Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation.

We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19-point fourth order discretization scheme to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equation with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact scheme (SOC), compared with the previously published fourth order compact scheme (FOC).

我们提出了一种六阶显式紧致有限差分格式来求解三维（3D）对流扩散方程。我们首先使用多尺度多重网格方法来求解由19点四阶离散格式产生的线性系统，以计算粗网格和细网格上的四阶解。然后，使用基于算子的插值格式结合外推技术来近似细网格上的六阶精确解。由于使用标准点松弛平滑器的多重网格方法在求解高雷诺数的对流扩散方程时可能无法达到最优的网格无关收敛速度，我们在多重网格求解器中实现了平面松弛平滑器以实现更好的网格无关性。给出了支持性的数值结果，以证明六阶紧致格式（SOC）与先前发表的四阶紧致格式（FOC）相比的效率和准确性。