The finite difference approximation for the second derivative at point xi using a Taylor series is as follows:

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Automatic Generation of an Accurate Numerical
Approximation of the Laplacian Operator
Replace w/ Logo
Alice DuVivier, Jette Petersen, David Randall,
Ross Heikes CMMAP
Purpose
Results
Mathematical Methods
The goal of this project was to write a program
in Mathematica to calculate the Laplacian for the
twisted icosahedral grid. This will improve
accuracy of the current climate models. In
addition, the calculated approximation can be
stored and kept for reference in future models.
Both of these will decrease the cost of a
particular model both in time and in the number
of processors required to run the model.
Finite Difference Approximation
Using Mathematica, we wrote a program to
simultaneously find the Laplacian at each point
on a 1D grid. This model gives promising results
for a third order accurate scheme and non uniform
point spacing.
For both uniform grid spacing and non uniform
grid spacing our approximation gives values of
the Laplacian at each point that are very close
to the true value when tested with a function.
Figures 2 shows the uniform spaced case.
Background
The finite difference approximation for the
second derivative at point xi using a Taylor
series is as follows
The Laplacian operator, ?2, is a second order
differential operator. The Laplacian operator was
chosen because it is frequently used in
atmospheric models and relatively
straightforward. However, because data on the
sphere are in the form of scalar or vector values
we must use finite difference quotients between
values at particular points to numerically
approximate the second derivative.
We sum over the neighboring points about the
center point. The value cj is the weight of a
particular particular point in relation to the
center of expansion. We solve for cj by imposing
conditions dictated by the desired order of
accuracy. For first order accuracy in one
dimension the necessary conditions are
Figure 3 shows the non-uniformly spaced case.
A twisted icosahedral grid can eliminate the so
called pole problem associated with solving
differential equations in spherical geometry.
This scheme is essentially a 2D hexagonal grid
that has been wrapped around an icosahedron
representing the sphere (see figure 1).
Figure 1 The twisted icosahedral grid. In the
process of being wrapped about the icosahedron,
the hexagonal grid cells become non-uniformly
spaced and twelve hexagons are introduced into
the grid. Image courtesy of Ross Heikes
Future Work
The weights are the solutions to the above set of
linear equations. Once these weights are known
the Laplacian at a particular point, i, is found
by using the following straightforward formula
The algorithm for finding the Laplacian has only
been written for the 1D case. We must now extend
the program to a 2D hexagonal grid. Ultimately,
the program will approximate the Laplacian on the
twisted icosahedral grid. Once work has been
completed for the Laplacian operator this method
can be extended to other advection operators.
To find the value of mathematical operators used
in the models, such as the Laplacian, at a
particular point on the grid we use weighted
numerical values from the surrounding grid
points. Using numerical approximation methods, we
can assign weights to these neighboring points
based on their distance from the point of
interest and then find a weighted average for our
point of interest. Developing a numerical method
for approximating these non-uniform values is
essential in improving models efficiency.
By imposing conditions on the coefficients of the
Taylor expansion, our program solves for these
weights and then finds the Laplacian at each
point on the surface to third order accuracy.
Thanks to CMMAP for supporting this work.