This paper presents a memory-reduction third-order compact gas-kinetic scheme (CGKS) designed to solve compressible Euler and Navier-Stokes equations on 3D unstructured meshes. Utilizing the time-accurate gas distribution function, the gas kinetic solver provides a time-evolution solution at the cell interface, distinguishable from the Riemann solver with a constant solution. With the time evolution solution at the cell interface, evolving both the cell-averaged flow variables and the cell-averaged slopes of flow variables becomes feasible. Therefore, with the cell-averaged flow variables and their slopes inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. However, the HWENO reconstruction method requires the storage of a reconstruction-coefficients matrix for the quadratic polynomial to achieve third-order accuracy, leading to substantial memory consumption. This memory overhead limits both computational efficiency and the scale of simulations.
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 3 × 3 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 memory reduction CGKS is tested for both inviscid and viscous flow at low and high speeds on hybrid unstructured meshes. The proposed new reconstruction technique can reduce the overall computational cost by about 20% to 30%. In the meantime, it also simplifies the algorithm. 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.
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