  We derive stability estimates for three commonly used radial basis function
(RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite
difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and
Kansa's (global) RBF method. We give the estimates in the discrete
$\ell_2$-norm intrinsic to each of the three methods.The results show that
Kansa's method and RBF-PUM can be $\ell_2$-stable in time under a sufficiently
large oversampling of the discretized system of equations. On the other hand,
the RBF-FD method is not $\ell_2$-stable by construction, no matter how large
the oversampling is. We show that this is due to the jumps (discontinuities) in
the RBF-FD cardinal basis functions. We also provide a stabilization of the
RBF-FD method that penalizes the spurious jumps. Numerical experiments show an
agreement with our theoretical observations.
