Simulating non-vacuum systems such as relativistic hydrodynamical ones using spectral methods
can be problematic, particularly when surfaces, shocks, or other non-smooth behavior appears
in the fluid. Without further processing, the fast convergence is lost, and Gibbs’ oscillations
can destabilize the simulation. A method that has been successfully used to overcome this in
general-relativistic hydrodynamics is evolving the spacetime metric and the fluid on two different
grids, each using different numerical techniques. The spacetime is evolved spectrally, while the
fluid is evolved using standard finite difference/finite volume shock-capturing techniques on a
separate uniform grid. The first code adopting this approach was described in [142], which is a
stellar-collapse code assuming a conformally-flat three-metric, with the resulting elliptic equations being
solved spectrally. The two-grid approach was adopted for full numerical-relativity simulations
of black-hole–neutron-star binaries in [150, 149, 168]. The main advantage of this method
when applied to binary systems is that at any given time the fluid evolution grid only needs to
extend as far as the neutron-star matter. During the pre-merger phase, then, this grid can be
chosen to be a small box around the neutron star, achieving very high resolution for the fluid
evolution at low computational cost. More recently, in [168] an automated re-gridder was added, so
that the fluid grid automatically adjusts itself at discrete times to accommodate expansion or
contraction of the nuclear matter. The main disadvantage of the two-grid method is the large amount
of interpolation required for the two grids to communicate with each other. Straightforward
spectral interpolation would be prohibitively expensive, but a combination of spectral refinement
and polynomial interpolation [69] reduces the cost to about 20 – 30 percent of the simulation
time.
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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