A multi-layer Boussinesq-type model with second-order spatial derivatives: Theoretical analysis and numerical implementation

Abstract The accuracy of the linear and nonlinear properties embodied in multi-layer Boussinesq-type models with the highest spatial derivative n being 2, as proposed by Liu et al. (2018), is theoretically and numerically investigated in this paper. Theoretical analysis shows that the four-layer model has the highest accuracy and is applicable up to kh = 179.3 (where k is the wavenumber, and h is a typical water depth) in phase celerity at a 1% tolerance error, kh = 69.7 in the linear velocity components and kh = 141.8 in linear shoaling amplitude. At the same tolerance error, the super- and sub-harmonic transfer functions are accurate up to kh = 138.5 and 80, respectively, and the third-order harmonics and amplitude dispersion are accurate up to kh = 127.2. A high accuracy of the linear velocity profiles is also achieved and is approximately kh = 4.5, 14.9 and 69.7 for the two-layer, three-layer and four-layer models, respectively. Vertical two dimensional (2D) numerical models are established with a composite fourth-order Adams-Bashforth-Moulton scheme in time integration. Numerical simulations, including linear shoaling of regular waves over a mild slope, nonlinear regular wave evolution over a submerged breakwater and focusing wave group evolution over a constant water depth, are carried out. The computed results are in reasonable agreement with the experimental data. Furthermore, the CPU times for the two-layer model with n = 2 and the two-layer model with n = 3 are compared, and the numerical efficiency of the two-layer model with n = 2 increases by approximately 1/3.