A Nyström method for a boundary value problem arising in unsteady water wave problems

Preston, M. D., Chamberlain, P. G. and Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 (2011) A Nyström method for a boundary value problem arising in unsteady water wave problems. IMA Journal of Numerical Analysis, 31 (3). pp. 1123-1153. ISSN 1464-3642 doi: 10.1093/imanum/drq009

Abstract/Summary

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/7718
Item Type	Article
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Uncontrolled Keywords	water waves; Nyström method; Laplace's equation; nonperiodic surfaces
Publisher	Oxford University Press
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