The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible cfl numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical

. The oscillations of a centered second order finite difference scheme and the excessive diffusion of a first order centered scheme can be overcome by global composition of the two, that is by performing cycles consisting of several time steps of the second order method followed by one step of the diffusive method. We show the effectiveness of this approach on some test problems in two and three dimensions. 1. Introduction For a system of conservation laws U t = f x (U ), it is well known that the LaxWendroff (LW) finite difference scheme produces oscillations behind shock waves while the Lax-Friedrichs (LF) method is excessively diffusive, smearing out the shocks more than is usually acceptable. Simple two-step versions of both schemes are defined as follows. For both schemes the first half step defines new values on a staggered dual grid as U n+1=2 i+1=2 = 1 2 [U n i + U n i+1 ] + \Deltat 2\Deltax [f(U n i+1 ) \Gamma f(U n i )]: (1) The second half step of the LF scheme is gi...

One- and two-dimensional shallow water ows are computed using the Legendre spectral viscosity method and post-processed with edge detection and local reconstruction algorithms. The examples include dam break problems and the bores generated by submerged obstacles. Discontinuities are accurately located and the reconstruction is demonstrated to be spectrally accurate where the solution is smooth. Keywords: Spectral viscosity, edge detection, shallow water, dam break. AMS(MOS) subject classication: 65M70, 86A10, 35L65. Anne Gelb was supported in part by the Sloan Foundation, and in part by NSF grant No. DMS01-07428. James P. Gleeson was supported by the Fulbright Commission. y Department of Mathematics, P.O. Box 871804 Arizona State University Tempe, AZ 852871804. Email: ag@math.la.asu.edu z Department of Applied Mathematics, University College Cork, Ireland. Email: j.gleeson@ucc.ie 1 1

Comparison of several difference schemes on 1D and 2D