On Monotonicity of Difference Schemes for Computational Physics

Criteria are developed for monotonicity of linear as well as nonlinear difference schemes associated with the numerical analysis of systems of partial differential equations, integrodifferential equations, etc. Difference schemes are converted into variational forms that satisfy the boundary maximum principle and also allow the investigation of monotonicity for nonlinear operators using linear patterns. Sufficient conditions are provided to review the monotonicity of single and coupled difference schemes. Necessary as well as necessary and sufficient conditions for monotonicity of explicit schemes are also developed. The notion of submonotone difference schemes is considered and the associated criteria are developed. We discuss the interrelationship between monotonicity, submonotonicity, and stability. Some known schemes serve as examples demonstrating the implementation of the developed approaches. Among these examples, we describe the possibility that stable schemes such as total variation diminishing (TVD) as well as monotonicity preserving can produce spurious oscillations.