Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator

TL;DR

This paper introduces adaptive high-order A-WENO schemes with a new local smoothness indicator for hyperbolic conservation laws, improving accuracy and robustness while reducing computational cost.

Contribution

The paper presents a novel simple and robust local smoothness indicator integrated into adaptive A-WENO schemes, enhancing shock detection and computational efficiency.

Findings

The adaptive schemes outperform fully limited counterparts in accuracy and robustness.

The new LSI effectively detects rough solution regions near shocks.

The schemes are less computationally expensive than traditional methods.

Abstract

We develop new adaptive alternative weighted essentially non-oscillatory (A-WENO) schemes for hyperbolic systems of conservation laws. The new schemes employ the recently proposed local characteristic decomposition based central-upwind numerical fluxes, the three-stage third-order strong stability preserving Runge-Kutta time integrator, and the fifth-order WENO-Z interpolation. The adaptive strategy is implemented by applying the limited interpolation only in the parts of the computational domain where the solution is identified as ``rough'' with the help of a smoothness indicator. We develop and use a new simple and robust local smoothness indicator (LSI), which is applied to the solutions computed at each of the three stages of the ODE solver. The new LSI and adaptive A-WENO schemes are tested on the Euler equations of gas dynamics. We implement the proposed LSI using the pressure, which remains smooth at contact discontinuities, while our goal is to detect other ``rough'' areas and apply the limited interpolation mostly in the neighborhoods of the shock waves. We demonstrate that the new adaptive schemes are highly accurate, non-oscillatory, and robust. They outperform their fully limited counterparts (the A-WENO schemes with the same numerical fluxes and ODE solver but with the WENO-Z interpolation employed everywhere) while being less computationally expensive.