A low-order hybrid method for the variable-density incompressible Navier--Stokes equations
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In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier--Stokes (VDINS) equations based on hybrid velocity and piecewise constant density. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or polyhedral elements as well as non-matching interfaces. Stability is achieved by a careful discretisation of the advective and unsteady terms. Specifically, a discrete counterpart of the maximum principle is obtained through an upwind discontinuous Galerkin discretisation of the density advection equation; energy estimates for the velocity are, on the other hand, obtained by discretising the unsteady term in the momentum balance equation in the spirit of Guermond and Quartapelle (2000). In addition, we show existence and uniqueness of a discrete solution, and convergence of the latter to a suitably defined weak solution of the continuous problem. Numerical tests validate the theoretical results.
