Divergence-free Preserving Schemes: what’s wrong in SUPG and how to fix it
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The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.