A single-stage high-order compact gas-kinetic scheme in arbitrary Lagrangian-Eulerian formulation

TL;DR

This paper develops a single-stage, high-order compact gas-kinetic scheme within an arbitrary Lagrangian-Eulerian (ALE) framework on structured meshes. By leveraging a time-accurate, third-order gas-kinetic flux and a simplified fourth-order compact reconstruction, the method achieves fourth-order spatial accuracy with significantly reduced computational cost, even as the mesh moves. The approach updates cell-averaged variables and their gradients directly from a gas-evolution model, enabling compact reconstruction and robust shock-capturing via GENO. Numerical experiments, including Riemann, Sedov, Noh, and Saltzmann problems, demonstrate high accuracy, robustness, and 2.4–3.0x speedups over prior reconstructions, validating the scheme’s efficiency and reliability for complex, moving-mesh compressible flows.

Abstract

This study presents the development of a compact gas-kinetic scheme using an arbitrary Lagrangian-Eulerian (ALE) formulation for structured meshes. Unlike the Eulerian formulation, the ALE approach effectively tracks flow discontinuities, such as shock waves and contact discontinuities. However, mesh motion alters the geometry and increases computational costs. To address this, two key strategies were introduced to reduce costs and enhance accuracy. The first strategy is to use the gas-kinetic scheme to construct a third-order gas-kinetic flux, rather than the Runge-Kutta method to achieve high-order time accuracy, which allows a single reconstruction and flux calculation per time step. This approach enables direct updates of both cell-averaged flow variables and their gradients using a time-accurate flux function, facilitating compact reconstruction. Second, the significant computational expense is spent on reconstruction, which requires recalculating the reconstruction matrix at each time step due to mesh changes. A simplified fourth-order compact reconstruction using a small matrix was used to mitigate this cost. The combination of fourth-order spatial reconstruction and third-order time-accurate flux evolution ensures both high resolution and computational efficiency in the ALE framework. The tests shows that the current reconstruction is 2.4x to 3.0x faster than the previous reconstruction. Additionally, a generalized ENO(GENO) method for handling discontinuities enhances the scheme's robustness. The numerical test cases, such as the Riemann problem, Sedov problem, Noh problem, and Saltzmann problem, demonstrated the robustness and accuracy of our method.

Figures (18)

• Figure 1: The reconstruction stencils of fourth-order reconstruction
• Figure 2: The solution of double shear layer with $192^2$ mesh elements at time $t=0.8$
• Figure 3: The vorticity distribution along the line $x=0.5$
• Figure 4: The density contours (contour levels from 0.2 to 3.0 with increments of 0.05) of a two-dimensional Riemann problem with contact discontinuities at time $t=0.4$
• Figure 5: The mesh distribution of two-dimensional Riemann problem with contact discontinuity at time $t=0.4$