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

Abstract

In the present work, we propose two new variants of fifth order finite difference WENO schemes of adaptive order. We compare our proposed schemes with other 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]. The first algorithm (WENO-AON(5,3)), involves the construction of a new simple smoothness indicator which reduces the computational cost of WENO-AO(5,3) scheme. Numerical experiments show that accuracy of WENO-AON(5,3) scheme is comparable to that of WENO-AO(5,3) scheme and resolution of solutions involving shock 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 an extra cubic polynomial reconstruction in the base WENO- AO(5,3) scheme, which leads to a more accurate scheme. Extensive numerical experiments in 1D and 2D are performed, which shows that WENO-AO(5,4,3) scheme has better resolution near shocks or discontinuities among the considered WENO schemes with negligible increase in computational cost.