On Stability, Monotonicity, and Construction of Difference Schemes I: Theory

The stability and monotonicity of nonlinear difference schemes are studied. The basic approach is to investigate a nonlinear scheme in terms of its corresponding scheme in variations. The advantage of such an approach is that the scheme in variations will always be linear and, hence, enables the investigation of the stability and monotonicity for nonlinear operators using linear patterns. In part I of our two-part paper, we focus on the theoretical background. We establish the notion that the stability (and monotonicity) of a scheme in variations implies the stability (and, respectively, monotonicity) of its original scheme, and that a nonlinear explicit scheme is stable iff (if and only if) the corresponding scheme in variations is stable. Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations (PDEs). The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with nonsymmetric matrices. High-order interpolation and employment of monotone piecewise cubics in construction of monotone central schemes are considered. As applied to hyperbolic conservation laws with, in general, stiff source terms, we construct a second-order staggered central scheme based on operator-splitting techniques.