For advection simulation, an improved r‐factor algorithm for total variational diminishing (TVD) schemes is proposed to extend the TVD schemes to non‐uniform unstructured grids. In the new algorithm, the further upwind node (called node U) location is modified based on the size differences of the related grids, that is, the formula of the relationship between two length ratios, L_UC/L_CD = L_Cf/L_fD, should be maintained to reveal the meaning of r‐factor correctly on non‐uniform unstructured grids. After that, the inverse‐distance weighting average method rather than the Gauss theory method is adopted to estimate the value of node U. Furthermore, the deviations between the value of cell centroids and of their corresponding auxiliary points are exploited to complete the algorithm. The new algorithm is utilized in two pure convection cases, including a double‐step profile and a sinusoidal profile. The algorithm was compared against the Hou's r‐factor algorithm by using Superbee and Van Leer limiters on two‐dimension non‐uniform unstructured grids. The results indicate that a monotonous behavior and a higher accuracy result can be obtained by using the new algorithm.

中文翻译：

---

二维非均匀非结构化网格上总变分递减（TVD）方案的改进r因子算法

对于平流模拟，提出了一种用于总变分递减（TVD）方案的改进的r因子算法，以将TVD方案扩展到非均匀非结构化网格。在新算法中，根据相关网格的大小差异（即两个长度比率之间的关系公式L _UC / L _CD  =  L _Cf / L），修改了更上风节点（称为节点U）的位置_fD，应保持揭示r的含义正确地在非均匀非结构化网格上进行分解。此后，采用反距离加权平均法而不是高斯理论法来估计节点U的值。此外，利用单元质心的值及其对应的辅助点之间的偏差来完成算法。新算法在两种纯对流情况下使用，包括双步曲线和正弦曲线。通过在二维非均匀非结构化网格上使用Superbee和Van Leer限制器，将该算法与Hou的r因子算法进行了比较。结果表明，通过使用新算法可以得到单调的行为和更高的精度结果。