Numerical Methods for the Shallow Water Equations

In the present work we discuss a special class of conservation laws, the one-dimensional Shallow Water Equations, with a geometrical source term (the bottom topography). Namely, Ut + F(U)x = S(U); where, U = · h hu Έ; F(U) = · hu hu2 + g 2h2 Έ; S(U) = · 0 ‘ghZ0 Έ: This system describes the flow at time t Έ 0 at point x 2 R where h(x; t) Έ 0 is the total water height above the bottom, u(x; t) is the average horizontal velocity, Z(x) is the bottom height function and g the gravitational acceleration. The most important property of this system is that they preserve steady states and satisfying an entropy condition. To discretize the above system we proceed as follows. First we introduce the relaxation approximation of JinXin for the shallow water equation. We proceed by discretizing the relaxation system using finite differences schemes of first and second order and implicitexplicit RungeKutta methods as time stepping mechamisms. Key words: shallow water equations; relaxation schemes; finite difference