An improved r‐factor algorithm for total variational diminishing (TVD) schemes on two‐dimension non‐uniform unstructured grids

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, LUC/LCD = LCf/LfD, 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. An improved r‐factor algorithm for TVD schemes is proposed to extend the TVD schemes to non‐uniform unstructured grids. The computational results show that the equation LUC/LCD = (LCf/LfD) is appropriate to determine the further upwind node U, comparing with Hou's method. After that, the inverse‐distance weighting average method is superior to the Gauss theory method to estimate the value of node U. Furthermore, the deviations between the value of cell centroids and of their corresponding auxiliary points need to be exploited. The numerical studies indicate that monotonous behavior and higher accuracy can be obtained by using the new algorithm, in comparison to Hou's r‐factor algorithm. Convection of a double‐step profile using Superbee scheme with coarse grids: (A) ϕ profile at y = 0.5, (B) ϕ profile at x = 0.75. Convection of a sinusoidal profile using Superbee scheme with coarse grids: (C) ϕ profile at y = 0.5, (D) ϕ profile at x = 0.75.