A pressure-based preconditioner for multi-stage artificial compressibility algorithms

A differential preconditioner is developed for accelerating the convergence of explicit, multi-stage, artificial compressibility algorithms using ideas from pressure-based methods. The velocity derivatives in the continuity equation and the pressure gradient terms in the momentum equations are discretized in time implicitly. The discrete system of equations is linearized in time producing a block implicit operator which is approximately factorized and diagonalized via a similarity transformation. The so derived diagonal operator depends only on the metrics of the geometric transformation and can, thus, be implemented in an efficient and straightforward manner. It is combined with the standard implicit residual smoothing operator and 'incorporated in a four-stage Runge-Kutta algorithm also enhanced with local time stepping and multigrid acceleration. Linear stability analysis, for the coupled Navier-Stokes equations, and calculations of three-dimensional laminar flows through strongly curved square ducts and pipes demonstrate the damping properties and efficiency of the proposed approach.