A strategy is devised to make the WENO interpolation in the point values
achieve optimal accuracy near the discontinuities. The classical WENO
interpolation ensures the optimal accuracy when all stencils are smooth
and ENO property when the discontinuity appears. When there exist more
than two successive smooth stencils, the maximum theoretical accuracy
near discontinuity is also preferred to be obtained. To achieve it, we
divide the classical WENO algorithm into several sub-WENO procedures. In
each sub-WENO procedure, only two stencils are used and the order of
accuracy grows one at most. If both stencils are smooth, then sub-WENO
procedure increases the order of accuracy by one. If there is a stencil
is smooth and the left one is non-smooth, then algorithm conserves the
order of interpolation by corresponding smooth stencil and keeps the ENO
property. If both stencils are non-smooth, then the value constructed by
sub-WENO procedure will be ignored in the latter procedures. The whole
of new WENO algorithm can be expressed as a tree structure. The
indicator of smoothness of every medium stencil in the tree structure is
defined by the indicators of smoothness of corresponding stencils on the
top of tree. Such definition is proved to be capable of obtaining the
optimal accuracy and keep the ENO property. And the new WENO algorithm
has almost the same computational cost as the classical WENO algorithm.