  The essentially non-oscillatory (ENO) method is an efficient high order
numerical method for solving hyperbolic conservation laws designed to reduce
the Gibbs oscillations, if existent, by adaptively choosing the local stencil
for the interpolation. The original ENO method is constructed based on the
polynomial interpolation and the overall rate of convergence provided by the
method is uniquely determined by the total number of interpolation points
involved for the approximation. In this paper, we propose simple non-polynomial
ENO and weighted ENO (WENO) finite volume methods in order to enhance the local
accuracy and convergence. We first adopt the infinitely smooth radial basis
functions (RBFs) for a non-polynomial interpolation. Particularly we use the
multi-quadric and Gaussian RBFs. The non-polynomial interpolation such as the
RBF interpolation offers the flexibility to control the local error by
optimizing the free parameter. Then we show that the non-polynomial
interpolation can be represented as a perturbation of the polynomial
interpolation. That is, it is not necessary to know the exact form of the
non-polynomial basis for the interpolation. In this paper, we formulate the ENO
and WENO methods based on the non-polynomial interpolation and derive the
optimization condition of the perturbation. To guarantee the essentially
non-oscillatory property, we switch the non-polynomial reconstruction to the
polynomial reconstruction adaptively near the non-smooth area by using the
monotone polynomial interpolation method. The numerical results show that the
developed non-polynomial ENO and WENO methods enhance the local accuracy.
