In the present study, the efficiency of preconditioners for solving linear systems associated with the discretized variable-density incompressible Navier-Stokes equations with semiimplicit second-order accuracy in time and spectral accuracy in space is investigated. The method, in which the inverse operator for the constant-density flow system acts as preconditioner, is implemented for three iterative solvers: the General Minimal Residual, the Conjugate Gradient and the Richardson Minimal Residual. We discuss the method, first, in the context of the one-dimensional flow case where a top-hat like profile for the density is used. Numerical evidence shows that the convergence is significantly improved due to the notable decrease in the condition number of the operators. Most importantly, we then validate the robustness and convergence properties of the method on two more realistic problems: the two-dimensional Rayleigh-Taylor instability problem and the three-dimensional variable-density swirling jet.

翻译：本研究探讨了在时间上采用半隐式二阶精度、空间上采用谱精度的离散变密度不可压缩Navier-Stokes方程求解中，预处理子对线性系统求解效率的影响。该方法以常密度流动系统的逆算子作为预处理子，针对三种迭代求解器（广义最小残差法、共轭梯度法和理查德森最小残差法）进行了实现。我们首先在密度呈顶帽型分布的一维流动情形下讨论了该方法。数值结果表明，由于算子条件数的显著降低，收敛性得到大幅改善。更重要的是，我们随后在两个更具实际意义的问题（二维Rayleigh-Taylor不稳定性问题和三维变密度旋流射流）中验证了该方法的鲁棒性和收敛特性。