Analysis of the second vorticity confinement scheme

The second vorticity confinement scheme proposed by Steinhoff is analysed in detail. First, starting from the 1D linear transport equation applied to a pulse, various formulations for the confinement are compared. The linear 2nd-order schemes (Warming-Beam and Lax–Wendroff schemes) give oscillatory solutions, while the nonlinear confinements of same accuracy behave as limiters with artificial compression. Not only these 2nd-order schemes with confinement conserve the pulse as accurately as a 3rd-order one for identical and sufficiently fine grids, but they remain stable with a global negative diffusion which allows them to conserve the pulse endlessly concentrated over 5–8 mesh cells in the computation, although the form of the equations is then lowered to 1st-order. Application of the energy method for stability analysis indicates that the signal relaxes towards a constant energy solution, for which the energy brought in by the anti-diffusive confinement is balanced by the energy removed from the solution by the diffusive terms. Application to the Euler equations for the advection of a 2D vortex proves that a similar approach can be applied to nonlinear problems. The most appropriate formulation of compressible vorticity confinement is to apply it identically to the incompressible formulation, i.e. without source term in the energy conservation equation.