Convergence of a Boundary Integral Method for Water Waves
Abstract
We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration.
Additional Information
©1996 Society for Industrial and Applied Mathematics. Received by the editors November 11, 1993; accepted for publication (in revised form) November 7, 1994. The research of this author [J.T.B.] was supported by NSF grant DMS-9102782. Research at the M.S.R.I. was supported in part by NSF grant DMS-8505550. The research of this author [T.Y.H.] was supported by a Sloan Foundation Research Fellowship, NSF grant DMS-9003202, and in part by AFOSR grant AFOSR-90-0090 and NSF grant DMS-9100383. The research of this author [J.L.] was supported by an NSF Postdoctoral Fellowship, the University of Minnesota Army High Performance Computing Research Center, and the Minnesota Supercomputer Center. The first author is grateful for the hospitality of the Courant Institute and the M.S.R.I. during his sabbatical year. The second and third authors wish to thank I.A.S. for its hospitality.Files
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