FORMULATIONS OF ARTIFICIAL VISCOSITY FOR MULTIDIMENSIONAL SHOCK-WAVE COMPUTATIONS

In this paper we present a new formulation of the artificial viscosity concept. Physical arguments for the origins of this term are given an d a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated. The first important property i s that by definition a viscosity must always be dissipative, transferr ing kinetic energy into internal energy, and must never act as a false pressure. The artificial viscous force should be Galilean invariant a nd vary continuously as a function of the criterion used to determine compression and expansion, and remain zero fur the latter case. These requirements significantly constrain the functional form that the arti ficial viscous force can have. In addition, an artificial viscosity sh ould be able to distinguish between shock-wave and adiabatic compressi on, and not result in spurious entropy production when only the latter is present. It must therefore turn off completely for self-similiar m otion, where only a uniform stretching and/or a rigid rotation occurs. An additional important, but more subtle, condition where the artific ial viscosity should produce no effect is alone the direction tangenti al to a convergent shock front, since the velocity is only discontinuo us 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 variable s that meets all of these standards, and without the need for problem dependent numerical coefficients that have in the past made the artifi cial viscosity method appear somewhat arbitrary. Our formulation and n umerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions. A centra l Feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to tu rn off the artificial viscosity for the above mentioned conditions, an d to substantially reduce its effect when strong velocity gradients ar e absent. (C) 1998 Academic Press.