A third order gas-kinetic scheme for unstructured grid

TL;DR

This paper introduces a third order gas-kinetic scheme for unstructured grids that combines compact least-square reconstruction with advanced iterative solvers, achieving high accuracy for various flow problems.

Contribution

It develops a novel third order gas-kinetic scheme using CLS reconstruction and efficient linear solvers, enhancing accuracy and computational efficiency for unstructured grid simulations.

Findings

Validated by multiple numerical tests including shock and vortex problems.

Demonstrated high accuracy in both incompressible and compressible flows.

Showcased advantages in simulating inviscid and viscous flows.

Abstract

In our study, a compact third order gas-kinetic scheme is constructed for unstructured grid which is combined the compact least-square reconstruction (CLS) method. The CLS method can achieve arbitrary high order compact reconstruction using the stencil from the whole computational domain implicitly. A large sparse linear system resulted from the CLS reconstruction method is solved by applying the generalized minimal residual algorithm (GMRES), and the Reverse-Cuthill-McKee (RCM) algorithm and the incomplete lower-upper (ILU) factorization method are implemented to accelerate the convergence of the iterative method. Different from the traditional flux solver, the BGK based gas-kinetic scheme is in the nature of spatial and temporal accuracy. Applying the second order expansion to the distribution function, the third order flux solver can be obtained directly. The accuracy of present method is validated by several numerical cases such as the advection of density perturbation problem, Sod shock wave problem, Lax shock tube test case, Shu-Osher problem, shock-vortex interaction, and lid-driven cavity flow. The advantages of this high order gas-kinetic scheme are exhibited in some benchmarks including incompressible flow and supersonic compressible flow, inviscid flow and viscous flow.