Numerical stability analysis and rapid algorithm for calculations of fully developed laminar flow through ducts using time-marching method

In the numerical calculations of parabolic partial differential equations using explicit-type finite-difference methods (FDMs), the most fundamental finite-difference scheme is the time-forward, centered-space scheme. Under this scheme, several theoretical stability analysis methods have been established to avoid oscillations and divergences in numerical solutions. In solving elliptic partial differential equations using FDMs, the point successive over-relaxation method is one means to rapidly obtain steady state solutions, a major concern being to derive the optimum relaxation parameter. These problems have been solved theoretically over regular calculation domains. To be able to use FDMs freely over irregular calculation domains, the preceding two problems need to be reinvestigated but within a new theoretical framework. A theoretical approach is of great importance; nevertheless, it becomes so cumbersome that a numerical experimentation is the more realistic approach. A numerical approach—numerical stability analysis—is proposed and both problems are solved using similar algorithms. Although only the parabolic partial and elliptic differential equations concerned with Poiseuille flows are investigated here, the conclusions have a wider generality.