This work pertains to the implicit high order compact discretization of the Navier-Stokes (N-S) equations on nonuniform grid. Subsequently, the discretization is used to approximate the Boussinesq equation as well. Contrary to earlier works on nonuniform grids this newly developed scheme is based on a comparatively smaller five-point stencil and leads to an algebraic system of equations with constant coefficients. The scheme carries the flow variable and its gradients as unknown and is seen to report back truncation accuracy of order four for linear flow problems even in a nonuniform mesh. Temporally the scheme is second-order accurate. Both primitive and vorticity-streamfunction formulations have been successfully tackled using the proposed formulation. Verification and validation studies were carried out to establish the efficiency of the formulation in conjunction with both Dirichlet and Neumann boundary conditions. Simulation of interior and exterior flow problems near-critical Hopf bifurcation points using a comparatively lesser number of grid points helps establish the robustness of the scheme. The numerical solution obtained by solving the Boussinesq equation for the problem of natural convection reveals the wider applicability of the scheme involving heat transfer.

这项工作涉及非均匀网格上Navier-Stokes（NS）方程的隐式高阶紧凑离散化。随后，离散化也用于近似Boussinesq方程。与先前在非均匀网格上的工作相反，该新开发的方案基于相对较小的五点模板，并导致了具有恒定系数的代数方程组。该方案携带未知的流量变量及其梯度，并且即使在非均匀网格中，也可以报告线性流量问题的四阶截断精度。该方案暂时是二阶准确的。使用提议的公式已成功解决了原始和涡流函数公式。进行了验证和确认研究，以结合Dirichlet和Neumann边界条件确定制剂的效率。使用相对较少的网格点来模拟接近临界Hopf分叉点的内部和外部流动问题，有助于建立该方案的鲁棒性。通过求解Boussinesq方程的自然对流问题而获得的数值解表明，该方案涉及传热，具有更广泛的适用性。