Throughout this article, we use the following notation and conventions. For a complex vector , we
denote by
its transposed, complex conjugate, such that
is the standard scalar product
for two vectors
. The corresponding norm is defined by
. The norm of a complex,
matrix
is
The transposed, complex conjugate of is denoted by
, such that
for all
and
. For two Hermitian
matrices
and
, the
inequality
means
for all
. The identity matrix is denoted by
.
The spectrum of a complex, matrix
is the set of all eigenvalues of
,
which is real for Hermitian matrices. The spectral radius of is defined as
Then, the matrix norm of a complex
matrix
can also be computed as
.
Next, we denote by the class of measurable functions
on the open subset
of
, which are square-integrable. Two functions
, which differ from each
other only by a set of measure zero, are identified. The scalar product on
is defined
as
and the corresponding norm is . According to the Cauchy–Schwarz inequality we
have
The Fourier transform of a function , belonging to the class
of infinitely-differentiable
functions with compact support, is defined as
According to Parseval’s identities, for all
, and the map
,
can be extended to a linear, unitary map
called the Fourier–Plancharel
operator; see, for example, [346
]. Its inverse is given by
for
and
.
For a differentiable function , we denote by
,
,
,
its partial derivatives with respect
to
,
,
,
.
Indices labeling gridpoints and number of basis functions range from to
. Superscripts and
subscripts are used to denote the numerical solution at some discrete timestep and gridpoint, as
in
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Living Rev. Relativity 15, (2012), 9
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