A fully discrete Crank--Nicolson Leap--Frog (CNLF) scheme is proposed and analyzed for the unsteady bioconvection flow problem with concentration-dependent viscosity. Spatial discretization is handled via the Galerkin finite element method (FEM), while temporal discretization employs the CNLF method for the linear terms and a semi-implicit approach for the nonlinear terms. The scheme is proven to be unconditionally stable, i.e., the time step is not subject to a restrictive upper bound. Using the energy method, $L^2$-optimal error estimates are derived for the velocity and concentration . Finally, numerical experiments are presented to validate the theoretical results.

翻译：本文针对具有浓度依赖粘性的非定常生物对流流动问题，提出并分析了一种全离散的Crank-Nicolson蛙跳（CNLF）格式。空间离散采用Galerkin有限元方法（FEM）处理，时间离散则对线性项采用CNLF方法，对非线性项采用半隐式方法。该格式被证明是无条件稳定的，即时间步长不受限制性上界的约束。利用能量方法，推导出了速度和浓度的$L^2$范数最优误差估计。最后，通过数值实验验证了理论结果。