Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes II: unsteady flows

TL;DR

This work develops an implicit high-order gas-kinetic scheme (HGKS) for compressible flows on three-dimensional unstructured meshes to overcome CFL-time-step restrictions in unsteady problems. It introduces a two-stage, third-order implicit time discretization solved via LUSGS or GMRES within both non-compact WENO and compact HWENO spatial reconstructions, achieving third-order temporal accuracy and high spatial fidelity. The framework is validated through extensive 3D tests (inviscid and viscous) showing accurate shock capturing, robustness across complex geometries, and substantial efficiency gains over explicit schemes, especially when mesh size variation is large. The approach thus offers a scalable, high-accuracy tool for unsteady, high-speed flows on unstructured meshes with practical performance improvements.

Abstract

For the simulations of unsteady flow, the global time step becomes really small with a large variation of local cell size. In this paper, an implicit high-order gas-kinetic scheme (HGKS) is developed to remove the restrictions on the time step for unsteady simulations. In order to improve the efficiency and keep the high-order accuracy, a two-stage third-order implicit time-accurate discretization is proposed. In each stage, an artificial steady solution is obtained for the implicit system with the pseudo-time iteration. In the iteration, the classical implicit methods are adopted to solve the nonlinear system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and generalized minimum residual (GMRES) methods. To achieve the spatial accuracy, the HGKSs with both non-compact and compact reconstructions are constructed. For the non-compact scheme, the weighted essentially non-oscillatory (WENO) reconstruction is used. For the compact one, the Hermite WENO (HWENO) reconstruction is adopted due to the updates of both cell-averaged flow variables and their derivatives. The expected third-order temporal accuracy is achieved with the two-stage temporal discretization. For the smooth flow, only a single artificial iteration is needed. For uniform meshes, the efficiency of the current implicit method improves significantly in comparison with the explicit one. For the flow with discontinuities, compared with the well-known Crank-Nicholson method, the spurious oscillations in the current schemes are well suppressed. The increase of the artificial iteration steps introduces extra reconstructions associating with a reduction of the computational efficiency. Overall, the current implicit method leads to an improvement in efficiency over the explicit one in the cases with a large variation of mesh size.

Figures (10)

• Figure 1: Accuracy test: the computational mesh distribution with $20^3\times6$ tetrahedral cells.
• Figure 2: One-dimensional Riemann problem: the density, velocity and pressure distributions for Sod problem (left) at $t=0.2$ and Lax problem (right) at $t=0.14$ at the center horizontal line.
• Figure 3: Two-dimensional Riemann problem: the density distribution of S2O4-E (top), S2O3-L (middle) and S2O3-G (bottom) methods $t=0.4$.
• Figure 4: Viscous shock tube: the density distributions with non-compact and compact S2O3-L at $t=1.0$.
• Figure 6: Viscous shock tube: the density distribution along the lower wall at $t=1.0$.