ANALYSIS OF CONVERGENCE ACCELERATION TECHNIQUES USED IN UNSTRUCTURED ALGORITHMS IN THE SOLUTION OF AERONAUTICAL PROBLEMS - PART I Edisson Sávio de Góes Maciel Rua Demócrito Cavalcanti, 152 - Afogados Recife – PE – Brazil - CEP 50750-080 e-mail: edissonsavio@yahoo.com.br Abstract. This work intends to study convergence accelerating techniques applied to the solution of steady state problems. Techniques like spatially variable time step, implicit residual smoothing, enthalpy damping and increase of the CFL number during iterative process are studied. The Jameson and Mavriplis explicit algorithm, with centered spatial discretization and the artificial dissipation operator of Mavriplis, was implemented to perform the two- dimensional numerical experiments. This scheme is second order accurate in space. The Euler equations in conservative form, employing a finite volume formulation and an unstructured spatial discretization are solved. The techniques were studied and compared with themselves in the solution of the physical problem of the transonic flow around a NACA 0012 airfoil, with zero angle of attack. The employed time march algorithm was the Runge-Kutta method of five stages and second order. The results have shown that the best technique studied was enthalpy damping, having better convergence gain than the others and have moderate computational cost. A final analysis of the technique computational performances is accomplished at the end (cost, maximum CFL number and iterations to convergence). Keywords: Spatially variable time step, Implicit residual smoothing, Enthalpy damping, Increase of CFL, Unstructured algorithm. 1. Introduction A point of intense research in CFD, Computational Fluid Dynamics, is the development of convergence acceleration techniques to steady state problems. The great amount of aerodynamics data required for aeronautical and aerospace vehicle design are obtained from these problems: Kutler (1985); Stiles and Hoffman (1985); and Nicola, Tognaccini, Visingardi and Paparone (1994); and others. Convergence acceleration techniques always were great objectives of CFD following the development of fluid mechanics equation solvers. A great amount of techniques were elaborated and several works tried to develop convergence acceleration computational tools which produce more efficient and, at the same time, less expensive codes. On the context of convergence gains, an efficient technique that raised to numerical calculation was the “multigrid” procedure. Such technique was initially presented to obtain solutions of problems involving elliptic partial differential equations (Brandt, 1981) and, posteriorly, was extended to hyperbolic partial differential equations. Some works that used this procedure were: Jameson and Mavriplis (1986); Mavriplis (1988); Radespiel (1989); and Mavriplis (1995). Techniques of convergence acceleration simpler and less expensive to explicit and implicit schemes, when applied to the solution of fluid mechanics equations, also were studied: spatially variable time step, enthalpy damping, residual smoothing and increase of CFL number during the iterative process. The spatially variable time step aims to use the maximum time step allowed by a local stability limit in each computational cell. This technique was initially applied by Jameson, Schmidt and Turkel (1981). Other works that used such technique were: Jameson and Mavriplis (1986); Mavriplis (1990); Swanson and Radespiel (1991); Luo, Baum and Löhner (1994); and Radespiel and Swanson (1995). The enthalpy damping strategy is based on the fact that the Euler equations are inviscid and the total enthalpy keeps constant in all computational domain when the steady state is reached and the mass and the energy equations are satisfied. Forcing terms proportional to the difference between local total enthalpy and the freestream total enthalpy are added to the mass, momentum and energy conservation equations, aiming to accelerate the convergence process to steady state condition. This procedure was suggested by Jameson, Schmidt and Turkel (1981). Others works were: Jameson and Mavriplis (1986); Baruzzi, Habashi and Hafez (1991) and Long, Khan and Sharp (1991). The maximum time step that can be used is limited by the CFL condition (Courant, Friedrichs and Lewy, 1928), which defines that the dependence domain of the discretized equations need no minimal contains the dependence domain of the original differential equation. To reduce this restriction, an implicit residual smoothing is developed to increase the scheme stability. Such strategy consists in determining an average residual value, weighted by neighbor residual values. This technique was introduced by Jameson and Mavriplis (1986) work. Others works were: Mavriplis (1990); Long, Khan and Sharp (1991); Luo, Baum and Löhner (1994); and Radespiel and Swanson (1995). Other convergence acceleration technique simpler than the others is the increase of the CFL number during the convergence process to reach steady state. This technique consists in increasing the CFL number all time that a prescribed number of iterations is reached. The user determines the number of iterations to increase the CFL as the increase value to be considered too. This work presents some convergence acceleration techniques generally applied to CFD community. The objective of this article is to present the best results obtained, in terms of convergence ratio and computational cost, for the studied techniques. The spatially variable time step, the implicit residual smoothing, the enthalpy damping and the