An Improved High-order Adaptive Mesh Refinement Framework for Shock-turbulence Interaction Problems based on cell-centered finite difference schemes

TL;DR

This work presents a high-order, cell-centered AMR–WENO framework for efficient, conservative shock–turbulence simulations on AMReX, using a staggered grid to ensure flux conservation at coarse–fine interfaces. A hybrid WENO–conservative interpolation strategy driven by troubled-cell detection balances accuracy in smooth regions with stability near discontinuities, and a sixth-order WENO-based restriction paired with a six-point prolongation enables robust inter-level communication. The method demonstrates high-order convergence on smooth problems, strong performance on challenging shock–interface tests, and robustness for multidimensional turbulence benchmarks, outperforming traditional non-conservative approaches. The framework offers significant practical impact for large-scale, turbulence-resolving simulations involving strong shocks, while acknowledging the need for formal stability analysis of the hybrid interpolation scheme.

Abstract

This work presents a high-order finite-difference adaptive mesh refinement (AMR) framework for robust simulation of shock-turbulence interaction problems. A staggered-grid arrangement, in which solution points are stored at cell centers instead of at the vertices, is presented to address the boundary conservation issues encountered in previous studies. The key ingredient in the AMR framework, i.e., the high-order nonlinear interpolation method applied in the prolongation step together with the determination of fine-grid boundary conditions, are re-derived for staggered grids following the procedures in prior work [1] and are thus used here. Meanwhile, a high-order restriction method is developed in the present study as the coarse and fine grid solutions are non-collocated in this configuration. To avoid non-conservative interpolation at discontinuous cells that could incur instabilities, a hybrid interpolation strategy is proposed in this work for the first time, where the non-conservative WENO interpolation is applied in smooth regions whereas the second-order conservative interpolation is applied at shocks. This significantly mitigates the numerical instabilities introduced by non-conservative interpolation and pointwise replacement. The two interpolation approaches are seamlessly coupled through a troubled-cell detector achieved by a scale-irrelevant Riemann solver in a robust way. The present work is developed on a publicly available block-structured adaptive mesh refinement framework AMReX [2]. The canonical tests demonstrate that the proposed method is capable of accurately resolving a wide range of complex shock-turbulence interaction problems that have been proven intricate for existing approaches

Figures (19)

• Figure 1: Algorithmic flowchart of the subcycling AMR framework. The highlighted rounded rectangles indicate the procedures where a high-order interpolation schemes are required.
• Figure 2: Different setups for high-order schemes in an AMR grid for a one-dimensional case. Fine and coarse solution points are illustrated by the green squares and blue circles, respectively, while the gray squares with dotted edges represent the ghost solution points of the fine grid.
• Figure 3: Fifth-order WENO interpolation schemes for staggered AMR grids during prolongation for a refinement ratio of (a) 2 and (b) 4. Fine and coarse solution points are shown as green squares and blue circles, respectively, while flux points are indicated by minor ticks along the axis.
• Figure 4: A blended WENO/ENO interpolation schemes for staggered AMR grids during restriction for (a) interior points and (b) points near the boundaries. The red circle denotes the target coarse grid point to be updated. Green and gray squares represent the valid and ghost fine grid solution points used for construction the candidate stencils, respectively. Blue circles indicate neighboring coarse grid points that provide additional information near the boundary.
• Figure 5: A typical grid-refinement case where a discontinuity is present with buffer cells surrounded.