On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

DOI

10.2991/jnmp.2005.12.s1.13How to use a DOI?

Abstract

An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N2 ) to O(N), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Roberto Camassa
AU  - Jingfang Huang
AU  - Long Lee
PY  - 2005
DA  - 2005/01/01
TI  - On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 146
EP  - 162
VL  - 12
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.s1.13
DO  - 10.2991/jnmp.2005.12.s1.13
ID  - Camassa2005
ER  -