### 11.4 Spectral multi-domain binary black-hole evolutions

In current binary black-hole evolutions using spectral collocation methods, there are typically three sets
of spherical shells, one around each black hole and one in the wave zone. These three shells are connected
by subdomains of various shapes and sizes. Figure 13 shows the global structure of one such
grid-structure, emphasizing the spherical patches in the wave zone and how the different domains are
connected. Inter-domain boundary conditions are set by the spectral penalty method described in
Section 10. The adaptivity provided by domain decomposition in addition to the spectral convergence
rate, has lead to the highest-accuracy binary black-hole simulations to date. Currently these
evolutions use a first-order symmetry-hyperbolic reduction of the harmonic system with constraint
damping as derived in [286] and summarized in Section 4.1, with constraint-preserving boundary
conditions, as designed in [286, 366, 363]. The field variables are expressed in an “inertial”
Cartesian coordinate system, which is related to one fixed to the computational domain through
a dynamically-defined coordinate transformation tracking the black holes (the “dual frame”
method) [384].
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.