A COMPARISON OF STABILITY AND ACCURACY OF SOME NUMERICAL-MODELS OF TWO-DIMENSIONAL CIRCULATION
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AbstractThe purpose of the paper is twofold: Firstly, we develop stream functionvorticity and primitive variable finite element models of twodimensional barotropic equations that satisfy the conservation of mean vorticity, mean squared vorticity (or enstrophy) and mean kinetic energy, and scondly, we present a comparative study of a number of numerical schemes for their accuracy in phase speed as well as in amplitude calculations for a twodimensional, timedependent, stream functionvorticity equation for periodic fluid motion in a channel. A circular vortex is placed in a uniform channel flow of a constant velocity (U) as an initial condition. An analytic solution exists for the problem such that the vortex moves with a constant speed U conserving the shape of the vortex: where U, 0a are constants. This example makes it easier to identify the cause of phase speed error, either due to linear or nonlinear processes, and furthermore, to find a satisfactory scheme for time integration. The numerical schemes compared include: Arakawa Jacobian,1 ArakawaMatsuno scheme, Galerkin finite element, LaxWendroff, leapfrog, and CrankNicholson. The effect of a variational adjustment (see Sasaki16) is also studied. Computational time, RMS errors in stream function and vorticity, and the conservation of the mean kinetic energy and enstrophy are compared at the end of 120 (one period) and 240 (two periods) time steps. The study indicates that the numerical scheme that employs finite elements in space (same as Arakawa Jacobian) and CrankNicholson in time is the most accurate among the schemes studied.
