Finite-difference schemes for first order differential equations having the correct linear stability properties, 1990

We prove that finite-difference schemes can be con-structed having exactly the same linear stability proper-ties as the corresponding differential equation. This holds for all step-sizes h > 0. The major implication of this result is that an important source of numerical instabilities is eliminated. The derived result is a conse�quence of using a generalized definition of the first derivative of a function