Relaxation Models and Finite Element Schemes for the Shallow Water Equations

Theodoros Katsaounis () and Charalambos Makridakis ()
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Theodoros Katsaounis: Ecole Normale Supérieure, Département de Mathématiques et Applications
Charalambos Makridakis: University of Crete, and Institute of Applied and Computational Mathematics, Department of Applied Mathematics

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2003, pp 621-631 from Springer

Abstract: Abstract We consider the one-dimensional system of shallow water equations (or SaintVenant system) with a source term (1a) $$ {{h}_{t}} + {{(hu)}_{x}} = 0, $$ (1b) $$ {{(hu)}_{t}} + {{(h{{u}^{2}} + \frac{g}{2}{{h}^{2}})}_{x}} = - ghZ', $$ which describes the flow at time t ≥ 0 at point x ε ℝ, where h(x, t) ≥ 0 is the height of water, u(x, t) is the velocity, Z(x) is the bottom height and g the gravity constant. In the sequel will denote Q = hu the discharge. System (1) belongs in the more general class of hyperbolic systems with source terms (2) $$ {{u}_{t}} + f{{(u)}_{x}} = q(u), $$ where u is a vector valued function and f, q are the given flux and source functions. In this paper we propose relaxation models and corresponding time discrete and finite element schemes for approximating (1). Our schemes can be formulated for the more general system (2) and special attention is given in the steady state approximations and their relation to the exact steady states especially for (1).

Keywords: Source Term; Hyperbolic System; Shallow Water Equation; Relaxation Model; Finite Element Space (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55711-8_58

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DOI: 10.1007/978-3-642-55711-8_58

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