ELSEVIER E~,lneert~Amdysis with Boundary ~ 13(1994) 181-191 O 1994 Elsevier Scieaee I.imlted Printedin CaeatBritain.All fishUfeatured 0955-7997/~t/s¢7.00 Quasi-singular integrals in the modeling of nonlinear water waves in shallow water Sttphan T. Grim & Ravishankar Subramanya Ocean Engineering Department, University of Rhode Island, Kingston, Rhode Island 02881, USA (Received 13 September 1993; accepted 17 November 1993) The model by Grilli eta/., 5,s based on fully nonlinear potential flow equations, is used to study propagation of water waves over arbitrary bottom topography. The model combines a higher-order boundary element method for the solution of Laplace's equation at a given time, and Lagrangian Taylor expansions for the time updating of the free surface position and potential. In this paper, both the accuracy and the efficiency of computations are improved, for wave shoaling and breaking over gentle slopes, in domains with very sharp geometry and large aspect ratio, by using quasi-singular integration techniques based on modified Telles 17 and Lutz H methods. Appfcations are presented that demonstrate the accuracy and the efllciencyof the new approaches. Key words: Boundary element method, quasi-singular integration, numerical integration, nonlinear wave modeling, free surface potential flow. 1 INTRODUCTION When water waves reach coastal regions it is observed that, for any given wave, both an increase in wave height H and a decrease in wavelength L occur due to the reduc- tion in water depth h. This phenomenon, referred to as wave shoaling, makes waves steeper towards the shore and, eventually leads to wave instability and breaking in the surf-zone. Knowledge of the shape and kinematics of waves close to breaking is of prime importance in coastal engineering, for these directly control sediment transport by coastal currents, shaping of beaches, and design of coastal structures used for beach protection. Over gently sloping beaches, wave breaking roughly occurs when wave height is equal to the local depth. Hence, small amplitude waves usually shoal-up and pro- pagate all the way up a beach before they break. Free surface waves are modeled quite well using potential flow equations. In general, these equations are further simplified or linearized into wave theories, using perturbation expansions based on small param- eters like the wave steepness, H/L, or the relative wave height, H/h. For waves close to breaking, how- ever, nonlinearities are large, and wave theories cannot in general be used. For such cases, it is necessary to solve directly fully nonlinear potential flow equations. Efficient two-dimensional models have been devel- oped over the last decade based on fully nonlinear 181 potential flow equations, for the solution of (mostly deep water) wave propagation and breaking (see Refs 1, 5, 10, 12 and 18 and Grilli eta/. 5 for a review of the literature to date). Most of the existing approaches have been based on solving Laplace's equation at a given time, using a boundary integral equation or a boundary element method (BEM), and on updating both the free surface geometry and potential, using a time stepping method based on an Eulerian-Lagran- gian description of the free surface. Applying such models to calculate wave shoaling and breaking in very shallow water, however, requires using computational domains with very sharp geometry and large aspect ratio (see, e.g., Figs 1 and 6). For such domains, in addition to corner problems (see Ref. 8), the narrowing of the geometry over the slope, the shallower the water, creates quasi-singular situations for the nodes that, on one side of the boundary, are getting closer to the other side of the boundary (e.g. bottom or free surface, Figs 6 and 8). For such cases, unless a refined discretization or special integration methods are used, quasi-singularities will always lead to a loss of accuracy of the BEM and to early numerical instability of the computations, s The formation of narrow jets in breakers, in which BEM nodes, equiva- lent to Lagrangian particles, accumulate, is another cause for such quasi-singular situations (see Figs 6(c) and 8(c)).