  A new adaptive weighted essentially non-oscillatory WENO-$\theta$ scheme in
the context of finite difference is proposed. Depending on the smoothness of
the large stencil used in the reconstruction of the numerical flux, a parameter
$\theta$ is set adaptively to switch the scheme between a 5th-order upwind and
6th-order central discretization. A new indicator $\tau^{\theta}$ measuring the
smoothness of the large stencil is chosen among two candidates which are
devised based on the possible highest-order variations of the reconstruction
polynomials in $L^2$ sense. In addition, a new set of smoothness indicators
$\tilde{\beta}_k$'s of the sub-stencils is introduced. These are constructed in
a central sense with respect to the Taylor expansions around the point $x_{j}$.
  Numerical results show that the new scheme combines good properties of both
5th-order upwind schemes, e.g., WENO-JS ([Jiang and Shu, JCP 126 (1996)]),
WENO-Z ([Borges et al., JCP 227 (2008)]), and 6th-order central schemes, e.g.,
WENO-NW6 ([Yamaleev and Carpenter, JCP 228 (2009)]), WENO-CU6 ([Hu el al., JCP
229 (2010)]). In particular, the new scheme captures discontinuities and
resolves small-scaled structures much better than the 5th-order schemes;
overcomes the loss of accuracy near some critical regions and is able to
maintain symmetry which are drawbacks detected in the 6th-order ones.
