ON THE TRUNCATION ERROR, STABILITY, AND CONVERGENCE OF DIFFERENCE SOLUTIONS OF THE BAROTROPIC VORTICITY EQUATION

Abstract

The truncation error, stability and convergence properties of various finite-difference formulations of the one-dimensional barotropic vorticity equation are considered, and analytic solutions of the difference equations for simple harmonic initial conditions are presented. With conventional centered space differences, the schemes considered may be classified according to the method of time differencing as the forward difference case (unstable), the first-forward-then-centered difference case (conditionally stable), and the implicit difference case (unconditionally stable). The first-forward-then-centered difference scheme, corresponding to that commonly employed in meteorological numerical integration, gives rise to an oscillation phenomenon in both the amplitude and phase speed of the solution, which is most serious for a small space mesh, a large time mesh, and for the shorter wavelength disturbances. In each difference scheme considered, the truncation error leads to a cumulative phase departure of the difference solution relative to the true solution, an effect which is approximately proportional to the cube of the wavelength.