Improved fully implicit discrete-velocity method for efficient simulation of flows in all flow regimes

Abstract

In this work, an improved fully implicit discrete-velocity method (DVM) is developed for flows over all Knudsen numbers. The improvements aim to overcome the drawbacks of the conventional semi-implicit DVM which performs with low accuracy and efficiency in the continuum flow regime. These defects originate from the explicit discretization of the equilibrium state in the collision term at the cell center and the ignorance of the particle collision effect in the calculation of numerical flux on the cell interface. To alleviate these drawbacks, the present method innovatively incorporates an implicit prediction step into the conventional model. In this step, the macroscopic governing equation is resolved to estimate the equilibrium state, and the local solution of the Boltzmann equation with the collision term is adopted to physically reconstruct the macroscopic numerical flux. Two major merits are brought by this strategy. On one hand, by using the predicted equilibrium state, the collision term in the discrete velocity Boltzmann equation can be discretized implicitly, which significantly improves the computational efficiency. On the other hand, the consideration of the collision effect in the physical reconstruction of the numerical flux on the cell interface benefits the solution accuracy, especially in the continuum flow regime. Meanwhile, the developed scheme well keeps the inherent simplicity of the conventional semi-implicit DVM by maintaining the basic resolving process of distribution functions. Numerical results show that in the rarefied flow regime, the improved scheme gives a similar solution as the conventional semi-implicit DVM with little extra computational efforts, while in the continuum flow regime, the present scheme shows higher efficiency and accuracy.