This dissertation presents numerical studies of gravitational

waves produced by black holes in two scenarios: perturbations

of a single black hole, and the collision of a binary pair.

Their detection plays a crucial roll in further testing General

Relativity and opens a whole new field of observational

astronomy.

First, a technique called Cauchy--perturbative matching is

revisited in one dimension through the use of

new numerical methods, such as high order finite

difference operators, constraint-preserving boundary conditions and,

most important, a multi-domain decomposition (also referred

to as multi-patch, or multi-block approach).

These methods are then used to numerically solve the fully non-linear

three-dimensional Einstein vacuum equations representing a

non-rotating distorted black hole. In combination with a

generalization of

the Regge-Wheeler-Zerilli formalism,

we quantify the effect of the background

choice in the wave extraction techniques. It is found that

a systematic error is introduced

at finite distances. Furthermore, such error is found to be

larger than those due to numerical discretization.

Subsequently, the first simulations ever of binary black

holes with a finite-difference multi-domain approach are presented.

The case is one in which the black holes

orbit for about twelve cycles before merging. The salient

features of this multi-domain approach are:

i) the complexity of the problem scales linearly with the size

of the computational domain, ii) excellent scaling, in

both weak and strong senses, for several thousand processors.

As a next step, binary black hole simulations from inspiral

to merger and ringdown are performed

using a new technique, turduckening, and a standard finite

difference, adaptive mesh-refinement code. The

computed gravitational waveforms

are compared to those obtained through evolution of

the same exact initial configuration but with a pseudo-spectral

collocation code.

Both the gravitational waves extracted at finite locations

and their extrapolated values to null infinity are compared.

Finally, a numerical study of generic second order perturbations

of Schwarzschild black

holes is presented using a new gauge invariant high order

perturbative formalism. A study of the self-coupling of first

order modes and the resulting radiated energy, in particular

its dependence on the type of initial perturbation, is detailed.