On monotonicity, stability, and construction of central schemes for hyperbolic conservation laws with source terms

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 for the accurate solution of hyperbolic conservation laws is presented. A monotone piecewise cubic interpolation is used to modify LxF scheme 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.