High-order finite volume WENO schemes for Boussinesq modelling of nearshore wave processes

ABSTRACT

This paper presents the formulation and validation of a shock‐capturing hybrid method using the WENO (weighted essential non‐oscillatory)‐HLL (Harten–Lax–Van Leer) scheme for the solution of the two-dimensional extended Boussinesq-type equations. The governing equations are reformulated in a conservative form, using high-order finite volume WENO schemes with HLL Riemann solver to handle the flux terms. The efficient third-order Runge–Kutta time-stepping scheme is used to advance the numerical solution through time. Wave breaking is predicted by applying a wave breaking criterion to automatically degenerate the Boussinesq equations into nonlinear shallow water equations. In order to test whether the present numerical model could accurately reproduce the processes of transformation, breaking, and run-up of nonlinear dispersive waves under different wave conditions in the nearshore region, five test cases have been simulated. The numerical results show that the present shock-capturing model could simulate the nearshore wave processes very well.

Keywords:

• Breaking waves
• Boussinesq equations
• nearshore wave processes
• shock capturing
• WENO scheme

Acknowledgements

We would like to thank the support of the Administrator Foundation of South Branch, SOA(1539).

Additional information

Funding

This work was supported by The Hong Kong Polytechnic University [grant number H-ZG68] and National Marine Public Welfare Research Projects of China [grant number 201005002].