  The stability of difference schemes for, in general, hyperbolic systems of
conservation laws with source terms are studied. The basic approach is to
investigate the stability of a non-linear scheme in terms of its cor-
responding scheme in variations. Such an approach leads to application of the
stability theory for linear equation systems to establish stability of the
corresponding non-linear scheme. It is established the notion that a non-linear
scheme is stable if and only if the corresponding scheme in variations is
stable. A new modification of the central Lax-Friedrichs (LxF) scheme is
developed to be of the second order accuracy. A monotone piecewise cubic
interpolation is used in the central schemes to give an accurate approximation
for the model in question. The stability of the modified scheme are
investigated. Some versions of the modified scheme are tested on several
conservation laws, and the scheme is found to be accurate and robust. As
applied to hyperbolic conservation laws with, in general, stiff source terms,
it is constructed a second order nonstaggered central scheme based on
operator-splitting techniques.
