Alternating Cell Direction Implicit Method Using Approximate Factorization on Hybrid Grids (Advances in Applied Mathematics and Mechanics, 2022)
In this study, a novel fast-implicit iteration scheme called the alternatingcell direction implicit (ACDI) method is combined with the approximate factorizationscheme. This application aims to offer a mathematically well-defined version of theACDI method and to increase the accuracy of the iteration scheme used for the numerical solutions of partial differential equations. The ACDI method is a fast-implicitmethod that can be used for unstructured grids. The use of fast implicit iteration methods with unstructured grids is not common in the literature. The new ACDI methodhas been applied to the unsteady diffusion equation to determine its convergence andtime-dependent solution ability and character. The numerical tests are conductedfor different grid types, such as structured, unstructured quadrilateral, and hybridpolygonal grids. Second, the ACDI was applied to the unsteady advection-diffusionequation to understand the time-dependent and progression capabilities of the presented method. Third, a full potential equation solution is created to understand thecomplex flow solving ability of the presented method. The results of the numericalstudy are compared with other fast implicit methods, such as the point Gauss–Seidel(PGS) and line Gauss–Seidel (LGS) methods and the fourth-order Runge-Kutta (RK4)method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the abilities of the ACDI method. Due to the newACDI method, the approximate factorization method, which is used only in structuralgrids that are known to be advantageous, can be applied to any mesh structure.