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

TL;DR

The paper tackles the robustness and efficiency drawbacks of very high-order finite-volume reconstructions by introducing ASE-DF, an adaptive stencil extension framework driven by a discontinuity feedback factor. DF estimates interface-discontinuity strength to adjust reconstruction order locally, replacing traditional smoothness indicators and enabling seamless extension to arbitrary orders from 5th to beyond 9th. Through extensive 1D/2D accuracy tests and challenging discontinuous flows (including a Mach 20000 astrophysical jet), ASE-DF demonstrates high accuracy, strong robustness, and favorable computational efficiency, with threshold tuning (e.g., \(\sigma_{thres}=2.0\)) balancing resolution and stability. The method eliminates the need for expensive smoothness indicators, improves shock handling, and provides a scalable path to robust, high-order schemes in complex CFD applications.

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.

Figures (11)

• Figure 1: x-direction reconstruction at the left side of the interface ${\rm \Gamma}_{i+1/2,j}$. Discontinuity strength of the interfaces to be considered for normal and tangential reconstruction(from left to right).
• Figure 2: A possible distribution of variables $\{W_{i-2},W_{i-1},W_i,W_{i+1},W_{i+2}\}$. Each sub-stencil has a discontinuity, and the WENO reconstruction can only select the relatively smooth sub-stencil by weights. Take stencil $\{W_{i-2},W_{i-1},W_i\}$ as an example, the green line shows the reconstructed polynomial in the domain $[x_{i-1/2},x_{i+1/2}]$ by WENO method, and the orange line shows the DF factor can automatically converge the reconstructed polynomial to 1st-order when stencil exist a discontinuity, which is more robust compared to the WENO method.
• Figure 3: Blast wave problem: the density distributions and local enlargement at $t=3.8$ with a cell size $\Delta x=1/400$. $c_1=0.05,c_2=5.0$. Different values of $\sigma_{thres}$ using the ASE-DF(5, 3) with the GKS solver. The reference solution is obtained by the 1-D 5th-order WENO-AO GKS with 4000 meshes.
• Figure 4: Configuration 3: the density distribution at $t=0.6$ with $500\times500$ meshes. $c_1=0.05,c_2=1.0$. This figure is drawn with 30 density contours. (a-d) Different values of $\sigma_{thres}$ using the ASE-DF(5, 3) with the GKS solver.
• Figure 5: Shu-Osher problem: the density distributions and local enlargement at $t=1.8$ with a cell size $\Delta x=1/20$. All results are calculated by the GKS solver, $c_1=0.05,c_2=1.0$. The reference solution is obtained by the 1-D 5th-order WENO-AO GKS with 10000 meshes.