Integral and integrable algorithms for a nonlinear shallow-water wave equation Roberto Camassa, Jingfang Huang, Long Lee * Department of Mathematics, University of North Carolina at Chapel Hill, Philips Hall cb 3250, Chapel Hill, NC 27599, United States Received 12 May 2005; received in revised form 18 December 2005; accepted 20 December 2005 Available online 8 February 2006 Abstract An asymptotic higher-order model of wave dynamics in shallow water is examined in a combined analytical and numer- ical study, with the aim of establishing robust and efficient numerical solution methods. Based on the Hamiltonian struc- ture of the nonlinear equation, an algorithm corresponding to a completely integrable particle lattice is implemented first. Each ‘‘particle’’ in the particle method travels along a characteristic curve. The resulting system of nonlinear ordinary dif- ferential equations can have solutions that blow-up in finite time. We isolate the conditions for global existence and prove l 1 -norm convergence of the method in the limit of zero spatial step size and infinite particles. The numerical results show that this method captures the essence of the solution without using an overly large number of particles. A fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from O(N 2 ) to O(N), where N is the number of particles. The method possesses some analogies with point vortex methods for 2D Euler equations. In particular, near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches. The second method is based on integro-differential formulations of the equation. Two different algorithms are proposed, based on different ways of extracting the time deriv- ative of the dependent variable by an appropriately defined inverse operator. The integro-differential formulations reduce the order of spatial derivatives, thereby relaxing the stability constraint and allowing large time steps in an explicit numer- ical scheme. In addition to the Cauchy problem on the infinite line, we include results on the study of the nonlinear equa- tion posed in the quarter (space-time) plane. We discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical examples. Ó 2006 Elsevier Inc. All rights reserved. 1. Introduction The nonlinear partial differential equation (PDE) of evolution u t þ 2ju x  u xxt þ 3uu x ¼ 2u x u xx þ uu xxx ð1:1Þ results from an asymptotic expansion of the Euler equations governing the motion of an inviscid fluid whose free surface can exhibit gravity driven wave motion [7]. The small parameters used to carry out the expansion 0021-9991/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2005.12.013 * Corresponding author. Tel.: +1 919 843 2218; fax: +1 919 962 9345. E-mail address: longlee@email.unc.edu (L. Lee). Journal of Computational Physics 216 (2006) 547–572 www.elsevier.com/locate/jcp