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A new adaptive weighted essentially non-oscillatory WENO-$\theta$ scheme inthe context of finite difference is proposed. Depending on the smoothness ofthe 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 and6th-order central discretization. A new indicator $\tau^{\theta}$ measuring thesmoothness of the large stencil is chosen among two candidates which aredevised based on the possible highest-order variations of the reconstructionpolynomials 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 ina central sense with respect to the Taylor expansions around the point $x_{j}$.  Numerical results show that the new scheme combines good properties of both5th-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., JCP229 (2010)]). In particular, the new scheme captures discontinuities andresolves small-scaled structures much better than the 5th-order schemes;overcomes the loss of accuracy near some critical regions and is able tomaintain symmetry which are drawbacks detected in the 6th-order ones.

A New Adaptive Weighted Essentially Non-Oscillatory WENO-$\theta$ Scheme for Hyperbolic Conservation Laws