A Matrix-free Implicit Solution Algorithm for Incompressible Flows on Hybrid Unstructured Grids A. G. Malan ⋆ 1 , J. P. Meyer ⋆ and R. W. Lewis † ⋆ Deptartment of Mechanical & Aeronautical Engineering, University of Pretoria, Pretoria 0002, Rep. of South Africa, Web page: {http://www.up.ac.za/academic/mae/} ‡ Department of Mechanical Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, United Kingdom, Web page: {http://www.engineering.swan.ac.uk/mech eng.htm} Abstract The complex flow regimes resulting from the intricate geometries prevalent in industrial incom- pressible flow processes necessitate the use of a numerical simulation tool which is applicable to hybrid unstructured type meshes while offering a high degree of computational efficiency. This paper assesses the convergence characteristics of an implicit matrix-free Generalized Minimal Residual (GMRES) method which is applied to a hybrid unstructured preconditioned artificial compressibility algorithm recently developed by the authors. The system of discrete equations is Newton linearized where analytical expressions for the Jacobian terms are employed. The convergence characteristics of the solver is compared to that of two other matrix free algorithms viz. explicit with local time-stepping and lower-upper symmetric Gauss-Seidel) (LU-SGS) by application to the solution of incompressible flow on a hybrid unstructured mesh. 1 Introduction Numerical simulation of incompressible fluid flow is of great practical importance due to its many industrial applications. These range from hydrodynamics and low speed aerodynamics to the natural and forced convection systems found in heat exchangers and electronic cooling devices. The involved spectrum of flow regimes and wide range of length scales has resulted in the need for an efficient numerical tool which may be readily applied to simulate the extensive range of flow conditions without prior ad hoc modifications. The use of unstructured meshes for fluid dynamics problems has become widespread due to their inherent suitability for the spatial decomposition of complex geometries often present in industrial applications. Unfortunately, a numerical solution on such grids is computationally not competitive with the structured counterparts in terms of storage and the number of operations required (Sbarbella and Imregyn [1] and Sørensen [2]). Another advantage of structured de- composition is the improved accuracy resulting from the ability to align edges in a preferential manner in the case of a field which contains large gradients in a particular direction (Sbar- bella and Imregyn [1] and Khawaja and Kallinderis [3]). The favorable characteristics of both decomposition systems may, however, be exploited by meshing different regions of a particular spatial domain with the most suitable type i.e. structured or unstructured. The result is a hybrid unstructured mesh. Early matrix free temporal discretization methods used in conjunction with unstructured spatial discretization algorithms focussed on purely explicit type schemes. Many researchers employed 1 Corresponding author.