Re: [Users] looking for help with CT_MultiLevel
Thanks again for the reply, Eloisa. I think I'm set now. Kyle On 06/24/2016 04:19 PM, Eloisa Bentivegna wrote: > On 23/06/16 22:06, Slinker, Kyle Patrick wrote: >> Thanks for the reply, Eloisa. >> >> I wasn't thinking of solving the elliptic equation in terms of the >> stationary state of a parabolic equation as you described in your paper. >> But, I see now how Gauss-Seidel for the elliptic equation can be derived >> from finite differencing the parabolic equation. Now that I think I'm on >> the same page in those terms, let me see if I can rephrase the issue I'm >> seeing. >> >> I tried a couple times to write something, but the best explanation I >> came up with is an example. I attached a short PDF walking through it. >> >> Thanks again for your help. > Dear Kyle, > > I've followed your reasoning, but I don't see where equation (2) comes > from. To the best of my knowledge, one is not free to construct an > iterative process by deforming the differencing stencils at will. The > existing recipes (like Gauss-Seidel) are carefully crafted to have > specific properties; you can, for instance, read on the Numerical > Recipes book (equation 20.5.4 and following, in the third edition) what > dtime needs to be set to for a stable evolution. This has to do with the > stability of the Forward-Time-Centered-Space representation of the equation. > > Notice, however, that what dtime is set to in CT_MultiLevel is only the > largest admissible value. One is free to decrease this number (although > that would require more iterations to relax to the same state); your > suggestion for the coefficient, for instance, would also work. And you > are right to point out that a change in dtime is equivalent to a change > in the SOR omega (which, however, also cannot be chosen arbitrarily). > > I hope this clarifies the issue! > > Eloisa > ___ Users mailing list Users@einsteintoolkit.org http://lists.einsteintoolkit.org/mailman/listinfo/users
Re: [Users] looking for help with CT_MultiLevel
On 23/06/16 22:06, Slinker, Kyle Patrick wrote: > Thanks for the reply, Eloisa. > > I wasn't thinking of solving the elliptic equation in terms of the > stationary state of a parabolic equation as you described in your paper. > But, I see now how Gauss-Seidel for the elliptic equation can be derived > from finite differencing the parabolic equation. Now that I think I'm on > the same page in those terms, let me see if I can rephrase the issue I'm > seeing. > > I tried a couple times to write something, but the best explanation I > came up with is an example. I attached a short PDF walking through it. > > Thanks again for your help. Dear Kyle, I've followed your reasoning, but I don't see where equation (2) comes from. To the best of my knowledge, one is not free to construct an iterative process by deforming the differencing stencils at will. The existing recipes (like Gauss-Seidel) are carefully crafted to have specific properties; you can, for instance, read on the Numerical Recipes book (equation 20.5.4 and following, in the third edition) what dtime needs to be set to for a stable evolution. This has to do with the stability of the Forward-Time-Centered-Space representation of the equation. Notice, however, that what dtime is set to in CT_MultiLevel is only the largest admissible value. One is free to decrease this number (although that would require more iterations to relax to the same state); your suggestion for the coefficient, for instance, would also work. And you are right to point out that a change in dtime is equivalent to a change in the SOR omega (which, however, also cannot be chosen arbitrarily). I hope this clarifies the issue! Eloisa ___ Users mailing list Users@einsteintoolkit.org http://lists.einsteintoolkit.org/mailman/listinfo/users
Re: [Users] looking for help with CT_MultiLevel
Thanks for the reply, Eloisa.
I wasn't thinking of solving the elliptic equation in terms of the
stationary state of a parabolic equation as you described in your paper.
But, I see now how Gauss-Seidel for the elliptic equation can be derived
from finite differencing the parabolic equation. Now that I think I'm on
the same page in those terms, let me see if I can rephrase the issue I'm
seeing.
I tried a couple times to write something, but the best explanation I
came up with is an example. I attached a short PDF walking through it.
Thanks again for your help.
Kyle
On 06/22/2016 03:41 PM, Eloisa Bentivegna wrote:
> On 22/06/16 16:42, Slinker, Kyle Patrick wrote:
>> I have been working with CT_MultiLevel in an attempt to solve the
>> non-conformally-flat constraint equations and I'm wondering if there is
>> an error in the thorn. Where I'm potentially seeing an issue is in
>> CT_SolvePsiEquation.cc and CT_SolveErrorEquation.cc in the definitions
>> of dtime (for both files this is line 33 in the Brahe release).
>>
>> First of all, having 0.5 in this definition means that the coefficient
>> for the central point in the finite difference is hard-coded to a second
>> order stencil. When using fourth order, this coefficient should be 0.4,
>> I believe. This effectively changes the SOR omega.
>>
>> More importantly, I don't think it accounts for ct_cxx, ct_cyy, or
>> ct_czz. So the central point in the finite difference is not getting
>> multiplied by whatever coefficient the user is setting.
>>
>> Am I misunderstanding how the thorn works or are these really problems?
>> I've attached what I believe is a way to fix them. Thanks for your help.
> Dear Kyle,
>
> thanks for the comments. I am not sure I understand your points though:
> in CT_MultiLevel, fd_order is used to control the approximation order of
> spatial derivatives, while dtime appears in the way the "time"
> derivative is discretized. These two are in principle independent.
>
> Can you clarify which derivative you believe is wrong?
>
> Thanks,
> Eloisa
> ___
> Users mailing list
> Users@einsteintoolkit.org
> http://lists.einsteintoolkit.org/mailman/listinfo/users
>
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Re: [Users] looking for help with CT_MultiLevel
On 22/06/16 16:42, Slinker, Kyle Patrick wrote: > I have been working with CT_MultiLevel in an attempt to solve the > non-conformally-flat constraint equations and I'm wondering if there is > an error in the thorn. Where I'm potentially seeing an issue is in > CT_SolvePsiEquation.cc and CT_SolveErrorEquation.cc in the definitions > of dtime (for both files this is line 33 in the Brahe release). > > First of all, having 0.5 in this definition means that the coefficient > for the central point in the finite difference is hard-coded to a second > order stencil. When using fourth order, this coefficient should be 0.4, > I believe. This effectively changes the SOR omega. > > More importantly, I don't think it accounts for ct_cxx, ct_cyy, or > ct_czz. So the central point in the finite difference is not getting > multiplied by whatever coefficient the user is setting. > > Am I misunderstanding how the thorn works or are these really problems? > I've attached what I believe is a way to fix them. Thanks for your help. Dear Kyle, thanks for the comments. I am not sure I understand your points though: in CT_MultiLevel, fd_order is used to control the approximation order of spatial derivatives, while dtime appears in the way the "time" derivative is discretized. These two are in principle independent. Can you clarify which derivative you believe is wrong? Thanks, Eloisa ___ Users mailing list Users@einsteintoolkit.org http://lists.einsteintoolkit.org/mailman/listinfo/users
