Abstract:
This paper firstly reviews the development of the high-order off-lattice Boltzmann method (OLBM). Based on the high-order flux reconstruction scheme (FR), we develop a high-order flux reconstruction lattice Boltzmann method (FRLBM). Two numerical methods are adapted. One is to directly solve the discrete velocity Boltzmann equation with collision term treated implicitly (direct method) and the other is to sequencially implement collision step and the convection term (two-step method). Through the convergence study, the accuracy and stability of the two methods are compared. It is found that, when the time step is small, the errors of the two methods are in similar order, and the high-order accuracy can be obtained. However, when the time step is increased, the error of the direct method is almost unchanged, while the error of the two-step method increases significantly. The results indicate that the direct method can use a larger time step to obtain higher accuracy to abtain the reasonable accuracy in comparing with two step method. Meanwhile, the direct method also has better stability. Then, the ability of FRLBM to capture the details of the flow field is verified by simulating the lid-driven cavity flow. In addition, we compare the maximum Courant-Friedrichs-Lewy (CFL) numbers at different Reynolds numbers by semi-implicit time-marching scheme, first, second and third-order implicit-explicit (IMEX) Runge-Kutta schemes, respectively. The numerical results indicate the second-order IMEX performs best. Finally, the numerical simulation of the flow over a circular cylinder verifies the reliability of the FRLBM for calculation of the flow around the body with complex geometry.