Numerical investigation of vacuum formation in compressible flows

Pierre Gremaud
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University

Can one "tear a hole" in a compressible fluid? This seemingly simple question has stymied mathematicians for quite some time. A numerical investigation will be presented in the case of compressible Navier-Stokes flows. We will start by very briefly reviewing what is known analytically about this question. The difficulties that have so far prevented its theoretical resolution contribute to making this problem a very challenging one numerically. The spatial discretization is based on pseudospectral methods; on the other hand, the temporal integration has to be able to overcome extreme stiffness. Those concepts will be reviewed in details. The numerical results seem to indicate that there may indeed be vacuum formation for multidimensional compressible Navier-Stokes flows. Those results are compared to benchmark exact results for both the inviscid case (Euler flow) for which vacuum formation is well known and for one-dimensional Navier-Stokes flows.

Joint work with K. DeVault and H.K. Jenssen.