An LDG numerical approach for Boussinesq type modelling

Abstract

This work describes the development of a Boussinesq type model based on a set of fully nonlinear and first order dispersive equations in one horizontal dimension. The numerical discretisation is carried out using a discontinuous Galerkin framework. In particular, the basis of the scheme is established through a local discontinuous Galerkin (LDG) method for the spatial discretisation while for the time discretisation a high order total variation diminishing Runge–Kutta procedure is utilised. The numerical tool is validated by means of the simulation of two classic benchmark test cases. Firstly, the propagation over a flat bed of three solitary waves with different nonlinearity is considered. The presence of a dispersive tail of short waves is detected behind nonlinear solitons and confirmed by a RANS model. Secondly, two wave dispersion scenarios over a submerged bar are reproduced. The impact of different orders of accuracy is assessed on the computed solution. The model shows good behaviour for weakly dispersive waves up to kh values close to π. Finally, the numerical efficiency of the code is illustrated by a CPU time analysis for given combinations of the order and the mesh resolution. Test cases for a multi-layer system will be considered in future work.