A Memory Reduction Compact Gas Kinetic Scheme on 3D Unstructured Meshes

TL;DR

This work tackles the memory burden of high-order HWENO-based compact gas-kinetic schemes on 3D unstructured meshes by introducing a memory-reduction CGKS with a two-step linear reconstruction that preserves $3^{\text{rd}}$-order spatial accuracy while avoiding a large reconstruction matrix. The approach uses a time-evolving gas distribution function from the BGK framework to obtain interface fluxes and to update cell-averaged slopes within a finite-volume formulation, enabling matrix-free computations. Extensive 3D tests—from subsonic to hypersonic regimes on hybrid meshes—demonstrate comparable accuracy to the original CGKS but with about $20$–$30\%$ lower computational cost and strong robustness on complex, large-scale grids. The method shows promise for industrial-scale simulations and can be extended with ALE and multi-GPU strategies in future work.

Abstract

This paper introduces a memory-reduction third-order compact gas-kinetic scheme (CGKS) for solving compressible Euler and Navier-Stokes equations on 3D unstructured meshes. The scheme utilizes a time-evolution gas distribution function to provide a time-evolution solution at cell interfaces, enabling the implementation of Hermite WENO techniques for high-order reconstruction. However, the HWENO method needs to store a coefficients matrix for the quadratic polynomial to achieve third-order accuracy, resulting in high memory usage. A novel reconstruction method, built upon HWENO reconstruction, has been designed to enhance computational efficiency and reduce memory usage compared to the original CGKS. The simple idea is that the first-order and second-order terms of the quadratic polynomials are determined in a two-step way. In the first step, the second-order terms are obtained from the reconstruction of a linear polynomial of the first-order derivatives by only using the cell-averaged slopes, since the second-order derivatives are nothing but the "derivatives of derivatives". Subsequently, the first-order terms left can be determined by the linear reconstruction only using cell-averaged values. Thus, we successfully split one quadratic least-square regression into several linear least-square regressions, which are commonly used in a second-order finite volume code. Since only a small matrix inversion is needed in a 3-D linear least-square regression, the computational cost for the new reconstruction is dramatically reduced and the storage of the reconstruction-coefficient matrix is no longer necessary. The proposed new reconstruction technique can reduce the overall computational cost by about 20 to 30 percent. The challenging large-scale unsteady numerical simulation is performed, which demonstrates that the current improvement brings the CGKS to a new level for industrial applications.

Figures (27)

• Figure 1: The procedure of memory reduction reconstruction for $p^2$.
• Figure 2: The framework of memory reduction CGKS.
• Figure 3: Meshes used in accuracy test. Left: hexahedron mesh. Right: tetrahedron mesh.
• Figure 4: Original CGKS versus Simplify CGKS of CPU time. Left: hexahedron mesh. Right: tetrahedron mesh.
• Figure 5: Subsonic flow around a cylinder. Left: hexahedron mesh. Right: Mach number contour with streamlines.