Nearshore Wave Modeling And Simulation Based On Simple Finite Volume Methods

Nearshore wave propagation and evolution is a very complicated process and often affected by a variety of factors simultaneously. For the accurate simulation of nearshore wave propagation process, the nonlinear and dispersive characteristics of water waves should usually be taken into full account. Derived from the basic equations of water wave problems, the Boussinesq-type equations(BTEs) not only be capable of describing the deformation of water waves in shallow water, but also offer a simple description of the problem. Based on the Finite Volume(FV) method, the present paper aims to establish some numerical models of higher-order nonlinear BTEs and simulate the propagation and evolution process of water waves in nearshore regions.The FV method is widely used for numerical solutions of BTEs. Unfortunately,there are three main di?culties for the present FV method. Firstly, extra efforts are needed to get the primitive variables from the computational ones which include dispersive terms. Secondly, a numerical imbalance which generates numerical noise arises for the different schemes used in discretizing the convective and source terms.Thirdly, high-order schemes are needed to reconstruct the convective terms, however,they are usually very complex. Addressing these three di?culties, two different models are proposed in present paper for nearshore water wave simulations.First, for solving the Nwogu type BTEs, a new conservative form based on primitive variables is proposed by using an approximate transformation method where the mixed partial derivative terms in the momentum equation are replaced by pure spatial ones. On this basis, a simple and e?cient FV scheme is presented, where the second-order upwinding total variation diminishing(TVD) technique is applied for solving the ?ux terms, and the third-order TVD Runge-Kutta method is used for solving the time integral terms, while a new spatial gradient reconstruction method is utilized for solving the rest terms. The advantages of this numerical scheme are its simplicity, strong stability and no need of adjustable parameters.Second, for solving the three typical BTEs(including the Madsen and S?rensen type, the Nwogu type, and the Ge Wei and Subramanya type), a uni?ed conservative form of equations is proposed. Using the new spatial gradient reconstruction method presented in current paper, a simple and e?cient numerical method is developed for solving the mixed partial derivative terms. On this basis, another FV scheme is presented based on the uni?ed conservative form. This FV scheme can be applied to a class of extended BTEs for their similar expressions.Last, the two models are implemented in several typical tests to verify the effectiveness of them. Compared with analytical solutions, measurements and counterparts respectively, good performances are observed for all of these tests,indicating that the two models are useful alternative approaches for nearshore water wave simulations.