  In this work, we investigate (energy) stability of global radial basis
function (RBF) methods for linear advection problems. Classically, boundary
conditions (BC) are enforced strongly in RBF methods. By now it is well-known
that this can lead to stability problems, however. Here, we follow a different
path and propose two novel RBF approaches which are based on a weak enforcement
of BCs. By using the concept of flux reconstruction and simultaneous
approximation terms (SATs), respectively, we are able to prove that both new
RBF schemes are strongly (energy) stable. Numerical results in one and two
spatial dimensions for both scalar equations and systems are presented,
supporting our theoretical analysis.
