Path-Following Methods for Calculating Linear Surface Wave Dispersion Relations on Vertical Shear Flows

Autor: Peter Maxwell, Simen Å. Ellingsen
Rok vydání: 2020
Předmět:
Zdroj: Water Waves
ISSN: 2523-3688
2523-367X
Popis: 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 $$F(\cdot )$$ 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 $$10^{-4}$$ 10 - 4 is sought.
Databáze: OpenAIRE
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