Issues and Methods in Numerical General Relativity  

In General > s.a. computational physics; numerical relativity [gauge choice, constraints].
* Form of the 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 LNP(06)gq/04 [examples]; Shinkai JKPS(09)-a0805-ln; Zumbusch CQG(09)-a0901; Bona et al PRD(10)-a1008 [action principle].
@ Boundary conditions: Sarbach JPCS(07)-a0708 [absorbing]; > s.a. numerical relativity.
@ 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-proc [waves]; Barreto et al PRD(05)gq/04 [Einstein-Klein-Gordon]; Winicour LRR(05)gq, LRR(09), LRR(12) [rev]; Kreiss & Winicour CQG(11)-a1010 [null-timelike boundary problem]; van der Walt & Bishop PRD(12) [and observational cosmology].
@ 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 PhD(00)gq; 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]; Winicour CQG(12) [boundary conditions, rev].
> Related topics: see initial-value formulation; models [collapse, binaries, cosmology, astrophysics]; regge calculus.

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-proc [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 Approaches and Methods > s.a. Symplectic Integrators.
@ 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 CQG(08)-a0705 [spherical]; Gentle IJMPD(10)-a0707 [nature of equations].
@ In higher dimensions: Yoshino & Shibata PRD(09)-a0907; Zilhão a1301-PhD; Sperhake IJMPD(13)-a1301; Witek IJMPA(13)-a1308-ln; Witek et al PRD(14)-a1406 [comparison of two codes].
@ Algebraic computing: Husa & Lechner gq/03-proc; > s.a. computation.
@ Connection variables: Salisbury et al CQG(94)gq; Shinkai & Yoneda gq/97-MG8, PRD(99)gq, CQG(00)gq [Ashtekar variables].
@ Spectral methods: Grandclément & Novak LRR(09)-a0706; Amorim et al CQG(09) [Gowdy spacetimes].
@ Other formulations and techniques: Teukolsky PRD(00)gq/99 [iterated Crank-Nicholson method]; Sperhake PhD(01)gq/02 [non-linear techniques]; Jansen et al PRD(06)gq/03 [stability, AA vs ADM & BSSN formulations]; Calabrese PRD(05)gq/04 [first- vs second-order, black holes]; Alic et al PRD(07)-a0706 [finite-volume methods]; Paschalidis PRD(08) [mixed hyperbolic–second-order-parabolic]; Garrett a0902 [direct finite differencing]; Mongwane GRG(15)-a1504 [mesh refinement algorithms]; Hamilton a1611 [covariant Hamiltonian tetrad approach].
@ Related topics: Stewart PRS(89) [Bondi mass]; Arbona et al PRD(99)gq; Papadopoulos & Sopuerta PRD(02)gq/01 [background geometry]; Bona et al PRD(02)gq [evolution systems], gq/02/PRL [extended constraint-free system]; Cordero-Carrión et al PRD(08); Jasiulek CQG(09)-a0906 [quasilocal invariants]; > s.a. differential forms [discrete]; models in numerical relativity [including adaptive mesh].

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