It has been pointed out some time ago by Yakimiw and Robert (Mon. Wea. Rev., 1986) that the method of fractional steps for the numerical solution of the shallow water equations has the advantage of reducing the multidimensional matrix inversion problem into an equivalent one-dimensional problem, so the technique becomes very simple and very attractive to apply provided it is accurate and stable enough.

We apply this fractional steps technique by splitting the shallow water equations, and successively integrating in every direction along the characteristics using the Riemann invariants (which are constant quantities along the characteristics), associated with cubic spline interpolation (Shoucri, J. Comp. Phys. ,1986). The method is tested on simple advection models as well as the full shallow-water equations, and shown to be accurate and stable. The linear analysis of the equations show the method is unconditionally stable, reproducing exactly the frequency of the slow mode, while the frequency of the fast modes is exact to second order.

Extension to the three-dimensional weather forecast equations will be discussed.