Highly Parallel Alternating Directions Algorithm for Time Dependent Problems

In our work, 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. For this problem, a parallel algorithm based on a novel 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 the direction splitting. The scheme used in the algorithm is composed of two parts: (i) velocity prediction, and (ii) pressure correction. This is a Crank‐Nicolson‐type two‐stage time integration scheme for two and three dimensional parabolic problems in which the second‐order derivative, with respect to each space variable, is treated implicitly while the other variable is made explicit at each time sub‐step. 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 secon...