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

TL;DR

This paper introduces two new fifth-order WENO schemes with adaptive order for hyperbolic conservation laws, improving accuracy and resolution near discontinuities while reducing computational cost.

Contribution

The paper presents a simple smoothness indicator for WENO schemes and a new adaptive order scheme that enhances accuracy and efficiency in solving hyperbolic conservation laws.

Findings

WENO-AON(5,3) has comparable accuracy to WENO-AO(5,3) with lower computational cost.

WENO-AO(5,4,3) achieves better resolution near shocks with minimal additional cost.

Numerical experiments confirm improved performance in 1D and 2D problems.

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.