HIGH-PRECISION CALCULATIONS OF VORTEX SHEET MOTION

The motion of a vortex sheet undergoing Kelvin-Helmholtz instability i s known to be ill-posed, causing deterioration in numerical calculatio ns from the rapid growth of round-off errors. In particular, it is the smallest scales (introduced by round-off ) that grow the fastest Kras ny ([12]) introduced a spectral filter to suppress the growth of round -off errors of the smallest scales. He was then able to detect evidenc e supporting asymptotic studies that indicate the formation of a curva ture singularity in finite time. We use high precision interval arithm etic, coded in C++ , to re-examine the evolution of a vortex sheet fro m initial conditions used previously by several researchers. Most impo rtantly, our results are free from the influence of round-off errors. We show excellent agreement between results obtained through high prec ision interval arithmetic and through the use of Krasny's spectral fil ter. In particular, our results support the formation of a curvature s ingularity in finite time. After the time of singularity formation, th e markers move in peculiar patterns. We rule out any possibility of th is motion resulting from round-off errors, but it does depend on the l evel of resolution. We find no consistent behavior in the motion of th e markers as we improve the resolution of the vortex sheet. Also, we f ind some disagreement between the results obtained through high precis ion interval arithmetic and through the use of the spectral filter. (C ) 1994 Academic Press, Inc.