On the Computation of Roll Waves

¹ Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. (jin@math.wisc.edu.)
² Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455-0436, USA. (yjkim@ima.umn.edu)

Received: 9 September 2000
Revised: 29 January 2001

Abstract

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 u_t + uu_x = 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 method.