Finite-Volume Models for Unidirectional, Nonlinear, Dispersive Waves

Several finite-volume schemes are developed and applied to simulate nonlinear, dispersive, unidirectional waves propagating over a flat bed. These schemes differ mainly in the treatment of advection, while dispersion is treated the same among the different models. Three methods—linear, total variation diminishing, and essentially nonoscillatory—are used to discretize the advection portion of the governing equation. The linear schemes are analyzed with Von Neumann's method to discern stability limits as well as their damping and dispersion characteristics. In addition, predictions from all of the models are compared with analytical solutions for solitary and cnoidal waves as well as experimental data for undular bores. The finite-volume methods are also compared with a second-order accurate finite-difference scheme. The results indicate that the finite-volume schemes yield more accurate solutions than the finite-difference scheme. In addition, the Warming-Beam and Fromm linear finite-volume schemes yielded the most accurate solutions and were among the most computationally efficient schemes tested.

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