非結構化增量修正多重網格法的概括性研究

Generalization of the Unstructured Additive Correction Multigrid Method

本篇論文將擴展目前增量修正多重網格法的處理方式，其中著重在計算變數的修正方向與方程式的結合方式。根據這樣的概念，推導出兩個有別於傳統的增量修正法：最小誤差與正交增量修正法。前者可以在給定變數修正方向的條件下，得到對應方程式最小誤差的結合權重值；相反地，後者可以在已知方程式的結合權重值，獲得與變數平面互為垂直的修正方向。另外，本文又提出根據格點相鄰幾何基礎的連結式Gauss-Seidel法。在實例驗證結果中可知，正交增量修正法可以在流場Reynolds 數較小時得到較快速的收歛結果，而使用最小誤差增量修正法即使在極高的Reynolds 數時，仍可獲得收歛結果；至於連結式Gauss-Seidel法在非結構化格點計算中則可以有效地增加求解效率。

The present study aims at generalizing the additive correction multigridmethod. Based on its underlying perceptions, one can derive twouseful alternatives: minimum-error and orthogonal additive corrections.The former determines the weighting factors for equation agglomerationwith a given solution correction direction; the latter yields a solutioncorrection direction normal to the solution hyper-plane provided that theweighting factors have been specified. Furthermore, a linked Gauss-Seidel scheme complying with solution information propagation is suggestedto improve convergence effectiveness. From numerical experiments,it is found that the multigrid strategy with orthogonal additivecorrection yields a more efficient solution procedure in the cases of lower Reynolds number flows. On the other hand, that with minimum-error additivecorrection still provides a convergent solution for higher Reynoldsnumber flows. Meanwhile, the solution convergent rate is effectivelyincreased by the linked Gauss-Seidel scheme, especially in problems formulatedwith unstructured grids.