(ed.). Gravitational waves (IOP, 2001)(422s).

Still newer formulations: towards a stable evolution system 377source term remains a large uncharted territory. As astrophysics of compactobjects that needs general relativity for its understanding is attracting increasingattention, general relativistic hydrodynamics will become an increasinglyimportant subject as astrophysicists begin to study more relativistic systems,as relativists become more involved in studies of astrophysical sources. Thispromises to be one of the most exciting and important areas of research inrelativistic astrophysics in the coming years.Previously, most work in relativistic hydrodynamics has been done on fixedmetric backgrounds. In this approximation the fluid is allowed to move in arelativistic manner in strong gravitational fields, say around a black hole, but itseffect on the spacetime is not considered. Over the last years very sophisticatedmethods for general relativistic hydrodynamics have been developed by theValencia group led by José MIbáñez [89–92]. These methods are based ona hyperbolic formulation of the hydrodynamic equations, and are shown to besuperior to traditional artificial viscosity methods for highly relativistic flows andstrong shocks.However, just fixed background approximation is inadequate in describinga large class of problems which are of most interest to gravitational-waveastronomy, namely those with substantial matter motion generating gravitationalradiation, like the coalescences of neutron star binaries. We are constructinga multipurpose 3D code for the NASA Neutron Star Grand Challenge Project[93] that contains the full Einstein equations coupled to general relativistichydrodynamics. The hydrodynamic part consists of both an artificial viscositymodule, [94] and a module based on modern shock capturing schemes [95],containing three hydroevolution methods [95]: a flux split method, Roe’sapproximate Riemann solver [96] and Marquina’s approximate Riemann solver[92, 97]. All are based on finite-difference schemes employing approximateRiemann solvers to account explicitly for the characteristic information of theequations. These schemes are particularly suitable for astrophysics simulationsthat involve matter in (ultra)relativistic speeds and strong shock waves.In the flux split method, the flux is decomposed into the part contributingto the eigenfields with positive eigenvalues (fields moving to the right) and thepart with negative eigenvalues (fields moving to the left). These fluxes are thendiscretized with one sided derivatives (which side depends on the sign of theeigenvalue). The flux split method presupposes that the equation of state of thefluid has the form P = P(ρ, ɛ) = ρ f (ɛ), which includes, for example, theadiabatic equation of state. The second scheme, Roe’s approximate Riemannsolver [96] is by now a ‘traditional’ method for the integration of nonlinearhyperbolic systems of conservation laws [90, 91, 98]. This method makes noassumption on the equation of state, and, is more flexible than the flux splitmethods. The third method, the Marquina’s method, is a promising new scheme[97]. It is based on a flux formula which is an extension of Shu and Osher’sentropy-satisfying numerical flux [99] to systems of hyperbolic conservationlaws. In this scheme there are no artificial intermediate states constructed at