The finite difference preconditioning for higher-order compact scheme discretizations of non separable Poisson’s equation is investigated. An eigenvalue analysis of a one-dimensional problem is detailed for compact schemes up to the tenth-order. The analysis concludes that the spectrum is bounded irrespective of the mesh size and the continuous variable coefficient. Hence, combined to a multigrid method, the preconditioned Richardson method shows a convergence rate which is independent from the mesh size and the variable coefficient. Several numerical experiments, including the simulation of a flow with large density variations, confirm that the spectrum of the preconditioned operator remains bounded.

研究了不可分离泊松方程高阶紧致格式离散化的有限差分预处理。一维问题的特征值分析详细介绍了直至十阶的紧凑型方案。分析得出结论，频谱是有界的，与网格大小和连续可变系数无关。因此，结合到多网格方法，预处理的Richardson方法显示出收敛速度，该收敛速度与网格大小和可变系数无关。几个数值实验，包括对密度变化较大的流的模拟，证实了预处理算子的谱仍然是有界的。