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).
* Status: 1989, 3D problems, like collapse and gravitational wave production, can be tackled, 1995, Still relatively few 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 de); > s.a. models in numerical relativity [binaries].
* 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.

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

Stability and Hyperbolicity > s.a. einstein's equation [various forms].
* Idea: To ensure stability of the evolution, a common strategy involves using symmetric hyperbolic formulations of Einstein's equation.
@ Stability: Alcubierre et al PRD(00)gq/99 [ADM]; Szilágyi et al PRD(00)gq/99; Frittelli & Gómez JMP(00)gq [ill-posedness]; Miller gq/00/PRD [ADM vs CT]; Laguna & Shoemaker CQG(02)gq [conformal-traceless]; Calabrese et al PRD(02)gq, PRD(02)gq, JCP(06)gq/05; O'Shaughnessy PRD(03)gq; Lehner et al CQG(06)gq/05 [higher accuracy].
@ Hyperbolic form: Bona et al PRL(95)gq/94, PRD(97)gq [first-order]; Scheel et al PRD(97)gq [black holes]; Yoneda & Shinkai CQG(01)gq/00; Shinkai & Yoneda gq/01-in [connection variables]; Buchman & Bardeen PRD(03)gq [tetrad variables]; Bona & Palenzuela PRD(04)gq [and dynamical shift].
@ Symmetric hyperbolicity: Tiglio et al PRD(04)gq/03 [3D simulations].

Other Issues and Methods > s.a. computational physics; initial value formulation; models [punctures]; regge calculus.
* Form of equations: 2008, The two main formulations used to produce well-posed initial value problems are the BSSN and the harmonic formulations.
@ General references: Alcubierre et al CQG(04)gq/03 [testbeds]; Neilsen et al gq/04-in [examples]; Shinkai a0805-ln.
@ Boundary conditions: Sarbach a0708-in [absorbing].
@ Characteristic problem: Stewart & Friedrich PRS(82); Corkill & Stewart PRS(83) [2 Killing vectors, vacuum]; Bishop CQG(93); Winicour PTPS(99)gq [binary black holes], gq/00-in [waves]; Barreto et al PRD(05)gq/04 [Einstein-Klein-Gordon]; Winicour LRR(05)gq [rev].
@ Cauchy + characteristic: Clarke & d'Inverno CQG(94); Clarke et al PRD(95); d'Inverno & Vickers PRD(96), PRD(97) [axial symmetry]; Papadopoulos & Laguna PRD(97)gq/96 [Einstein-Klein-Gordon]; Dubal et al PRD(98) [spherical + fluid]; Bishop et al gq/98-in; d'Inverno et al CQG(00)gq; Szilágyi gq/00-PhD; Winicour LRR(01)gq.
@ Cauchy + boundary: Stewart CQG(98); Szilágyi & Winicour PRD(03)gq/02; Frittelli & Gómez PRD(03) [boundary conditions]; Nagy & Sarbach CQG(06)gq [variational problem for lapse]; Babiuc et al PRD(06) [harmonic].
@ Conformal form: Shibata & Nakamura PRD(95); Frauendiener PRD(98)gq/97, PRD(98)gq/97; Baumgarte & Shapiro PRD(99)gq/98; Alcubierre et al PRD(00)gq; Lehner et al gq/00 [causal differencing excision]; Gourgoulhon & Novak IJMPA(02).
@ BSSN formulation: Sarbach et al PRD(02)gq; Yoneda & Shinkai PRD(02)gq; Beyer & Sarbach PRD(04)gq; Brown a0705 [spherical]; Gentle a0707 [nature of equations].
@ Algebraic computing: Husa & Lechner gq/03-in; > s.a. computation.
@ Connection variables: Salisbury et al CQG(94)gq; Shinkai & Yoneda gq/97-MG8, PRD(99)gq, CQG(00)gq [Ashtekar].
@ Other approaches and topics: Stewart PRS(89) [Bondi mass]; Arbona et al PRD(99)gq; Teukolsky PRD(00)gq/99 [iterated Crank-Nicholson method]; Papadopoulos & Sopuerta PRD(02)gq/01 [background geometry]; Sperhake gq/02-PhD [non-linear techniques]; Bona et al PRD(02)gq [evolution systems], gq/02/PRL [extended constraint-free system]; Jansen et al PRD(06)gq/03 [stability, AA vs ADM & BSSN]; Calabrese PRD(05)gq/04 [first- vs second-order, black holes]; Alic et al a0706 [finite volume methods]; Grandclément & Novak a0706 [spectral methods]; Cordero-Carrión et al PRD(08); > s.a. models [including adaptive mesh]; > s.a. differential forms [discrete].

References
@ Books and collections of papers: Centrella ed-86; Evans et al ed-88; d'Inverno 92; Hehl et al ed-96; Bona & Palenzuela-Luque 05; issue CQG(06)#16, CQG(07)#12.
@ General: Hobill & Smarr in(89); Choptuik et al CQG(92) [spherical, scalar + gravity, 2 codes]; Anninos et al PW(96) [II, black holes]; Lehner CQG(01)gq, gq/02-GR16, van Putten gq/02-in [rev]; Alcubierre gq/04-GR17; Shapiro gq/05-in [rev]; Andersson CQG(06)gq [and mathematical relativity]; Babiuc et AppleswithApples a0709 [standard testbeds].
@ Computational aspects: Suen gq/99-in [and TeraFlop machines].

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 12 jun 2008