Hyperbolicity of BSSN

March 23, 2011

by Ben

So, from what I understand it seems that strong and symmetric hyperbolicity of equations is only defined for systems with first order derivatives. The significance is that strong or symmetric hyperbolic systems of differential equations are “well posed” that is, roughly speaking, that they have unique solutions that depend continuously on the initial data (i.e. a small change in the initial data produces a small change in the solution). Of cause this is very important if one is running some numerical simulation since it ensures that no problems in the simulation can be the result of a problem with the equations (although there can be issues with the representation of the equations on the computer).

There is a formulation of Einstein’s equations, called the Baumgarte-Shapiro-Shibata-Nakamura or BSSN system. It is a system involving first order time derivatives and second order spatial derivatives and therefore it is not immediately obvious that the system should behave well in numerical simulations. But it does, the question is why?

Sarbach, Calabrese, Pullin and Tiglio solved this by showing in “Hyperbolicity of the Baumgarte-Shapiro-Shibata-Nakamura system of Einstein evolution equations” that the BSSN is equivalent to a subfamily of the Kidder-Scheel-Teukolsky (KST) formulation for the Einstein equations. Since the KST systems have first order spatial and first order time derivatives this equivalence allows one to state that the BSSN system has, in some sense, the strongly and symmetrically hyperbolic properties as the KST system. Thus for certain values of certian variables the BSSN system is well posed, that is the BSSN system is a theoretically a good system to work with. The paper thus supports a variety of numerical evidence showing this (check out the paper for references).

All that being said I stumbled across Gundlach and Martin-Garcia’s, “Symmetric hyperbolic form of systems of second-order evolution equations subject to constraints” which attempts to give a definition of strong and symmetric hyperbolicity for such systems as the BSSN system. A quick search on ISI of the citing papers doesn’t bring anything up so it seems at first glance that their paper has gone no where at least with regards to the theory (Correct me if I’m wrong!).

In any case I find all this very interesting and am constantly surprised by the depth of complexity which means that no general theory for this kind of stuff has been worked out.

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Analytic work on constraint quantities lacking?

March 8, 2011

by Ben

For the record I make no claims about the title but I do get that impression from the paper “Algebraic stability analysis of constrain propagation” by Frauendiener and Vogel.

The paper provides a general introduction to how to perform analysis of the attractors of some dynamical system (with specified initial conditions) and then applies this to a particular initial value formulation of Einstein’s equations. The analysis turns out to be very complicated so two techniques are used; assume that the Christoffel symbols vanish and use numerical investigation. I find the initial results intriguing and worthy of more than a close second look.

They show that in order for the Constraint quantities to tend to zero in some numerical approximation various inequalities must be satisfied. These inequalities are linked directly to the geometrical formulation of the evolution scheme! Sections 6.2 and 6.3 give two common formulations of evolution schemes in Minkowski space and show that for only one of these schemes does violation of the constraint equations reduce over time. As it happens this turns out to be the hyperboloidal formulation, which I have a fondness for. The consequences are that for a numerical evolution of Einstein’s equations to behave nicely with respect to the evolution of the constraint quantities certain common choices must be avoided.

The paper does leave me with the feeling that this kind of topic hasn’t been looked at much. Moreover, since this is the first paper I’ve read about such things, my own comments should be taken with more than just a grain of salt. In any case I want more…

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MOND v LambdaCDM

March 2, 2011

by Ben

I have a soft spot for MOND. Yes, it’s a silly sounding name but I kind of like it. I must admit that this is because \LambdaCDM feels a bit contrived. The \LambdaCDM models requires dwarf galaxies to be missing approximately 99% of the baryons associated to their dark matter halo. A bit unnecessary perhaps? Maybe there is a simpler answer?

For the record, however, \LambdaCDM is an established model that has convinced many people are who more technically orientated to cosmology than myself and therefore due respect is needed.

In any case McGaugh’s new paper, “A novel test of the modified newtonian dynamics with gas rich galaxies” hits my soft spot. It’s a nicely written, 4 page, paper that demonstrates that as a phenomenological model MOND should at the least be assumed to be giving some accurate picture of gravity at very large scales. The reduced \chi^2 for the prediction of MOND for the data used in the paper, depending on some technical assumptions about what counts as a degree of freedom, is between 0.96 and 0.99. So the data fits nicely. Figure 2 of the paper demonstrates that \LambdaCDM is substantially worse (the \LambdaCDM prediction doesn’t lie in the error bars of any of the data points).

More kudos to McGaugh, as the paper contains a nicely balanced discussion of this which doesn’t jump to sudo-crackpot accusations and gives \LambdaCDM the respect it deserves. And it’s readable by lay-physicists!

So you know… I kind of like the paper.

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Summation by parts (SBP) operators and interface boundary treatments

February 25, 2011

by Ben

Finished of Carpenter, Nordstrom and Gottlieb’s paper, “A stable and conservative interface treatment of arbitrary spatial accuracy”. I like papers like this. It ‘feels’ clean, organised self contained. Writing such papers takes work so when I find ones like this I take a bit more time over them.

The crux of the paper is a penalty term in the difference approximation to the time derivative to implement the interface. In principle this seems the same as the SAT boundary conditions (see Carpenter, Gottlieb and Ararbanel’s paper, “Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems” – which is another pleasurable paper to read). Given the success of SAT it’s no surprise that penalty boundary conditions perform better than some other standard interface treatments, particular for SBP operators.

In any case I’m most interested in the presentation of a specific 4th order SBP operator in the last section of the appendix. Note that they reference Gustafsson’s paper to justify the loss of one order of accuracy close to the boundary. This makes me feel uneasy, but it seems to work.

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Rate of convergence not affected by lower order convergence near boundaries

February 24, 2011

by Ben

I’ve just finished up my second reading of Gustafsson’s, “The convergence rate for difference approximations to mixed initial boundary value problems” and I’m still abit confused.

The paper is referenced by (what seems to me) almost everyone and their dog as justification that a finite difference scheme can have one order of convergence less on boundary points than the design convergence rate and still achieve design. It strikes me that the paper works with multi-step schemes while it is sometimes referenced by papers using a semi-continuous or Runge-Kutta approach to time integration. This seems a little contradictory to me and while the bounds derived can probably be applied to these cases there does, in principle, seem to me a large difference between an operator used to estimate a derivative and an operator used to construct the next time level. Admittedly I’m not on top of the paper as it’s quite dense and assumes knowledge of the equally dense, but far larger, paper by Gustafsson, Kreiss and Sundstrom, so there could be something I’m missing here. Perhaps it’s “obvious” to the experts?

The main theorems, 2.1 and 2.2, both assume that every solution to the difference scheme is bounded by an appropriate norm applied to the initial conditions.

The techniques used to prove the theorems are quite nice, and follow the general structure used in Gustafsson, Kriess and Sundstrom: application of Fourier transforms and similarity transformations. Indeed compared to the difficult to swallow definitions the proofs don’t take much work. It makes me reflect that in more modern papers I haven’t seen much of this technique, perhaps there are better ways, or perhaps I’m looking in the wrong places.

Lastly, there is a nice example at the back of the paper which gives a much needed example.

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