Towards generalization and optimization of implicit methods

A general implicit (GI) method for solving iteratively the algebraic system arising from a finite difference approximation of an elliptic partial differential equation is formulated. Under certain assumptions this method can be reduced to the already known implicit techniques. It is shown that the GI method has a very special physical meaning when solving fluid flow problems. It is shown also how this method can be optimized to achieve the maximum rate of convergence. Finally, it is shown how this new strategy is applied by solving some classical numerical fluid dynamics problems.