Internal wave reflections and transmissions arising from a non‐uniform mesh. Part III: The occurrence of evanescent waves in the Crank‐Nicolson linear finite element scheme

Cathers, B.; Bates, Susan M.; O’Connor, Brian A.

doi:10.1002/fld.1650090706

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“…The wave equation is obtained by taking the x-derivative of equation (l), the y-derivative of equation (2), the time derivative of equation (3) and eliminating all mixed derivatives.…”

Section: Continuummentioning

confidence: 99%

Spurious numerical refraction

Cathers

Bates²

1995

Numerical Methods in Fluids

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SUMMARYThe purpose of this paper is to investigate the effect of a non-uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model.The domain consists of two conterminous regions discernible only by their different nodal spacings in the xdirection. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank-Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and transmitted waves shows that energy is conserved across the interface between the two regions.KEYWORDS: spurious wave refraction; total internal reflection INTRODUCTKlNAn appreciation of the effects of a non-uniform mesh is fundamental to a good understanding of the processes occumng in numerical models with varying mesh sizes. This is particularly relevant to time dependent finite element (FE) models and interactively nested finite difference (FD) models (in contradistinction to passively nested FD models). Practioners are often faced with questions such as 'how much can I change the mesh size and what are the consequences?'In 1D linear systems, these questions have been partly addressed.'-3 The analyses were completed for the simplified situation of a 1D domain consisting of two semi-infinite regions abutting at a common interfacial node. The two regions were discernible on numerical grounds (e.g. different mesh spacings or different numerical algorithms) or physical grounds (effected by an abrupt change in coefficient in the governing equation). The latter case is not considered herein and hence the inclusion of the word 'spurious' in the title of the present paper.The magnitude of the effects of a change in 1D mesh size were quantified by determining the amplitudes of the transmitted and (unwanted) reflected waves, and their associated energy fluxes due to an incident wave. In a more general sense, the incident wave may be interpreted as one of many Fourier components which can be superimposed to make up a general waveform.In ID, a change in mesh size results in both wave rejection and wave transmission. In 2D however, there is an additional process at work viz wave refraction. The interface between the two conterminous

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“…The wave equation is obtained by taking the x-derivative of equation (l), the y-derivative of equation (2), the time derivative of equation (3) and eliminating all mixed derivatives.…”

Section: Continuummentioning

confidence: 99%