4 Time-Dependent Problems
From a relativistic point of view, the time coordinate could be treated in the same way as spatial
coordinates and one should be able to achieve spectral accuracy for the time representation of a spacetime
function 
and its derivatives. Unfortunately, this does not seem to be the case and we are
neither aware of any efficient algorithm dealing with the time coordinate, nor of any published successful
code solving any of the PDEs coming from Einstein’s equations, with the recent exception of the 1+1
dimensional study by Hennig and Ansorg [113
]. Why is time playing such a special role? It is not easy to
find in the literature on spectral methods a complete and comprehensive study. A first standard explanation
is the difficulty, in general, of predicting the exact time interval on which one wants to study the time
evolution. In addition, time discretization errors in both finite difference techniques and spectral methods
are typically much smaller than spatial ones. Finally, one must keep in mind that, contrary
to finite difference techniques, spectral methods store all global information about a function
over the whole time interval. Therefore, one reason may be that there are strong memory and
CPU limitations to fully three-dimensional simulations; it is already very CPU and memory
consuming to describe a complete field depending on 3+1 coordinates, even with fewer degrees of
freedom, as is the case for spectral methods. But the strongest limitation is the fact that, in the
full 3+1 dimensional case, the matrix representing a differential operator would be very big;
it would therefore be very time consuming to invert it in a general case, even with iterative
methods.
More details on the standard, finite-difference techniques for time discretization are given in Section 4.1.
Due to the technical complexity of a general stability analysis, we restrict the discussion of this
section to the eigenvalue stability (Section 4.1) with the following approach: the eigenvalues of
spatial operator matrices must fall within the stability region of the time-marching scheme.
Although this condition is only a necessary one and, in general, is not sufficient, it provides
very useful guidelines for selecting time-integration schemes. A discussion of the imposition
of boundary conditions in time-dependent problems is given in Section 4.2. Section 4.3 then
details the stability analysis of spatial discretization schemes, with the examples of heat and
advection equations, before the details of a fully-discrete analysis are given for a simple case
(Section 4.4).
