Relaxation Models and Finite Element Schemes for the Shallow Water Equations

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).