Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations

Abstract

In this paper we present a new formulation of the artificial viscosity concept. Physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated. The first important property is that by definition a viscosity must always be dissipative, transferring kinetic energy into internal energy, and must never act as a false pressure. The artificial viscous force should be Galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion, and remain zero for the latter case. These requirements significantly constrain the functional form that the artificial viscous force can have. In addition, an artificial viscosity should be able to distinguish between shock-wave and adiabatic compression, and not result in spurious entropy production when only the latter is present. It must therefore turn off completely for self-similiar motion, where only a uniform stretching and/or a rigid rotation occurs. An additional important, but more subtle, condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front, since the velocity is only discontinuous in the normal direction. Our principal result is the development of a new formulation of an edge-centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards, and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary. Our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions. A central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions, and to substantially reduce its effect when strong velocity gradients are absent.