A comparison of stability and accuracy of some numerical models of two‐dimensional circulation
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The purpose of the paper is two‐fold: Firstly, we develop stream function‐‐vorticity and primitive variable finite element models of two‐dimensional 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 two‐dimensional, time‐dependent, stream function‐‐vorticity 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: (Formula Presented.) where U, ψ0 a are constants. This example makes it easier to identify the cause of phase speed error, either due to linear or non‐linear processes, and furthermore, to find a satisfactory scheme for time integration. The numerical schemes compared include: Arakawa Jacobian,1 Arakawa–Matsuno scheme, Galerkin finite element, Lax–Wendroff, leap‐frog, and Crank–Nicholson. 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 Crank–Nicholson in time is the most accurate among the schemes studied. Copyright © 1980 John Wiley & Sons, Ltd
author list (cited authors)
Sasaki, Y. K., & Reddy, J. N.