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In the present work, we propose two new variants of fifth order finitedifference WENO schemes of adaptive order. We compare our proposed schemes withother variants of WENO schemes with special emphasize on WENO-AO(5,3) scheme[Balsara, Garain, and Shu, {\it J. Comput. Phys.}, 326 (2016), pp 780-804]. Thefirst algorithm (WENO-AON(5,3)), involves the construction of a new simplesmoothness indicator which reduces the computational cost of WENO-AO(5,3)scheme. Numerical experiments show that accuracy of WENO-AON(5,3) scheme iscomparable to that of WENO-AO(5,3) scheme and resolution of solutions involvingshock or other discontinuities is comparable to that of WENO-AO(5,3) scheme.The second algorithm denoted as WENO-AO(5,4,3), involves the inclusion of anextra cubic polynomial reconstruction in the base WENO- AO(5,3) scheme, whichleads to a more accurate scheme. Extensive numerical experiments in 1D and 2Dare performed, which shows that WENO-AO(5,4,3) scheme has better resolutionnear shocks or discontinuities among the considered WENO schemes withnegligible increase in computational cost.

Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws