Singular matrix inversion in fluid dynamics computations

In certain types of hydrodynamic calculation the solution of a field equation, after finite differencing, resolves into the inversion of a singular matrix though with a consistent source. There are no physical or mathematical objections to this in principle, but in terms of computer application round-off error can make the set inconsistent. In this case no solution would be possible and this would indeed be discovered with direct methods of matrix inversion.However, with an iterative method such as ADI well-behaved ‘solutions’ are obtained, leading to satisfactory convergence of the overall problem. This article explains why this is so: it is shown that the difference in the computed field variable from the value appropriate to a consistent source is related to the error in the source by a finite matrix multiplier. Hence if the inconsistency in the source is small, e.g. owing to round-off, the error in the computed field variable is likely to be acceptable.