An Adaptive Reconstruction Method for Arbitrary High-Order Accuracy Using Discontinuity Feedback

TL;DR

This paper presents an adaptive high-order reconstruction method using a discontinuity feedback factor that enhances robustness and reduces computational costs in high-order schemes, especially above 7th-order.

Contribution

The method adaptively increases accuracy order based on local smoothness, avoiding expensive smoothness indicators and improving robustness over traditional schemes.

Findings

Effective in high Mach number jet simulations

Reduces computational complexity in high-order reconstructions

Demonstrates superior robustness in challenging test cases

Abstract

This paper introduces an effcient class of adaptive stencil extension reconstruction methods based on a discontinuity feedback factor, addressing the challenges of weak robustness and high computational cost in high-order schemes, particularly those of 7th-order or above. Two key innovations are presented: The accuracy order adaptively increases from the lowest level based on local stencil smoothness, contrasting with conventional methods like Weighted Essentially Non-Oscillatory (WENO) and Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL)limiters, which typically reduce order from the highest level. The Discontinuity Feedback Factor (DF) serves a dual purpose: detecting sub-cell discontinuity strength and explicitly incorporating into the reconstruction process as a local smoothness measure. This approach eliminates the need for computationally expensive smoothness indicators often required in very high-order schemes, such as 9th-order schemes, and can be easily generalized to arbitrary high-order schemes. Rigorous test cases, including a Mach 20000 jet, demonstrate the exceptional robustness of this approach.