The phenomenon of roll waves occurs in a uniform open-channel
flow 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 water
equations. The main difficulty associated with the numerical approximation of
this problem is its linear instability. Numerical round-off error
can easily overtake the numerical solution and yields false roll wave
solution at the steady state.
In this paper, we first study the analytic behavior of the solution to the above
model. We then discuss the numerical difficulty, and introduce a numerical
method that predicts precisely the evolution and steady state of its
solution. Various numerical experiments are performed to illustrate
the numerical difficulty and the effectiveness of the proposed numerical