Fully discrete stability and dispersion analysis for a linear dispersive internal wave model

Abstract

In the context of modeling internal water waves, some strongly nonlinear reduced models exhibit a nonlocal term involving a Hilbert-type transform. These models accurately account for the physical dispersion regarding the Euler equations. To perform a careful stability analysis and detect whether numerical solutions are reliable or spurious, it is necessary to adapt the classical von Neumann analysis to account for nonlocal dispersive terms. We address this issue by a fully discrete analysis of the one-dimensional linear model, in the flat bottom case. We find a formula for the amplification factor that provides estimates concerned with numerical stability and dispersion (namely, phase errors). Subsequently, we contrast the numerical properties of the original dispersive problem with that of the underlying non-dispersive case, namely a linear hyperbolic system. The stability estimates corroborate the fact that physical dispersion provided by a nonlocal (singular integral) term allows for less restrictive stability conditions.