. II: The discrete-time case along characteristics. (English) Zbl 1145.76404
Summary: The mixed finite element (MFE) methods for a shallow water equation system consisting of water dynamics equations, silt transport equation, and the equation of bottom topography change were derived. A fully discrete MFE scheme for the discrete-time along characteristics is presented and error estimates are established. The existence and convergence of MFE solution of the discrete current velocity, elevation of the bottom topography, thickness of fluid column, and mass rate of sediment is demonstrated.
[For Part I, see ibid. No. 1, 80–92 (2004; Zbl 1141.76424).]

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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[1] Boris J P, Book D L. Flux corrected transport[J].SHASTA J Comput Phys, 1973,2(1):36–69. · Zbl 0251.76004
[2] Pironneau O. On the transport-diffusion algorithm and its applications to the Navier-Stokes equations [J].Numer Math, 1982,38(2):309–332. · Zbl 0505.76100
[3] Bermúdez A, Rodriguez C, Vilar M A. Solving shallow water equations by a mixed implicit finite element method[J].IMA J Numer Anal, 1991,11(1):79–97. · Zbl 0713.76069
[4] LUO Zhen-dong, ZHU Jiang, ZENG Qing-cun,et al. Mixed finite element methods for the shallow water equations including current and silt sedimentation (I)–The continuous-time case[J].Applied Mathematics and Mechanics (English Edition), 2003,24(1):80–92. · Zbl 1079.76601
[5] XIN Xiao-kang, LIU Ru-xun, JIANG Bo-cheng.Computational Fluid Dynamics [M]. Changsha: National Defence Science Technical Press, 1989. (in Chinese)
[6] ZENG Qing-cun. Silt sedimentation and relevant engineering problem–an example of natural cybernetics[A]. In:Proceeding of the Third International Congress on Industrial and Applied Mathematics[C]. ICIAM95 held in Hamburg, Academic Verlag, 1995, 463–487. · Zbl 0849.76091
[7] Chipada S, Dawson C N, Martinez M L,et al. Finite element approximations to the system of shallow water equations–I: Continuous-time a priori error estimates[J].SIAM J Numer Anal, 1998,35(2):692–711. · Zbl 0910.76034
[8] Adams R A.,Sobolev Spaces [M]. New York: Academic Press 1975.
[9] Ciarlet P G.The Finite Element Method for Elliptic Problems [M]. Amsterdam: North-Holland, 1978. · Zbl 0383.65058
[10] LUO Zhen-dong.Theory Bases and Applications of Finite Element and Mixed Finite Element Methods, Evolutions and Applications[M]. Jinan: Shandong Educational Press, 1996. (in Chinese)
[11] Brezzi F, Fortin M.Mixed and Hybrid Finite Element Methods [M]. Berlin, Heidelberg, New York: Springer-Verlag, 1991. · Zbl 0788.73002
[12] Girault V, Raviart P A.Finite Element Approximations of the Navier-Stokes Equations[M]. Berlin Beidelberg, New York: Springer-Verlag, 1979. · Zbl 0396.65070
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