The ENO Algorithm for Reconstruction

The ENO reconstruction algorithm is based on a special interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth and avoids Gibbs phenomena.

The key idea of ENO schemes relies in the adaptive stencil which automatically selects the interpolating values in the locally smoothest region. This strategy has been proven to strongly inhibit differencing across discontinuities. In fact, near discontinuities this technique automatically switches to one-sided approximations, thus avoiding the use of discontinuos data which brings about spurious numerical oscillations.

The one-dimensional ENO reconstruction must specifically satisfy three conditions:

1. at all points where is smooth:

   

2. the conservation in the sense of ``cell-averages'':

   

   or written extensively:

   

   for any cell of the computational domain

3. the reconstruction must be ``Essentially Non-Oscillatory'':

   

   where is the total variation defined for unbounded and all possible partitions as

   

Two different approaches to design ENO reconstruction have been developed in the first papers by Harten et al. (1986) [10] and Harten et al. (1987) [9]: the reconstruction via primitive variables and the reconstruction via deconvolution. In the present work only the first strategy of reconstruction has been applied and will be presented.