We propose a simple numerical method for capturing thesteady state solution of hyperbolic systems with geometricalsource terms. We usethe interface value, rather than the cell-averages, for the source terms that balance the nonlinear convectionat the cell interface, allowing the numerical capturing of the steadystate with a formal high order accuracy. This method applies to Godunovor Roe type upwind methods butrequires no modification of the Riemann solver. Numerical experiments on scalar conservationlaws and the one dimensional shallow water equationsshow much better resolution of the steady state than the conventionalmethod, with almost no new numerical complexity.

The phenomenon of roll waves occurs in a uniform open-channelflow down an incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation ut + uux = u,u(x,0) = u0(x), which arises as a weakly nonlinear approximation of the shallow waterequations. The main difficulty associated with the numerical approximation ofthis problem is its linear instability. Numerical round-off errorcan easily overtake the numerical solution and yields false roll wavesolution at the steady state.In this paper, we first study the analytic behavior of the solution to the abovemodel. We then discuss the numerical difficulty, and introduce a numericalmethod that predicts precisely the evolution and steady state of itssolution. Various numerical experiments are performed to illustratethe numerical difficulty and the effectiveness of the proposed numericalmethod.