On the Computation of Roll Waves

Abstract

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 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 method.Mathematics Subject Classification. 35L65, 65M06, 76B15.

A Steady-State Capturing Method for Hyperbolic Systems with Geometrical Source Terms

“…Consequently, the solution of the cell-average method eventually becomes unbounded and can never converge to the steady state solution. Such a numerical instability is well-documented in [12].…”

“…Roll wave is a series of periodic traveling waves connected by shocks, and the solution of (2.20), starting with a zero mass, periodic initial data will tend to the steady state roll wave [12]. The problem (2.20) is linearly unstable due to the exponential growing mode from the source term, which inherits the physical instability of the roll wave in shallow water.…”

“…All computations are made using double precision arithmetic. It is easy to check that, when one uses odd number of cells over each period of the sine wave, the Roe flux, given in (2.14), is perfectly symmetric with respect to the middle cell over one period, at which u j = 0, thus gives [12] …”

“…Since the initial data contains four sine wave lengths over the domain [0, 1], the exact solution should have four roll waves over [0, 1] with slope one, peak value 0.125 with jumps at x = 1/8, 3/8, 5/8 and 7/8 [12]. In order to obtain the correct steady state, we need odd number of cells in each of the wave length.…”

A Well-Balanced Gas-Kinetic Scheme for the Shallow-Water Equations with Source Terms

“…With the capturing of particle accelerating or decelerating mechanisms in the flux evaluation of the kinetic BGK scheme, the BGK current method should be a physically reliable one. The implementation of other physical terms in it for the complicated wave phenomena, such as resonance [15], bed friction [22], and roll waves problems [13], will be investigated in the near future.…”