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Fourier analysis of the Eulerian–Lagrangian least squares collocation method
Author(s) -
Bentley L. R.,
Aldama A.,
Pinder G. F.
Publication year - 1990
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.1650110406
Subject(s) - mathematics , discretization , collocation (remote sensing) , mathematical analysis , eulerian path , fourier series , spectral method , courant–friedrichs–lewy condition , fourier transform , wavenumber , fourier analysis , physics , remote sensing , lagrangian , optics , geology
A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian–Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C 1 ‐continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one‐dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability. Unconditional stability for time‐stepping weights greater than or equal to 0‐5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δ x . As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher‐wave‐number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank–Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.