13 December 1999
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Physics Letters A 264 1999 36–44
www.elsevier.nlrlocaterphysleta
Area preservation in computational fluid dynamics
Robert I. McLachlan
1
Mathematics, Institute of Fundamental Sciences, Massey UniÕersity, Palmerston North, New Zealand
Received 7 June 1999; accepted 20 October 1999
Communicated by A.R. Bishop
Abstract
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Incompressible two-dimensional flows such as the advection Liouville equation and the Euler equations have a large
family of conservation laws related to conservation of area. We present two Eulerian numerical methods which preserve a
discrete analog of area. The first is a fully discrete model based on a rearrangement of cells; the second is more
conventional, but still preserves the area within each contour of the vorticity field. Initial tests indicate that both methods
suppress the formation of spurious oscillations in the field. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
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When a smooth field
v
x, y is advected by an
area-preserving flow, the area within each contour of
v
is preserved. This is seen in pure advection and in
the Euler equations, for example, and is important in
the numerical solution of two-phase free boundary
problems, where the total volume of each fluid should
be preserved. Yet, although the advection problem
has been addressed in probably thousands of papers,
and very accurate, stable, and efficient methods are
known, no existing numerical methods take the
area-preservation property into account. In this Letter
we present an initial study containing two methods
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which do preserve a discrete analog of area. Al-
though they are not, presumably, competitive with
1
E-mail: r.mclachlan@massey.ac.nz
the best existing methods for advection, the results
are extremely promising.
The configuration space of an inviscid incom-
pressible fluid is the group D
D
of volume-preserving
m
diffeomorphisms of the fluid’s domain; the ‘Arnold’
picture, in which the Euler equations are geodesic
equations on this group equipped with the kinetic
Ž. wx
energy L metric, is treated in Ref. 2 . The config-
2
uration at any time is a volume-preserving rearrange-
ment of the initial condition. Existing Eulerian
numerical methods do not preserve any discrete ana-
logue of this property. This is particularly relevant in
two dimensions, where area preservation leads to an
infinite number of conserved quantities, the general-
ized enstrophies.
We consider a two-dimensional fluid with diver-
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gence-free velocity field u s u,Õ , stream function
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c
i.e. us
c
, Õsy
c
, and some quantity
v
,
yx
which we call the vorticity, which is advected by the
fluid:
v
qu P=
v
s
v
qJ
v
,
c
s0, 1
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˙˙
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
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PII: S0375-9601 99 00763-X