SM stability for time-dependent problems
This work provides a framework for selecting optimal stable finite difference schemes for time-dependent PDEs, but it is incremental as it applies known Padé approximations to a specific class of problems.
The paper introduces the concept of SM-stable finite difference schemes for time-dependent problems, focusing on selecting unconditionally stable schemes that preserve key features of differential equations like diffusion and convection. It uses Padé approximations to construct such schemes for one-dimensional parabolic equations.
Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study, we use a simple boundary value problem for a one-dimensional parabolic equation to discuss the selection of an approximation with respect to time. We consider the pure diffusion equation, the pure convective transport equation and combined convection-diffusion phenomena. Requirements for the unconditionally stable finite difference schemes are formulated that are related to retaining the main features of the differential problem. The concept of SM stable finite difference scheme is introduced. The starting point are difference schemes constructed on the basis of the various Pad$\acute{e}$ approximations.