The design of high-order accurate numerical schemes for hyperbolic
equations and system of hyperbolic equations concerns with the
following problem of the approximation of functions: building a
high-order accurate interpolant of a piecewise smooth function from its
cell averages while avoiding the tipical order O(1) oscillations
known as Gibbs phenomenon at discontinuities.
This polynomial is built by a non-linearly
stable interpolation of the discrete set of cell-averaged data.
The order of this polynomial gives the formal spatial order of accuracy
of the resulting numerical method. That is, the spatial order of accuracy
whenever the solution is smooth. As Harten (1991) [8] points out,
there are basically two great classes of reconstruction schemes : TVD and ENO.
The property which distinguises
TVD from ENO schemes is related to the total variation
of the reconstructed function, see LeVeque (1990) [14].
Let us define the total variation for a discrete function 
as :
For a continous function 
one possible way for defining the
Total Variation is:
for any unbounded 
and all possible partitions 
.
For a TVD scheme the total variation of the reconstructed function
is non-increasing with respect to the cell-averaged
values :
This same relation holds by a factor 
where n is the formal order of accuracy of the reconstruction for
ENO schemes. Hence, as discussed by Harten et al. (1986) in [10],
ENO schemes are not completely oscillation-free
but oscillations are allowed up to the truncation error level.
Both MUSCL-TVD and ENO polynomials can be built on conservative or characteristic variables. These latter are defined as
where 
is the set of the conservative variables,
and 
is the matrix
of the eigenvectors of the Jacobian of the inviscid part of the
equations, which is a function of the flow state and is
generally computed using the value at the centroids.
