Answering von Neumann's conjecture on convergence of averages and analyzing the von Neumann-Richtmyer scheme via a material averaging method

摘要

A material averaging method is introduced and used to material-average certain partial differential equations that model the conservation laws of material dynamics. The resulting noncontinuum (semidiscrete) model for material dynamics is shown to lead to globally well-posed problems. Solutions of a time-discretization of the material-averaged equations are proved convergent. These results include the cases of the ideal gas (and generalizations thereof) with and without viscous stresses (i.e., both the viscous and the inviscid cases are included here). The material averaging method is so designed that a certain time-discretization of the material-averaged equations yields the widely known and much used scheme of von Neumann and Richtmyer. A convergence result for the von Neumann-Richtmyer scheme follows from the main theorem here. In addition to providing an answer to von Neumann's conjecture on the convergence of averages, these results also provide some progress on the von Neumann-Richtmyer problem.