Student(s):
Tracy Babb
Dates of Involvement:
June 2009 to May 2010
Faculty Advisor(s):
Gunnar Martinsson
Graduate Mentor:
Patrick Young
Numerical Evaluation Of Lattice Green's Functions
Introduction
This research project is aimed at finding efficient techniques for evaluating the lattice Green’s function related to the solution of the discrete Laplacian. Given a function 
where Z is the set of all integers, define the discrete Laplace operator A as
where 
are the coordinates of 
in the plane, and 
and 
. Then the solution to the equation
, 
(assuming 
is of the proper form) is given by
where 
is the lattice Green’s function,
When 
and 
are large, asymptotic expansions exist for determining . However, when and are small, numerical techniques must be utilized. [1] This is the focus of this project.
Gaussian Quadrature Technique:
To evaluate the lattice Green’s function a 15 point Gaussian quadrature is used. A couple things to note for evaluating the integral is that the integrand is even about the T1 and T2 axes, and therefore the integral may be evaluated in one quadrant and multiplied by four. Also of note is that the integrand is very oscillatory and has a singularity at the origin. Therefore the numerical integration will integrate the region while leaving out a small area around the origin. The difficulties encountered in the numerical integration may be seen in figure 1.
To deal with the aforementioned issues, a method will be used where the region is split up into many square subregions which are made small enough such that the integral is accurate to the desired tolerance. The exact size of each region is determined in the following manner. The total region is split up into a 4 by 4 grid, with each region being equivalent in area. In one of the quadrants, the function is evaluated using the Gaussian quadrature over the three subregions on the edge of the quadrant (not the one in contact with the origin). The value for any one of the subregions is not very accurate, so each subregion is split into a 2 by 2 grid of subregions, and each one of these is evaluated, and the total value is compared to the value obtained from integrating the region without subdividing it. If this new value is within 10-15 of the first value, the value is suitable. However, it is likely that this won't be suitable. So the region may now be divided into a 4 by 4 grid and each subregion may be evaluated. The region is continuously divided until subsequent values are within 10-15 of each other. However, it is very likely that if a region is divided over and over again, the subregions get so small that the roundoff error becomes significant and subsequent values will never be within such a small tolerance. Therefore a maximum grid of 16 by 16 is allowed. After this the method moves to the inner region and integrates this in the same way the quadrant was integrated. This method is continually repeated until the value of the lattice Green’s function converges.
Acceleration With Richardson Extrapolation:
One downside to the method used is that it becomes extremely slow as the region being left out approaches the origin. To expedite the process, a Richardson extrapolation may be used. Using the interpolating polynomial for the value I as a function of the space around the origin being left out, the values of I may be stored in a matrix where yk=.5ky0 and A(k,1)=I(yk). Then the formula for the rest of the matrix can be shown to be
The most accurate values for the function will be in the I(1,j) slot where j is the maximum value of j so far computed. The lattice Green’s function converges immensely faster when a Richardson extrapolation is performed. In fact, the Richardson extrapolation will converge to the correct value by approximately I(.4), but without using a Richardson extrapolation the value never converges. Values obtained for the lattice Green’s function via this method are all within our desired tolerance of 10-15 of the values obtained via other methods. This is an extremely good method since it is guaranteed to be accurate since the value over each region is refined until it converges and the speed is still very fast compared to other methods. Other methods required a very small box size through the whole region since it was unknown what box size the values converged at, but this method stops when it hits the converging box size value for each region. The method spends time evaluating the function over a region using larger box sizes and then not using these values but the time needed to do the first several refinements in extremely small and does not slow the process down.
Future research directions:
Now that this problem has been completed a different project is being started. The most likely project is to take a galaxy and consider all the objects in the galaxy. This is an extraordinary number and each one contributes a force of gravity to every other object. Trying to determine all the forces on all the objects is an extremely difficult problem.
About Tracy Babb:
Tracy Babb is a sophomore pursuing a double degree in Applied Math and Aerospace Engineering. Tracy doesn’t know what he wants to do with his future but he loves math and is interested in aerospace and he hopes he can find interesting ways to apply his love of math to Aerospace Engineering.
References:
1. http://amath.colorado.edu/faculty/martinss/Pubs/2002_phdthesis.pdf
