A robustness-enhanced reconstruction based on discontinuity feedback factor for high-order finite volume scheme

TL;DR

The paper tackles robustness challenges in high-order finite volume methods for 2-D compressible flows, where traditional WENO reconstruction can fail near strong discontinuities. It introduces a hybrid reconstruction that combines discontinuity feedback factor (DF) with WENO-AO, using DF to trigger a reduced-order, more robust stencil when discontinuities are detected. The method is implemented with two flux solvers—a high-order gas-kinetic scheme (GKS) using a two-stage fourth-order temporal discretization (S2O4) and a Lax-Friedrichs solver with SSP-RK3—and validated across accuracy and robustness tests, including Shu-Osher, shock interactions, and hurricane-like flows. Results show that the hybrid approach preserves high-resolution capabilities in smooth regions while significantly improving robustness in discontinuous regions, with DF values localizing where shocks and rarefactions occur, and it remains amenable to extension to other high-order schemes and mesh types.

Abstract

In this paper, a robustness-enhanced reconstruction for the high-order finite volume scheme is constructed on the 2-D structured mesh, and both the high-order gas-kinetic scheme(GKS) and the Lax-Friedrichs(L-F) flux solver are considered to verify the validity of this algorithm. The strategy of the successful WENO reconstruction is adopted to select the smooth sub-stencils. However, there are cases where strong discontinuities exist in all sub-stencils of the WENO reconstruction, which leads to a decrease in the robustness. To improve the robustness of the algorithm in discontinuous regions in two-dimensional space, the hybrid reconstruction based on a combination of discontinuity feedback factor(DF) \cite{ji2021gradient} and WENO reconstruction is developed to deal with the possible discontinuities. Numerical results from smooth to extreme cases have been presented and validate that the new finite volume scheme is effective for robustness enhancement and maintains high resolution compared to the WENO scheme.

Figures (14)

• Figure 1: Three possible distributions of variables $\mathbf{Q}_{i-2}\cdots\mathbf{Q}_{i+2}$. (a) Smooth sub-stencils exist, and the WENO reconstruction will automatically approximated to $\{\mathbf{Q}_{i-2},\mathbf{Q}_{i-1},\mathbf{Q}_{i}\}$ by weights. (b-c) Each sub-stencil has a discontinuity, and the WENO reconstruction can only select the relatively smooth sub-stencil by weights, which means the effectiveness of the reconstruction polynomial for each sub-stencil is reduced.
• Figure 2: x-direction reconstruction at the left side of the interface ${\rm \Gamma}_{i+1/2,j}$. Normal velocity and tangential velocity(from left to right).
• Figure 3: Shu-Osher problem: the density distributions and local enlargement at $t=1.8$ with a cell size $\Delta x=1/40$. (a-b) Different reconstruction methods using the S2O4 GKS solver. (c-d) Different reconstruction methods using the SSP-RK3 L-F solver. The reference solution is obtained by the 1-D fifth-order WENO-AO GKS with 10000 meshes.
• Figure 4: Interaction of planar shocks problem: the density distributions and local enlargement at $t=0.6$ with $500\times 500$ meshes. (a-b) Different reconstruction methods using S2O4 GKS solver. (c-d) Different reconstruction methods using SSP-RK3 L-F solver.
• Figure 5: Interaction of planar contact discontinuous: the density distributions at $t=1.6$ with $800\times 800$ meshes. (a-b) Different reconstruction methods using S2O4 GKS solver. (c-d) Different reconstruction methods using SSP-RK3 L-F solver.