Discrete unified gas kinetic scheme for incompressible flow

In this work, an improved discrete unified gas kinetic scheme (DUGKS) is proposed. Like the original DUGKS and the optimized DUGKS, the improved DUGKS uses trapezoidal rule for the integration of the collision term, and introduces two auxiliary distribution functions with the inclusion of collision effect. Different from the original DUGKS and the optimized DUGKS, the improved DUGKS uses rectangular rule to implicitly integrate the convection term, and adopts trapezoidal rule and Taylor expansion to introduce the distribution function at the node to evaluate the interface flux at the next time step, which can be considered as a new version of the optimized DUGKS with implicit treatment. Combining the benefits of the choice of integration method, the design of implicit scheme, and the construction of interface flux, numerical results demonstrate that the improved DUGKS has a much better numerical stability than the original DUGKS and the optimized DUGKS while still maintaining better accuracy and higher efficiency. In terms of computational efficiency, the improved DUGKS improves the efficiency by fast convergence and reducing the number of auxiliary points. Among the three kinds of DUGKS, the improved DUGKS requires the fewest iteration steps and the least calculation time to reach convergence. Moreover, with the increase of the number of grids, the iteration steps and computation time decrease more. In terms of accuracy, the improved DUGKS has some improvements. Numerical tests of Couette flow and Taylor-Green vortex flow show that the error between the results obtained by the improved DUGKS and the analytical solution is the least. In terms of numerical stability, the significant improvement of the improved DUGKS is shown. In numerical tests of Couette flow, Poiseuille flow, and lid-driven cavity flow, the original DUGKS and the optimized DUGKS do not converge with a low-resolution mesh and high Re number, but the improved DUGKS can still reach convergence while still maintaining good accuracy. The ratio of time step to relaxation time is about 8×10^{6} times larger than that of the original DUGKS and the optimized DUGKS, which greatly enlarges the stability region.