[Users] McLachlan test suite failures

[Ian Hinder]

Hi,

I have solved the mystery of why the McLachlan test suites were  
failing on Kraken.  The root cause of the problem was, again, the fact  
that the Exact thorn which sets up the gauge wave initial data  
calculates the extrinsic curvature imprecisely.  The way the Exact  
thorn works is that the 4-metric is given in the code and is then  
finite differenced with a 2nd order accurate centered difference to  
compute the extrinsic curvature.  It does not use the analytic  
derivatives.  In order to make this accurate, a small step size, eps =  
1e-6, is chosen for the finite differencing.

Unfortunately, this small value means that g(t+eps) and g(t-eps) are  
very similar to each other, and when they are subtracted from each  
other, the least significant bits in g become very significant in K.   
As Peter pointed out, this can be many orders of magnitude.  So an  
error of 1e-15 caused by roundoff differences due to compilers, math  
libraries, etc in g can lead to an error of 1e-11 in K.  This is  
exactly what was happening with the McLachlan test suite.  kxx on the  
initial data, which has a maximum value of ~ 0.2, had differences  
between the test suite result and Kraken of O(1e-11).  This led to a  
similar difference in the BSSN RHS computation, and from iteration 0  
to iteration 1, the difference in gxx jumped 4 orders of magnitude.

To avoid this problem, Erik wrote a patch to implement fourth order  
accurate finite differencing in Exact.  The idea is that you can get  
the same truncation error in the computation of K by using a higher  
order scheme with a larger step size.  Since you are subtracting  
numbers which are not so similar to each other, the cancellation above  
does not occur, and roundoff errors are not amplified.

I have tested the patch with the gauge wave initial data by computing  
kxx with order = 4 and eps = 1e-1, 1e-2, 1e-3 and 1e-4.  The  
convergence with eps is 4th order, and a Richardson extrapolation  
reveals that the 1e-4 result has an error which is O(1e-15); i.e.  
using order = 4 and eps = 1e-4 for this initial data yields a result  
for kxx which is correct up to roundoff error.

I attach a plot of the difference in gxx beween using order = 2, eps =  
1e-6 and using order = 4, eps = 1e-4 between my laptop and Kraken.  We  
see that after the initial data, the difference jumps from 1e-15 to  
1e-12 with order = 2, eps = 1e-6, whereas with order = 4, eps = 1e-4,  
the difference remains small.

Since we are close to a release, Erik doesn't want to commit his patch  
at this point.  We understand why the McLachlan tests fail on kraken,  
and I have verified that with the patch, they can be made to pass.   
After the release, we will commit Erik's patch and modify the test  
suites to use it.

We are not sure how to proceed in general with Exact.  It is not clear  
that a single value of eps is suitable for all of the models in  
Exact.  The current patch could be improved to perform a Richardson  
extrapolation to estimate the error, but it is then not clear how to  
reliably calculate the relative error that we need to minimise.

-- 
Ian Hinder
ian.hinder at aei.mpg.de
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