Convergence of a boundary integral method for water waves


Journal Article

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.

Full Text

Duke Authors

Cited Authors

  • Beale, JT; Hou, TY; Lowengrub, J

Published Date

  • January 1, 1996

Published In

Volume / Issue

  • 33 / 5

Start / End Page

  • 1797 - 1843

International Standard Serial Number (ISSN)

  • 0036-1429

Digital Object Identifier (DOI)

  • 10.1137/S0036142993245750

Citation Source

  • Scopus