Numerical
General Relativity |

**In General**

* __Motivation__: Make realistic astrophysical predictions; Use to look
for chaos (the first positive claims were wrong because the constraints were
not preserved, and error propagation simulated negative energy density).

* __History, status__:
1960s, First attempts at solving a binary black hole spacetime by Hahn and Lindquist; 1976, B DeWitt coins the expression "numerical relativity"; 1989, Some
3D problems, like collapse and gravitational wave production, can be tackled;
1992, First qualitatively new solution found by numerical methods, Choptuik's
critical collapse; 1994-1999, Binary-black-hole grand challenge; Still
relatively few 3D problems done; Also, better understanding of the convergence
of Regge
calculus,
theoretically,
but in
practice method
for Regge calculus not as developed (choosing initial data involves solving
elliptical differential equations); 2005, F Pretorius breakthrough and stable
simulation of black-hole inspiral and merger; 2009, About 11 groups worldwide
can now do full merger simulations; New results have been obtained (gravitational
recoil "kicks", black-hole triplets, gravitational-wave production); Gravity
is in the process of becoming data-driven.

* __Data__: One way of handling the fact that the region is finite is
to give data on a finite spacelike region, and then free data on the outgoing
light front from its boundary.

__Related topics__:
see issues and methods; models
in numerical relativity [collapse, binaries, cosmology, astrophysics].

**Gauge and Coordinate Choices** > s.a. coordinates; gauge
choices.

* __Idea__: It looks like the best gauge choices are dynamical ones.

@ __Choices and effects__: Alcubierre & Massó PRD(98)gq/97 [gauge
problems]; Garfinkle & Gundlach CQG(99)gq [approximate
Killing vector field]; Garfinkle PRD(02)gq/01 [harmonic
coordinates]; Reimann et al PRD(05)gq/04,
Alcubierre CQG(05)gq [gauge
shocks].

@ __BCT gauge (minimal strain equations)__: Brady et al; Gonçalves PRD(00)gq/99;
Garfinkle et al CQG(00)gq.

@ __Special cases__: Gentle et al PRD(01)gq/00 [constant *K* and
black holes].

**Constraints** > s.a. Symplectic
Integrators.

* __Idea__: Due to finite
precision errors, constraints in numerical relativity are never exactly satisfied,
so one can solve them initially and then simply monitor them as a check on
the evolution (unconstrained evolution), or somehow enforce them as part of
the evolution; 2008, Recent simulations use initial data generated by constraint
solvers that differ by the amount of gravitational radiation they include in
the initial configuration.

@ __General references__: Detweiler PRD(87);
Cook LRR(00)gq;
Tiglio gq/03 [control];
Fiske PRD(04)gq/03 [as
attractors]; Gentle et al CQG(04)gq/03 [as
evolution equations]; Baumgarte PRD(12)-a1202 [Hamiltonian constraint, alternative approach]; Okawa IJMPA(13)-a1308-ln [elliptic differential equations].

@ __And boundary conditions__: Calabrese et al PRD(02)gq/01;
Calabrese & Sarbach JMP(03)
[ill-posed]; Sarbach & Tiglio JHDE(05)gq/04;
Kidder et al PRD(05)gq/04;
Rinne et al CQG(07)-a0704 [comparison
of methods]; > s.a. methods in numerical relativity.

@ __Enforcement and violations__: Siebel & Hübner PRD(01)gq [effects
of enforcement]; Lindblom & Scheel PRD(02)gq [violations
and stability]; Berger GRG(06)gq/04-fs;
Matzner PRD(05)gq/04 [hyperbolicity
and constrained evolution]; Marronetti CQG(05)gq [Hamiltonian relaxation], CQG(06)gq/05,
gq/06-MGXI
[constraint relaxation]; Paschalidis et al PRD(07)
[well-posed evolution].

**References**

@ __Books and collections of papers__: Centrella ed-86; Evans et al ed-88;
d'Inverno 92; Hehl et al ed-96; issue CQG(06)#16,
CQG(07)#12,
CQG(09)#11; Alcubierre 08;
Bona et al 09 [and
relativistic astrophysics]; Baumgarte & Shapiro 10; issue CQG(10)#11; Shibata 16.

@ __Reviews__: Lehner CQG(01)gq, gq/02-GR16;
van Putten gq/02-conf; Rezzolla in(14)-a1303-proc; Cardoso et al LRR(15)-a1409 [fully non-linear evolutions and perturbative approaches, applications to new physics]; Garfinkle RPP(16)-a1606 [applications beyond astrophysics]; Tichy RPP(17)-a1610 [initial-value problem].

@ __Other general references__: Hobill & Smarr in(89); Choptuik
et al CQG(92)
[spherical, scalar + gravity, 2 codes]; Anninos et al PW(96)
[II, black holes]; Alcubierre
gq/04-GR17;
Shapiro PTPS(06)gq/05-proc
[rev]; Andersson CQG(06)gq [and
mathematical relativity]; Babiuc et AppleswithApples CQG(08)-a0709 [standard
testbeds]; Sekiguchi CQG(10)-a1009 [taking microphysics into account]; Cardoso et al CQG(12)-a1201 [NR/HEP Workshop summary]; Zilhão a1301-PhD [extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory].

@ __Computational aspects__: Suen gq/99-rp
[and TeraFlop machines]; Löffler et al CQG(12)-a1111, Zilhão & Löffler IJMPA(13)-a1305-ln [Einstein Toolkit, based on Cactus].

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send feedback and suggestions to bombelli at olemiss.edu – modified 9
jul
2017