To tackle the complexities of many practically relevant transport problems, we often encounter mathematical systems involving more than a few partial differential equations, and discretized on a large number of grid points. In order to find ways to solve such large number of matrix systems efficiently, two approaches can be devised, namely (i) to improve the convergence rate of an iterative method employed to solve large linear matrix systems and/or to handle the nonlinearity of the physical phenomena, and (ii) to expedite the rate of the floating-point computation. In this chapter we will present both the multigrid method to accelerate convergence rate and the parallel computing technique to expedite the data processing speed. We will discuss parallel computing first, then the multigrid technique in the context of a parallel computing environment.
Parallel computing introduces new issues distinct from the physical modeling issues and the numerical accuracy, stability, and consistency issues that are fundamental to all scientific computing problems. The goal of this chapter is to illustrate some of the important current issues that arise in the context of pressure-based methods for incompressible fluid dynamics, in particular for data-partitioned problems.
Data-partitioned problems arise naturally in many scientific computing problems because the laws of physics can be cast in terms of differential equations for scalar and vector field quantities, and the same equations apply concurrently at all points of the domain.