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Internal wave reflections and transmissions arising from a non‐uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme
Author(s) -
Cathers B.,
Bates S.,
O'Connor B. A.
Publication year - 1989
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650090704
Subject(s) - polygon mesh , finite element method , mathematics , crank , wavelength , reflection (computer programming) , mathematical analysis , geometry , context (archaeology) , contraction (grammar) , mesh generation , physics , optics , computer science , geology , medicine , cylinder , thermodynamics , programming language , paleontology
This is the first of a series of three related papers dealing with some of the consequences of non‐uniform meshes in a numerical model. In this paper the accuracy of the Crank–Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths (2Δ x , 4Δ x 8Δ x ) in three ‘hot‐start’ numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.