Modeling Interactions and Shoaling of Solitary Waves Using a Hybrid Finite Volume and Finite Difference Solver

Kehhan Wang Burak Turan

doi:10.4172/2168-9679.1000173

Abstract

This paper presents a mixed finite volume and finite difference solver with results showing the solitary wave interactions and shoaling process by solving a set of conservative forms of Boussinesq equations. A second order accurate finite volume scheme is applied to the conservative terms of the governing equations while up to the second order finite difference formulations are used to discretize the dispersive source terms with higher order derivatives. The limiters and surface gradient method are implemented in the model to remove the unwanted spurious oscillations and preserve the still water condition without introducing errors at the interfaces. The performance of the present numerical solver is tested with results of head on collisions and shoaling of solitary waves compared against those from finite element models that were developed based on fully nonlinear weakly dispersive and weakly nonlinear weakly dispersive forms of the Boussinesq equations as well as analytical solutions and experimental observations.

Highlights

Modeling the propagation and shoaling transformation of solitary waves in shallow water regions is practically important to the study of impacts of nonlinear waves on coastal environments
A combined finite volume and finite difference model by solving the Madsen et al, [3] Bousinessq equations (BE) with a second order accurate FV scheme applied to the conservative terms and FD scheme on the dispersive source terms is developed in this study to model propagation of solitary waves, head-on collisions of solitary waves, and solitary wave shoaling over a submerged shoal
The present FV-FD model is simulated for head-on collision of two solitary waves and the results are compared with those computed using the models developed by Zhong and Wang [9,25]

Summary

Introduction

Modeling the propagation and shoaling transformation of solitary waves in shallow water regions is practically important to the study of impacts of nonlinear waves on coastal environments. This paper focuses on modeling cases of solitary wave transformation by discretizing the improved form of BE with a developed finite volumefinite difference solver. A combined finite volume and finite difference model by solving the Madsen et al, [3] BE with a second order accurate FV scheme applied to the conservative terms and FD scheme on the dispersive source terms is developed in this study to model propagation of solitary waves, head-on collisions of solitary waves, and solitary wave shoaling over a submerged shoal. The results obtained from the present FVFD model are compared with those from the FE solvers developed by Zhong and Wang [9,25] and other published solutions

Governing Equations

Hybrid Finite Volume and Finite

FNWD WNWD Present Model

Shoaling of a Solitary Wave over a Sloping Bottom

Conclusions