Hello Ian,
While working on a hydro code for CarpetX the people involved (in cc via the mailing list) ran into the issue of how to handle the lapse the show up in the flux of the fluid equations.
E.g. in GRHydro (nee Whisky) one uses a numerical flux of (say for D) in https://www.einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentat...
as F^i_D = D (\alpha v^i - \beta^i) yet the flux expressions used in the Riemann problem are computed as:
\tilde F^i_D = D (v^i - \beta^i/\alpha)
ie a factor of the lapse has been divided out.
It gets put back when the fluxes are combined to compute a right hand side by using:
RHS = \alpha_{i-1/2} F^j(x_{i-1/2}) - \alpha_{i+1/2} F^j(x_{i+1/2})
None of the CarpetX developers (with the possible exception of Erik, but his expertise is in a different area and in any case he does not remember) were involved in the original Whisky code.
So: do you remember if there was a specific reason why the lapse was divided out in the expressions used in the Riemann problem?
Yours, Roland
(To clarify, I am not one of the original authors of Whisky.)
-erik
On Tue, Oct 19, 2021 at 3:01 PM Roland Haas rhaas@illinois.edu wrote:
Hello Ian,
While working on a hydro code for CarpetX the people involved (in cc via the mailing list) ran into the issue of how to handle the lapse the show up in the flux of the fluid equations.
E.g. in GRHydro (nee Whisky) one uses a numerical flux of (say for D) in https://www.einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentat...
as F^i_D = D (\alpha v^i - \beta^i) yet the flux expressions used in the Riemann problem are computed as:
\tilde F^i_D = D (v^i - \beta^i/\alpha)
ie a factor of the lapse has been divided out.
It gets put back when the fluxes are combined to compute a right hand side by using:
RHS = \alpha_{i-1/2} F^j(x_{i-1/2}) - \alpha_{i+1/2} F^j(x_{i+1/2})
None of the CarpetX developers (with the possible exception of Erik, but his expertise is in a different area and in any case he does not remember) were involved in the original Whisky code.
So: do you remember if there was a specific reason why the lapse was divided out in the expressions used in the Riemann problem?
Yours, Roland
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Hello Ian,
Did you already have time to try and see if you remember any of this anymore?
If you already responded and were e.g. blocked by the mailing list: my apologies. In that case if you were to respond to me (rhaas@illinois.edu) I will forward to to the list (and add you to the allowed senders list).
Yours, Roland
Hello Ian,
While working on a hydro code for CarpetX the people involved (in cc via the mailing list) ran into the issue of how to handle the lapse the show up in the flux of the fluid equations.
E.g. in GRHydro (nee Whisky) one uses a numerical flux of (say for D) in https://www.einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentat...
as F^i_D = D (\alpha v^i - \beta^i) yet the flux expressions used in the Riemann problem are computed as:
\tilde F^i_D = D (v^i - \beta^i/\alpha)
ie a factor of the lapse has been divided out.
It gets put back when the fluxes are combined to compute a right hand side by using:
RHS = \alpha_{i-1/2} F^j(x_{i-1/2}) - \alpha_{i+1/2} F^j(x_{i+1/2})
None of the CarpetX developers (with the possible exception of Erik, but his expertise is in a different area and in any case he does not remember) were involved in the original Whisky code.
So: do you remember if there was a specific reason why the lapse was divided out in the expressions used in the Riemann problem?
Yours, Roland
My memory is that the idea is related to the smoothness of the metric/gauge terms: these should be C^2, so are sufficiently differentiable to be "best" approximated (at the second order that Whisky/GRHydro worked to) by the midpoint rule. However, if included within the fluxes then they would implicitly go through the reconstruction/Riemann solve stage, so would - in places - be limited to lower accuracy.
I would not do this now - keeping the fully densitized terms together is cleaner and a better way to extend to high order.
Ian ________________________________ From: Roland Haas rhaas@illinois.edu Sent: 31 October 2021 20:28 To: Ian Hawke I.Hawke@soton.ac.uk Cc: carpetx-developers@einsteintoolkit.org carpetx-developers@einsteintoolkit.org Subject: Re: lapse terms in Whisky's fluxes
CAUTION: This e-mail originated outside the University of Southampton.
Hello Ian,
My memory is that the idea is related to the smoothness of the metric/gauge terms: these should be C^2, so are sufficiently differentiable to be "best" approximated (at the second order that Whisky/GRHydro worked to) by the midpoint rule.
Ok, that is partially what we are doing (ie we do not reconstruct lapse explicitly (it is not stored as a volume average anyway) and compute it as just the next neighbour average. The confusion for us was that essentially GRHydro / Whisky do something like this:
num_flux_(i-1/2) = FOO_(i-1/2)(prims, gxx, beta/lapse) num_flux_(i+1/2) = FOO_(i+1/2)(prims, gxx, beta/lapse)
where num_flux is the result of the Rieman solver and reconstruction etc.
*Then* when it computes the actual right hand side terms for MoL it uses:
(1)
RHS_i = 0.5*(lapse_(i) + lapse_(i+1)) num_flux(i+1/2) - 0.5*(lapse_(i) + lapse_(i-1)) num_flux(i-1/2)
instead of the more natural (given the papers and also GRHydro docs):
(2)
RHS_i = num_flux(i+1/2) - num_flux(i-1/2)
(the two num_flux in method (1) and method (2) would differ in that one absorbed the extra lapse term while the other did not). There are always some metric terms in the fluxes (eg shift is always present and so of course is a component of the 3-metric itself).
However, if included within the fluxes then they would implicitly go through the reconstruction/Riemann solve stage, so would - in places - be limited to lower accuracy.
I would not do this now - keeping the fully densitized terms together is cleaner and a better way to extend to high order.
Ok, so from this I understand that we are ok doing things "different" from GRHydro and use method (2), at least until we have evidence for one method being clearly superior to the other. At least there is no clear "you must never do this" no go claim.
Yours, Roland
Yes; the "difference" is that the "num_flux" in the two cases are different (one will contain the lapse, the other not, as undensitized) [I think]. So the implemented case with the explicit averaged lapse at each interface should directly approximate the standard form, but effectively uses a different reconstruction approach for gauge vs the rest of the flux.
I have memories of trying both approaches when playing with WENO reconstruction early on, and it making little difference. There should be no problem with scaling by any term that does not explicitly depend on terms within the hydro state vector. ________________________________ From: Roland Haas rhaas@illinois.edu Sent: 03 November 2021 15:31 To: Ian Hawke I.Hawke@soton.ac.uk Cc: carpetx-developers@einsteintoolkit.org carpetx-developers@einsteintoolkit.org Subject: Re: lapse terms in Whisky's fluxes
CAUTION: This e-mail originated outside the University of Southampton.
Ian, thanks a lot for your help.
Cheers, Bruno
Il giorno mer 3 nov 2021 alle ore 16:39 Ian Hawke I.Hawke@soton.ac.uk ha scritto:
Yes; the "difference" is that the "num_flux" in the two cases are different (one will contain the lapse, the other not, as undensitized) [I think]. So the implemented case with the explicit averaged lapse at each interface should directly approximate the standard form, but effectively uses a different reconstruction approach for gauge vs the rest of the flux.
I have memories of trying both approaches when playing with WENO reconstruction early on, and it making little difference. There should be no problem with scaling by any term that does not explicitly depend on terms within the hydro state vector.
*From:* Roland Haas rhaas@illinois.edu *Sent:* 03 November 2021 15:31 *To:* Ian Hawke I.Hawke@soton.ac.uk *Cc:* carpetx-developers@einsteintoolkit.org < carpetx-developers@einsteintoolkit.org> *Subject:* Re: lapse terms in Whisky's fluxes
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