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

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.