Approximate factorization method using alternating cell direction implicit method: comparison of convergence characteristics using basic model equations

Abstract

A fast-implicit iteration scheme called Alternating Cell Directions Implicit method is combined with Approximate Factorization scheme. Usage of fast implicit iteration methods with unstructured grids is not absent. The proposed method allows fast implicit formulations to be used in unstructured meshes, thus revealing the advantages of fast implicit schemes in unstructured meshes. Fast implicit schemes used in structural meshes have evolved considerably and are much more accurate, robust and faster than explicit schemes. It is a very important novelty development that such developed schemes can be applied to unstructured schemes. In unstructured meshes, implicit schemes have proceeded only to form a matrix filled with a full implicit approach. Steady incompressible potential flow problem is also solved with the scheme and the convergence characteristics for both structured and unstructured grids are observed. Working with Unsteady diffusion equation is subject of another publication. In Steady incompressible potential flow; Convergence character of the scheme compared with Runge-Kutta order 4 method, Laasonen method and Point Gauss Seidel iteration method. It is seen that scheme behaves like an approximation of fully implicit method (Laasonen Method) up to an optimum pseudo-time step size. This is a highly anticipated result because the approximate factorization method is an approach to fully implicit formulation. The results of the numerical study are compared with other fast implicit methods like Point Gauss Seidel and Line Gauss Seidel methods as well as Runge-Kutta order 4 (RK4) method which is an explicit scheme and Laasonen Method which is a fully implicit scheme. It is seen that the study increased the accuracy of the Alternating Cell Directions Implicit Method. As a result; The new ACDI method has been shown to be faster in unstructured grids than other methods, and It has been proven that it can be used for any kind of mesh construction.