Implicit methods for two-dimensional unsteady free-surface flows
Beam and Warming implicit Unite difference schemes are introduced to integrate the equations describing the two-dimensional unsteady free-surface flows. These schemes are second-order accurate in space and time, allow sharp discontinuous initial conditions, and do not require isolation of a bore. Subcritical and supercritical flows may be present indifferent parts of the channel or they may occur in sequence. Depending upon the Courant number used in the computations, sharp discontinuities may be slightly smeared since they are not isolated. Different procedures for incorporating the boundary conditions are discussed. The analysis of two-dimensional unsteady free-surface (lows resulting from the breaking of a dam are presented to illustrate the aplication of the methods.
In this paper, we study the asymptotical optimal solutions for the general airline revenue management problem, considering both customer cancelations and no-shows. We study a simple case where there is only one fare-class; the management controls the opening and closing of the fare-class, with the objective of maximizing the total revenue. We first develop a fluid control problem, where the customer arrival and cancelation processes are treated as deterministic, whereas the no. of no-shows is random. The optimal fluid solution is shown to have a simple structure of accepting up to a single time point t* and rejecting afterwards. We show that this policy acts as an upper bound for the MDP problem. Applying the optimal fluid solution to the stochastic environment, we show that this policy is asymptotically optimal on the fluid scaling.