Multilevel schemes for the shallow water equations

In this paper, we study a number of multilevel schemes for the numerical solution of the shallow water equations; new schemes and new perspectives of known schemes are examined. We consider the case of periodic boundary conditions. Spatial discretization is obtained using a Fourier spectral Galerkin method. For the time integration, two strategies are studied. The first one is based on scale separation, and we choose the time scheme (explicit or semi-implicit) as a function of the spatial scales (multilevel schemes). The second approach is based on a splitting of the operators, and we choose the time integration method as a function of the operator considered (multistep or fractional schemes). The numerical results obtained are compared with the explicit reference scheme (Leap-Frog scheme), and with the semi-implicit scheme (Leap-Frog scheme with Crank-Nicholson scheme for the gravity terms), both computed with a similar mesh. The drawback of the explicit reference scheme being the numerical stability constraint on the time step, and the drawback of the semi-implicit scheme being the dispersive error, the aim with the new schemes is to obtain schemes with less dispersive error than the semi-implicit scheme, and with better stability properties than the explicit reference scheme. The numerical results obtained show that the schemes proposed allow one to reduce the dispersive error and to increase the numerical stability at reduced cost.