  The monotonicity and 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 and monotonicity of a non-linear
scheme in terms of its corresponding 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. The main
methodological innovation is the theorems establishing the notion that a
non-linear scheme is stable (and monotone) if the corresponding scheme in
variations is stable (and, respectively, monotone). Criteria are developed for
monotonicity and stability of difference schemes associated with the numerical
analysis of systems of partial differential equations. The theorem of
Friedrichs (1954) is generalized to be applicable to variational schemes with
non-symmetric matrices. 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 and monotonicity 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 scheme based on
operator-splitting techniques.
