Towards Stable Radial Basis Function Methods for Linear Advection Problems

Abstract

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.

Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions

“…Future work will focus on the application of the proposed weak RBF method to nonlinear problems and, in particular, on the adaptation of different methods [99,71,52,69,88,45,42,40] from DG and related methods to further stabilize the weak RBF method in the presence of (shock) discontinuities. Moreover, in a forthcoming work [41], the weak enforcement of BCs was also investigated in the context of RBF methods for linear advection problems based on their strong form. Finally, in addition to the energy stability analysis provided here, it would be useful to perform a (linear) eigenvalue stability analysis.…”