Implicit Gas Kinetic Lax–Wendroff Scheme for Steady Compressible Flows

This paper proposes an implicit gas kinetic Lax–Wendroff scheme (GKLWS)-based finite-volume Boltzmann method for steady compressible flows. The implicit framework couples the discrete velocity kinetic equation (DVKE) with the associated conservation moment equation (CME), both solved by a matrix-free lower–upper symmetric Gauss–Seidel method. In the present scheme, the GKLWS is employed to reconstruct the fluxes on the cell interface of the DVKE. Owing to the collision-invariance property of the projection matrix, the fluxes of the CME are obtained directly from the DVKE fluxes, circumventing the need for complicated reconstruction. The implicit CME system serves to predict the equilibrium distribution functions at the new time level, thereby removing the implicit dependence on the equilibrium distribution functions in the implicit DVKE system and providing strong convergence acceleration for the DVKE. Moreover, since the Jacobian matrix of the collision terms is not required, the implicit DVKE system can be closed and solved in a lightweight manner. To assess the accuracy and high efficiency of the present scheme, several benchmark test cases are conducted. Numerical results show that the proposed scheme retains the same numerical accuracy as its explicit counterpart, while achieving one to two orders of magnitude speedup for steady compressible flows.