<html>#2368: Instabilities with Summation by Parts thorn 4th order operator
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<tr><td style='text-align:right'> Reporter:</td><td>Miguel Zilhão</td></tr>
<tr><td style='text-align:right'> Status:</td><td>new</td></tr>
<tr><td style='text-align:right'>Milestone:</td><td></td></tr>
<tr><td style='text-align:right'> Version:</td><td></td></tr>
<tr><td style='text-align:right'> Type:</td><td>bug</td></tr>
<tr><td style='text-align:right'> Priority:</td><td>major</td></tr>
<tr><td style='text-align:right'>Component:</td><td>EinsteinToolkit thorn</td></tr>
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<p>Comment (by Peter Diener):</p>
<p>I’ve checked the optimized diagonal 4th order second derivative operator against the mathematica notebook originally used to derive it and didn’t find any mistakes. However, the 4th order 2nd derivative operator is the only 2nd derivative operator that has both a minimal bandwidth and an optimized operator. The other 2nd derivative operators have only minimal bandwidth operators. In order to explain further, when the SBP operators are derived many have free parameters that can be chosen freely. For the 1st derivative diagonal operators the 2-1 and 4-2 operators are unique, while the 6-3 and 8-4 operators have free parameters. One way to choose the parameters is to set as many coefficients to zero (called minimal bandwidth) , another is to choose the parameters so the sum of the squares of the truncation error coefficients is minimized (called optimal). For the 2nd derivative operators the 4-2, 6-3 and 8-4 operators have free parameters. For some reason an optimized ope
rator was only found for the 4-2 operators, while only minimal bandwidth operators were implemented for the 6-3 and 8-4 cases. Default parameter choices selects the optimized operators where implemented but uses the minimal bandwidth operators where they are not. I tried used the minimal bandwidth diagonal 2nd derivative operator (by setting SummationByParts::operator_type = "Minimal Bandwidth") and that seems to be stable.</p>
<p>It should be noted that stability of any of these operators are not guaranteed (i.e. proven) in multiple dimensions, so it is not impossible that SPB satisfying operators leads to unstable evolutions. However, it is surprising to me that using the free parameter to minimize the sum of squares of the truncation error coefficients leads to such dramatically different stability properties.</p>
<p>Miguel, can you confirm that you also get stable evolution for the minimal bandwidth 4th order diagonal operator by setting SummationByParts::operator_type = "Minimal Bandwidth"?</p>
<p>--<br/>
Ticket URL: <a href='https://bitbucket.org/einsteintoolkit/tickets/issues/2368/instabilities-with-summation-by-parts'>https://bitbucket.org/einsteintoolkit/tickets/issues/2368/instabilities-with-summation-by-parts</a></p>
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