Application of the full approximation storage method to the numerical simulation of two-dimensional steady incompressible viscous multiphase flows

Abstract

SUMMARY In recent years multigrid algorithms have been applied to increasingly difficult systems of partial differential equations and major improvements in both speed of convergence and robustness have been achieved. Problems involving several interacting fluids are of great interest in many industrial applications, especially in the process and petro-chemical sectors. However, the multifluid version of the Navier‐Stokes equations is extremely complex and represents a challenge to advanced numerical algorithms. In this paper, we describe an extension of the full approximation storage (FAS) multigrid algorithm to the multifluid equations. A number of special issues had to be addressed. The first was the development of a customised, non-linear, coupled relaxation scheme for the smoothing step. Automatic differentiation was used to facilitate the coding of a robust, globally convergent quasi-Newton method. It was also necessary to use special inter-grid transfer operators to maintain the realisability of the solution. Algorithmic details are given and solutions for a series of test problems are compared with those from a widely validated, commercial code. The new approach has proved to be robust; it achieves convergence without resorting to specialised initialisation methods. Moreover, even though the rate of convergence is complex, the method has achieved very good reduction factors: typically five orders of magnitude in 50 cycles. © 1998 John Wiley & Sons, Ltd.