[Users] Improve finite differencing in Exact

[Erik Schnetter]

Peter

Thanks for the pointer. I attach a patch that corrects this problem.

-erik

On Wed, Jun 2, 2010 at 10:45 AM, Peter Diener <diener at cct.lsu.edu> wrote:
> Hi Erik,
>
> the patch is not correct in the case of 4th order derivatives. One of dt,
> dx, dy and dz are set to eps (the rest to zero) and then when you call
> Exact_metric you pass in points with an extra eps multiplied (x-2*dx*eps for
> example). The end result is that the extrinsic curvature is eps times to
> small.
>
> Cheers,
>
>  Peter
>
>
> On Tue, 1 Jun 2010, Erik Schnetter wrote:
>
>> The enclosed patch improves the finite differencing that is used in Exact
>> to calculate the extrinsic curvature.  These finite differences are
>> independent of the Cactus grid spacing.  This is prompted by a remark by
>> Peter Diener that the current finite differencing accuracy for the extrinsic
>> curvature from thorn Exact is only about nine digits.
>>
>> It introduces new parameters "exact_order" and "exact_eps" that choose the
>> order (2nd or 4th) and stencil size (default 1e-6).  It also introduces new
>> functions Exact__metric_deriv and Exact__metric_deriv2 that evaluate these
>> finite differences.
>>
>> Some finite differencing terms for the spatial derivative of the shift
>> have to be rewritten, which may lead to differences on the order of the
>> finite differencing accuracy.
>>
>> The code could be rewritten to evaluate each stencil point only once in
>> all calls to the new finite differencing operators.  This would reduce the
>> number of evaluations.  Since thorn Exact is often very fast, this does not
>> seem necessary at this point.
>>
>> Unfortunately, this patch causes five of the AHFinderDirect test cases to
>> fail.
>>
>> -erik
>>
>> --
>> Erik Schnetter <schnetter at cct.lsu.edu>   http://www.cct.lsu.edu/~eschnett/
>>
>>
>



-- 
Erik Schnetter <schnetter at cct.lsu.edu>   http://www.cct.lsu.edu/~eschnett/
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