Performance Analysis of Parallel Alternating Directions Algorithm for Time Dependent Problems

We consider the time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. A parallel algorithm based on a new direction splitting approach is developed. Here, the pressure equation is derived from a perturbed form of the continuity equation in which the incompressibility constraint is penalized in a negative norm induced by direction splitting. The scheme used in the algorithm is composed by: pressure prediction, velocity update, penalty step, and pressure correction. In order to achieve a good parallel performance the solution of the Poison problem for the pressure correction is replaced by solving a sequence of one-dimensional second order elliptic boundary value problems in each spatial direction. The parallel code is developed using the standard MPI functions and tested on modern parallel computer systems. The performed numerical tests demonstrate the level of parallel efficiency and scalability of the direction-splitting based algorithm.