Shocks, Detonations, and Deflagrations

Abstract

For a contact discontinuity we separated the variables into two sets based on their continuity across the interface. The continuous variables were copied into the ghost fluid in a node-by-node fashion, capturing the correct interface values, while the discontinuous variables were extrapolated in a one-sided fashion to avoid numerical dissipation errors. In order to apply this idea to a general interface moving at speed D in the normal direction, we need to correctly determine the continuous and discontinuous variables across the interface. For example, consider a shock wave where all variables are discontinuous, and extrapolation of all variables for both the preshock and postshock fluids obviously gives the wrong answer, since the physical coupling is ignored. We generally state, For each degree of freedom that is coupled across a discontinuity, one can define a variable that is continuous across the discontinuity, and all remaining degrees of freedom can be expressed as discontinuous variables that can be extrapolated across the discontinuity in a one-sided fashion, as the key to extending the GFM. For the Euler equations, conservation of mass, momentum, and energy can be applied to any discontinuity in order to abstract continuous variables; i.e., the Rankine-Hugoniot jump conditions always dictate the coupling between the prediscontinuity and postdiscontinuity fluids. This idea was proposed by Fedkiw et al.