6 Dynamic Evolution of Relativistic Systems
The modeling of time-dependent physical systems is traditionally the ultimate goal in numerical
simulations. Within the field of numerical relativity, the need for studies of dynamic systems is even more
pronounced because of the search for gravitational wave patterns. Unfortunately, as presented in
Section 4.1, there is no efficient spectral time discretization yet and one normally uses finite-order
time-differentiation schemes. Therefore, in order to get high temporal accuracy, one must use
high-order explicit time-marching schemes (e.g., fourth or sixth-order Runge–Kutta [49
]). This
requires quite a lot of computational power and might explain why, except for gravitational
collapse [95, 157], very few studies using spectral methods have dealt with dynamic situations until the
Caltech/Cornell group began to use spectral methods in numerical relativity in the early part of this
century [128, 127]. This group now has a very well-developed pseudospectral collocation code,
“Spectral Einstein Code” (SpEC), for the solution of full three-dimensional dynamic Einstein
equations.
In this section, we review the status of numerical simulations that use spectral methods in some fields
of general relativity and relativistic astrophysics. Although we may give at the beginning of
each section a very short introduction to the context of the relevant numerical simulations,
our point is not to give detailed descriptions of them, as dedicated reviews exist for most of
the themes presented here and the interested reader should consult them for details of the
physics and comparisons with other numerical and analytic techniques. Among the systems that
have been studied, one can find gravitational collapse [84] (supernova core collapse or collapse
of a neutron star to a black hole), oscillations of relativistic stars [206, 130] and evolution
of “vacuum” spacetimes. These include the cases of pure gravitational waves or scalar fields,
evolving in the vicinity of a black hole or as (self-gravitating) perturbations of Minkowski flat
spacetime. Finally, we will discuss the situation of compact binary [175, 31] spectral numerical
simulations.
