Finite-difference scheme with improved dispersion properties for nonlinear dispersive shallow water equations
Z.I. Fedotova, O.I. Gusev, G.S. Khakimzyanov
Federal Research Center for Information and Computational Technologies, Novosibirsk, Russia
Keywords: long surface waves, nonlinear dispersive equations, Rusanov scheme, dispersion, stability, phase error
Abstract
This paper presents the results of a dissipation and dispersion analysis of a linearized version of a novel finite difference scheme for the fully nonlinear weakly dispersive shallow water equations. It is a modification of the Rusanov scheme, which is a third-order approximation of the gas dynamics equations. We derive stability conditions, phase change and phase error formulas, and investigate the behavior of the harmonic attenuation coefficient. The study uncovers some previously unknown properties of the Rusanov scheme. Notably, both the original and the modified schemes exhibit high order smallness in the phase error for long waves. Furthermore, we identify the optimal parameter values that ensure monotonicity of the harmonic attenuation coefficient, strong suppression of short-wave harmonics, and small change in their phase.
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