Wavelet Galerkin scheme for solving nonlinear dispersive shallow water waves: Application in bore propagation and breaking

An accurate and fast numerical method is developed to investigate the nonlinear (linear) shallow water wave propagation over flat (depth-varying) topography in one space dimension within an irrotational and inviscid flow. As physical model, we use a dispersive Boussinesq-type (BT) system for small-amplitude long waves with weak transverse variation. The problem is discretised in space using a wavelet-Galerkin method based on one-periodic Daubechies scaling functions. Assuming periodic boundary conditions, the discretised operators in spatial domain are circulant and skew-symmetric. These characteristics of discretised differential operators allow us to incorporate the Fast Fourier Transformation (FFT) in the matrix operations which results in a substantial improvement in the computational efficiency and accuracy of the numerical solver compared with the conventional finite difference or finite volume methods. We use a four-stage Runge-Kutta method to temporally discretise the governed spatially discretised differential equations. Several comparative test cases are conducted to validate the performance and efficiency of the proposed wavelet-Galerkin scheme for the BT model over flat beds relative to some existing analytical solutions and numerical results from a second-order finite difference method. We examine the numerical results of the BT system to investigate the two-way propagation of waves for some large L2-norm profiles of the initial free-surface elevation. We also assess the ability of the proposed method to predict the evolution and breaking of undular bores over a flat bed with a simple kinematic criterion. Moreover, we study (tsunami) wave runup and propagation by incorporating the effects of depth-varying topography in a simplified BT system in order to check the applicability of the approach to capture the interactions between the bathymetric features and the wet cells.