Abstract
An unsteady flow modelling system is often used to obtain steady conditions in open channels. To be more precise, if the boundary conditions are fixed, initial flow perturbations are allowed to dissipate or propagate out of the system. In this paper, the optimum difference scheme for that technique, dynamic relaxation, and the conditions to maximize the convergence rate are discussed.
The theoretical analysis shows the convergence rate of dynamic relaxation depends on absolute value, (λ), of the eigenvalue of the amplification matrix of a difference scheme and smaller (λ) accelerates the convergence rate. In this meaning, Preissmann sheme (weighting coefficient θ=1) with unconditional stability has a very good convergence rate because (λ) can be close to zero by taking sufficient large time step Δt in comparison to space interval Δx. Moreover, it shows good accuracy because it can compute the boundary points with second order accuracy in regard to Δx.
The time required for computations per Δt is almost the same as that with explicit schemes, for reasons of adopting the double-sweep method as the technique for solving the linearized difference equations derived from the Preissmann scheme. As a result of numerical experiments, the necessary number of time steps to get a steady-state solution using the Preissmann scheme isless than onetenth of that with explicit schemes.
