Lax–Wendroff Schemes with Polynomial Extrapolation and Simplified Lax–Wendroff Schemes for Dispersive Waves: A Comparative Study

Abstract

One of the features of Boussinesq-type models for dispersive wave propagation is the presence of mixed spatial/temporal derivatives in the partial differential system. This is a critical point in the design of the time marching strategy, as the cost of inverting the algebraic equations arising from the discretization of these mixed terms may result in a nonnegligible overhead. In this paper, we propose novel approaches based on the classical Lax–Wendroff (LW) strategy to achieve single-step high-order schemes in time. To reduce the cost of evaluating the complex correction terms arising in the Lax–Wendroff procedure for Boussinesq equations, we propose several simplified strategies which allow to reduce the computational time at fixed accuracy. To evaluate these qualities, we perform a spectral analysis to assess the dispersion and damping error. We then evaluate the schemes on several benchmarks involving dispersive propagation over flat and nonflat bathymetries, and perform numerical grid convergence studies on two of them. Our results show a potential for a CPU reduction between 35 and 40% to obtain accuracy levels comparable to those of the classical RK3 method.