  In this paper we discuss three symbolic approaches for the generation of a
finite difference scheme of a partial differential equation (PDE). We prove,
that for a linear PDE with constant coefficients these three approaches are
equivalent and discuss the applicability of them to nonlinear PDE's as well as
to the case of variable coefficients. Moreover, we systematically use another
symbolic technique, namely the cylindrical algebraic decomposition, in order to
derive the conditions on the von Neumann stability of a difference s cheme for
a linear PDE with constant coefficients. For stable schemes we demonst rate
algorithmic and symbolic approach to handle both continuous and discrete di
spersion. We present an implementation of tools for generation of schemes,
which rely on Gr\"obner basis, in the system SINGULAR and present numerous e
xamples, computed with our implementation. In the stability analysis, we use
the system MATHEMATICA for cylindrical algebraic decomposition.
