An Adaptive Smoothing-Based Multigrid Algorithm for Flow

D. Drikakis
UMIST, Mechanical Engineering Department, United Kingdom

O. P. Iliev
ITWM, University of Kaiserslautern, Germany, and
Institute of Mathematics and Informatics, BAS, Bulgaria

D.P. Vassileva
Institute of Mathematics and Informatics, BAS, Bulgaria


Numerical simulation of real flow problems often requires significant CPU time even on powerful computers. Therefore accelerating the computations is a critical point in solving such problems.

Accelerating incompressible flow computations with an existing 3D CFD code is discussed here. The single grid code was developed for the artificial compressibility formulation of the incompressible Navier-Stokes equations by the first author (Drikakis et al., 1992,1993), and it was based on a characteristics-based method in conjunction with high-order upwind spatial discretization. The discretization with respect to time was realized by an explicit Runge-Kutta method. Furthermore, a non-linear multigrid algorithm was developed and implemented aiming at accelerating the computations (Drikakis, Iliev, Vassileva 1997,1998). The algorithm combined the full multigrid and full approximation storage (FMG-FAS) schemes. Different prolongation operators were implemented and investigated.

The above mentioned multigrid algorithm was further modified by implementing adaptive smoothing and this modification is the objective of the present talk. By adaptive smoothing we mean that the smoother acts only on an adaptively formed subset omegaS of the grid omega. The choice of the adaptivity criterium is an open question, and it might be problem dependent. We restrict our discussion here to solution of steady problems through an unsteady procedure. Currently we use the following criteria: define rP=uPnew-uPold, then omegaS = {P: |rP| > gamma |rmax|, P in omega }. The parameter gamma satisfies the condition 0 < gamma < 1. It is obvious, that the subset omegaS is identical with the full grid omega if gamma=0. In other words, the smoothing is performed adaptively in the subregions where the solution changes rapidly. In addition, we perform a complete smoothing after any nu adaptive smoothings. This is done in order to better propagate the information between different subregions.

The adaptive smoothing can be viewed as a further development of at least two approaches: i) so called "local solution" method (Drikakis 1994) which is based on reducing the computational domain through the computations; ii) Southwell method for hand-solving systems of linear algebraic equations. The last can be viewed as a variant of Gauss-Seidel method, exploiting adaptive ordering of unknowns, based on the range of residuals.

Results from numerical experiments are presented and discussed in order to demonstrate the impact of the adaptive smoothing on the acceleration of flow computations.