A preconditioned, time-iterative method for the shallow water equations
by
Steve Bova
Corps of Engineers Waterways Experiment Station Major Shared Resource Center\\Engineering Research Center, Mississippi State University

Under certain assumptions, the shallow water equations define a nonlinear system of hyperbolic partial differential equations. Consequently, a convenient approach to obtain steady-state solutions is to use a time-iterative method wherein arbitrary initial data are marched through time to convergence. Because the system is hyperbolic, information propagates in well-defined directions at characteristic speeds. It is well-known that the convergence rate of this class of solvers is relatively fast for supercritical (supersonic) flows, but can decrease dramatically for subcritical (subsonic) flows. This decrease can result in unacceptable solution times. In recent years, the aerospace engineering community has addressed this problem as it occurs in the Euler equations of gas dynamics, a related hyperbolic system. The basic problem is that there are three characteristic speeds: the local flow speed; the local flow speed plus the speed of gravity waves; and the local flow speed minus the speed of gravity waves. For supercritical flows, all three speeds are of the same order. However, for subcritical flows, the local flow speed can be orders of magnitude less than that of gravity waves. It is this disparity that leads to ill-conditioning of the time-iterative equations. If the transient behavior of the system is not of interest, then time may be viewed strictly as an iteration parameter. Hence, the time derivatives may be modified in any convenient way that would result in accelerated convergence. (The well-known local time-step calculation is a simple example of this technique.) The present paper describes a modification of the time derivatives that controls the characteristic speeds and ameliorates the ill-conditioning associated with subcritical flows.

http://www.erc.msstate.edu/~swb

Date received: April 14, 1999