Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows
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Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows Peter Maxwell1
· Simen Å. Ellingsen1
Received: 31 May 2019 / Accepted: 23 March 2020 © The Author(s) 2020
Abstract The path-following scheme in Loisel and Maxwell (SIAM J Matrix Anal Appl 39(4):1726–1749, 2018) is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current. This is equivalent to solving the Rayleigh stability equation with linearized free-surface boundary condition for each sought point on the curve. Taking advantage of the analyticity of the dispersion relation, a path-following or continuation approach is adopted. The problem is discretized using a collocation scheme, parametrized along either a radial or angular path in the wave vector plane, and differentiated to yield a system of ODEs. After an initial eigenproblem solve using QZ decomposition, numerical integration proceeds along the curve using linear solves as the Runge–Kutta F(·) function; thus, many QZ decompositions on a size 2N companion matrix are exchanged for one QZ decomposition and a small number of linear solves on a size N matrix. A piecewise interpolant provides dense output. The integration represents a nominal setup cost whereafter very many points can be computed at negligible cost whilst preserving high accuracy. Furthermore, a two-dimensional interpolant suitable for scattered data query points in the wave vector plane is described. Finally, a comparison is made with existing numerical methods for this problem, revealing that the path-following scheme is the most competitive algorithm for this problem whenever calculating more than circa 1,000 data points or relative normwise accuracy better than 10−4 is sought. Keywords Path-following method · Rayleigh stability equation · Free surface · Quadratic eigenproblem · Dispersion relation · Numerical continuation
This work was funded by the Norwegian Research Council (FRINATEK) #249740.
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Peter Maxwell [email protected] Department of Energy and Process Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway
P. Maxwell, S. Å. Ellingsen
1 Introduction We develop a path-following method to numerically calculate the dispersion relation curve for linear surface waves atop a horizontal current of arbitrary depth dependence, an adaptation of the scheme developed by Loisel and Maxwell [29]. 1.1 Background and Motivation We consider a classical problem from the study of wave-current interactions [38–40], [36, sec. IV], [12,34,35,43], that of linear surface waves upon a vertical shear current. In other words, waves are considered as perturbations to first-order in wave steepness = ka upon some zeroth-order depth-dependent horizontal background flow that, in general, has non-constant vorticity. The problem geometry is shown in Fig. 1. Waves in this regime are dispersive with their behaviour being entirely characterized by the dispersion relation, i.e. the dependence of phase vel
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