Discrete maximum principle and the Ultraviolet Catastrophe of finite difference schemes on staggered Cartesian grids for heterogeneous and anisotropic diffusion equations.

摘要

The maximum principle, closely related to the non-negativity property, is a basic characteristic of second order PDEs of parabolic type. Its preservation for solutions to corresponding discretized problems is a natural requirement in reliable and meaningful numerical modeling of various real-life phenomena. Finite difference or finite volume methods on staggered Cartesian grids have the advantage of being easily parallelizable, for example in CUDA GPUs, with several processes performing at the same time to greatly decrease the computational costs. The aim of this report is to give a uniform introduction to finite difference/volume schemes for approaching anisotropic and heterogeneous diffusion equations, for which the validity of the discrete maximum/minimum principle is satisfied. Details are provided about the stability analysis for one-dimensional and two-dimensional problems, through the algebraic theory of positive matrices, to determine the range of numerical parameters under which fundamental properties are fulfilled. An extensive series of numerical tests is proposed to experimentally validate the theoretical results.