Convergence and stability in numerical relativity

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 formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used throughout the years, this procedure leads to nonconvergent 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.