Localization and spreading of contact discontinuity layers in simulations of compressible dissipationless flows

We introduce a systematic approach to examine the localization and width of contact discontinuity layers (CDLs) in simulations of the compressible Euler equations. We study the Piecewise Parabolic Method (PPM) [4] and WENO [3], by simulating a single diffusing or shock-accelerated CDL in 1D and 2D (inclined planar interface). Here the density jump η is greater or less than unity (i.e., fast/slow {f/s} and slow/fast {s/f}, respectively). We advocate a point-wise algorithm for width extraction when the flows are nearly-discontinuous. We examine the CDL localization under mesh refinement and the CDL width spreading at long times. We find that, PPM has an asymmetrically developing CDL width. PPM introduces an artificial steepening for the f/s case and a spreading width CDL for s/f∝t1/3. The median point of the interface depends on the order of accuracy r. For PPM, r=2, it is located at a density, (ρ1+2ρ2)/3, consistent with analysis in [14]. For WENO, r=3, (5th order accurate in smooth monotone regions) the width of the CDL increases as t1/4, although there is a slight dependence on translation speed of the CDL. These observations are essential for establishing the validity of simulations of accelerated flows of high-gradient stratified and compressible media (Rayleigh–Taylor and Richtmyer–Meshkov environments), particularly for reshock configurations in 2D and 3D.