  It is often the case in numerical relativity that schemes that are known to
be convergent for well posed systems are used in evolutions of weakly
hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show
that with several of the discretizations that have been used through out the
years, this procedure leads to non-convergent schemes. That is, arbitrarily
small initial errors are amplified without bound when resolution is increased,
independently of the amount of numerical dissipation introduced. The lack of
convergence introduced by this instability can be particularly subtle, in the
sense that it can be missed by several convergence tests, especially in 3+1
dimensional codes. We propose tests and methods to analyze convergence that may
help detect these situations.
