Self-adjusting,positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics

Several computational problems in science and engineering are stringent enough that maintaining positivity of density and pressure can become a problem. We build on the realization that positivity can be lost within a zone when reconstruction is carried out in the zone. We present a multidimensional, self-adjusting strategy for enforcing the positivity of density and pressure in hydrodynamic and magnetohydrodynamic (MHD) simulations. The MHD case has never been addressed before, and the hydrodynamic case has never been presented in quite the same way as done here. The method examines the local flow to identify regions with strong shocks. The permitted range of densities and pressures is also obtained at each zone by examining neighboring zones. The range is expanded if the solution is free of strong shocks in order to accommodate higher order non-oscillatory reconstructions. The density and pressure are then brought into the permitted range. The method has also been extended to MHD. It is very efficient and should extend to discontinuous Galerkin methods as well as flows on unstructured meshes.Media player