Internal wave reflections and transmissions arising from a non‐uniform mesh. Part II: A generalized analysis for the Crank–Nicolson linear finite element scheme

Abstract

SUMMARYInternal wave reflections and transmissions are examined for the Crank-Nicolson linear finite element scheme applied to the linear shallow water equations in a 1D domain containing an abrupt change in nodal spacing. In Part I of this series the reflection/transmission analysis was verified by some 'hot-start' numerical experiments. Here in Part 11, however, that analysis is found wanting when it comes to providing a description of the pseudo-steady state wave configuration which develops with some 'cold-start' experiments. It is shown that the analysis of Part I can be extended to take in both the 'hot-' and 'cold-start' experimental results such that four essentially different wave configurations can be identified. The four configurations are discernible on the basis of group velocity. In order to be sustained, two of the configurations require one energy source whereas the other two require two energy sburces. Numerical experiments confirmed the analysis.