A variational free‐Lagrange method for shallow water

Abstract

This article presents a variational free-Lagrange (VFL) method for rotating shallow water. This method was first derived by Augenbaum (2) who discretised Hamilton's action principle with a free-Lagrange data structure. The novel feature of our approach is to exploit the geometric structure preserved by the VFL method to the effect of conserving energy over long-time simulations. Numerical results demonstrate this property and motivate the extension of this approach to other variational formulations of climate models.

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Proceedings in Applied Mathematics and Mechanics, 3 January 2008
A Variational Free-Lagrange Method for Shallow Water
Matthew F Dixon∗1and Todd Ringler∗∗2†
1Institute for Mathematical and Computational Engineering, Durand Building, 496 Lomita Mall, Stanford University, Stan-
ford, CA 94305-4042, USA
2Climate, Ocean, and Sea Ice Modeling Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM ,
USA
This article presents a variational free-Lagrange (VFL) method for rotating shallow water. This method was first derived by
Augenbaum (2) who discretised Hamilton’s action principle with a free-Lagrange data structure. The novel feature of our
approach is to exploit the geometric structure preserved by the VFL method to the effect of conserving energy over long-time
simulations. Numerical results demonstrate this property and motivate the extension of this approach to other variational
formulations of climate models.
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1Introduction
The variational approach
challenging and active area of research. The traditional approach to deriving such methods requires the numerical analyst to
exercise some level of their own bias. To remedy this and systematically derive the numerical method from a conferred math-
ematical understanding, Salmon (1) pursues the variational approach for geophysical modeling. The power of the variational
approach rests upon Noether’s theorem which states that each continuous symmetry of Hamilton’s action principle exhibits a
corresponding conservation law. Consequently, the numerical analyst need only choose a suitable discretisation of Hamilton’s
action principle, with the confidence that not only will the resulting semi-discrete equations of motion preserve geometric
structure, manifesting in energy conservation, but that these equations will conserve additional quantities corresponding to the
symmetries of the discrete action principle.
The derivation and analysis of conservative numerical methods for global climate modeling is a
Voronoi diagram
A distinguishing feature of the free-Lagrange method is the use of a Voronoi diagram to represent the layer thickness. The
Voronoi diagram is attractive from both a practical and theoretical perspective. Firstly the approach is completely free from
mesh-tangling problems (see 3) which compromise the performance of many of the mesh-based Lagrangian methods. Addi-
tionally, (4) shows that the corresponding discrete Hamilton’s action principle permits translational and rotational symmetries
and hence respective linear and angular momentum conservation laws.
Following (2), we firstly outline a variational free-Lagrange (VFL) method for rotating shallow water.
Symplectic integrator
symplectic time-stepper (see 5, for a review of symplectic methods for particle methods). We demonstrate by numerical
experiment that the combination of the VFL method with an explicit symplectic time-stepper exhibits no secular drift in
the energy error. This property together with the computational tractability renders this approach suitable for long-time
geophysical simulations.
Our contribution is to exploit the symplectic structure preserved by the VFL method by using a
2 The Free-Lagrange Method
Consider a particle representation of a fluid in which N2particles or sites X = {X1,...,XN2} are individually labelled
by α. Each particle is inside a polygon containing the set of points Xβcloser to Xαthan to any other site. The polygon is
referred to as a Voronoi cell and the set of all these closed cells is the Voronoi diagram. A hexagonal Voronoi cell is shown in
Figure 1 together with the specification of a local index for referring to neighbouring particles.
The free-Lagrange method simply uses the particles to represent the material velocity field Uαand the Voronoi diagram to
construct the piecewise constant cellular layer thickness¯h(Xα,t) from the principle of cell mass mαconservation. Discreti-
sation of Hamilton’s action principle for rotating shallow water with this free-Lagrange fluid representation is the critical step
in deriving a VFL method.
∗Corresponding author: e-mail: mfrdixon@stanford.edu, Phone: +001 650 353 6949, Fax: +001 650 723 1778.
∗∗e-mail: ringler@lanl.gov, Phone: +001 505 667 7744.
†Los Alamos Report LA-UR 07-7482.
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PAMM header will be provided by the publisher2
3A Variational Free-Lagrange method for 2D Shallow Water
The semi-discrete material description of Hamilton’s action principle for 2d rotating shallow water in a f-plane is expressed in
terms of the particle material velocities and piecewise constant cellular layer thickness.
Definition 3.1 (The Semi-Discrete Hamilton’s Action Principle for Shallow Water)
Sd=1
2
?tb
ta
dt
?
α
mα
?|Uα(t)|2+ 2Rα· Uα
?− V,V =g
2
?
α
mα
?¯h(Xα,t) + 2¯bα).
(1)
Under the f-plane approximation, ∇ × R = f0ˆk,ˆk is the unit vector in the direction of gravity and f0is the Coriolis
parameter which is given by f0 = 2|ω|, where ω is the angular velocity of rotation relative to an inertial frame. g is the
gravitation constant,¯bαis the bottom topography, assumed to be piecewise constant over cell α and V is the potential energy.
Stationarity of the discrete action principle gives the semi-discrete Euler-Lagrange particle equations
g
mα
i
˙Uα=
?
?m¯h?βi
αdnβi
α− f0ˆk × Uα,
˙Xα= Uα,
(2)
where dnβi
a scalar quantity at the βth
By preserving symplectic structure, it follows that these semi-discrete Euler-Lagrange particle equations conserve energy.
A symplectic time-stepper, such as the explicit second order St¨ ormer-Verlet scheme, in particle position and canonical mo-
mentum co-ordinates (Xn
α:= ˆ nβi
α∆lβi
α, ∆lβi
i edge of cell α as the mean of that quantity over cell α and cell βi.
αis the length of the side indexed by βiof cell α and the operator [·]βi
α=1
2(·α+ ·βi) evaluates
α,Pn
α= mαUn
α) at time tn
[1]
Pn+1
α
2
= Pn
α−∆t
2VX(Xn
α), [2]
Xn+1
α
= Xn
α+∆tPn+1
2
α
mα
, [3]
Pn+1
α
= Pn+1
2
α
−∆t
2VX(Xn+1
α
),
preserves the symplectic two-form ωn=?N2
quently, this time-stepper will ensure the absence of secular drift in the energy error of the fully discrete equations over long
simulations as the graph in Figure 2 shows.
α=1dXn
α∧ dPn
α, where VXdenotes the gradient of the potential energy. Conse-
ˆ nβ5
α
β = 1
Xα
β = 2
β = 3
β = 4
β = 5
β = 6
∆lβ1
α
Xβ
Fig. 1
α. Each cell edge is of length ∆lβi
A hexagonal (ne = 6) Voronoi cell containing the particle with label
α and labelled by βi.
012345
t
678910
x 105
−6
−4
−2
0
2
4
6
8x 10−4
Energy Error
E−E0
Fig. 2
time steps of size t = 0.01 using 128 cells. The shallow water covers a peri-
odic domain [0,2π) rotating with f0 = 2π and is initialized with a gaussian
perturbation of the layer thickness. The error exhibits no secular drift in energy.
The graph shows the energy error of a 1D simulation over one million
Acknowledgements
The authors would like to thank Mark Petersen for organising the minisymposium.
References
[1] R. Salmon, J. Fluid Mech. 132, 431–444 (1983).
[2] J. Augenbaum, J. Comp. Phys. 53, 240–265 (1984).
[3] O. Harlen, J. Rallison, and P. Szabo, J. Non-Newt. Fluid Mech. 60, 81–104 (1995).
[4] M. Serrano, P. Espa˜ nol, and I. Z´ u˜ niga, J. Stat. Phys. 121, 133–147 (2005).
[5] M. Dixon and S. Reich, GAMM 27(1), 9–24 (2004).
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