Computation of the shallow water equations using the unified coordinates

Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, in the spirit of the arbitrary Lagrangian--Eulerian (ALE) approach, W.H. Hui, P.Y. Li, and Z.W. Li, [J. Comput. Phys., 153 (1999), pp. 596--637] have introduced a unified coordinate system which moves with velocity hq, q being the velocity of the fluid particle. It includes the Eulerian system as a special case when h=0, and the Lagrangian when h=1, and was shown for the two-dimensional Euler equations of gas dynamics to be superior to both Eulerian and Lagrangian systems. The main purpose of this paper is to adopt this unified coordinate system to solve the shallow water equations. It will be shown that computational results using the unified system are superior to existing results based on either the Eulerian system or Lagrangian system in that it (a) resolves slip lines sharply, especially for steady flow, (b) avoids grid deformation and computation breakdown in Lagrangian coordinates, and (c) avoids spurious flow produced by Lagrangian coordinates.