  We present adaptive finite difference ENO/WENO methods by adopting infinitely
smooth radial basis functions (RBFs). This is a direct extension of the
non-polynomial finite volume ENO/WENO method proposed by authors in
\cite{GuoJung} to the finite difference ENO/WENO method based on the original
smoothness indicator scheme developed by Jiang and Shu \cite{WENO}. The
RBF-ENO/WENO finite difference method slightly perturbs the reconstruction
coefficients with RBFs as the reconstruction basis and enhances accuracy in the
smooth region by locally optimizing the shape parameters. The RBF-ENO/WENO
finite difference methods provide more accurate reconstruction than the regular
ENO/WENO reconstruction and provide sharper solution profiles near the jump
discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in
the existing regular ENO/WENO code. The numerical results in 1D and 2D
presented in this work show that the proposed RBF-ENO/WENO finite difference
method better performs than the regular ENO/WENO method.
