An active-flux-type scheme for ideal MHD with provable positivity and discrete divergence-free property

Abstract

We develop a positivity-preserving (PP) PAMPA (Point-Average-Moment PolynomiAl-interpreted) scheme that enforces a discrete divergence-free (DDF) magnetic field for ideal MHD on Cartesian grids. Extending our 1D invariant-domain-preserving (IDP) PAMPA framework (Abgrall, Jiao, Liu, Wu, SIAM J. Sci. Comput., to appear) to multidimensional, multiwave MHD, the method combines a limiter-free PP update of interface point values via a new nonconservative reformulation with a local DDF projection. Cell averages are provably PP under a mild a~priori positivity condition on one cell-centered state, using: (i) DDF-constrained interface values, (ii) a PP limiter only at the cell center, (iii) a PP flux with appropriate wave-speed bounds, and (iv) a suitable discretization of the Godunov--Powell source term. The PP proof employs geometric quasi-linearization (GQL; Wu & Shu, SIAM Review, 2023), which linearizes the pressure constraint. The scheme avoids explicit polynomial reconstructions, is compatible with arbitrarily high-order strong-stability-preserving (SSP) time integration, and is simple to implement. Robustness and resolution are enhanced by a problem-independent Lax-type entropy troubled-cell indicator using only two characteristic speeds and a convex oscillation elimination (COE) mechanism with a new intercell-difference norm. Tests -- including a blast wave with plasma $\beta \approx 2.51\times 10^{-6}$ and jets up to Mach $10^{4}$ -- show high-order accuracy, sharp MHD-structure resolution, and strong-shock robustness. To our knowledge, this is the first active-flux-type ideal-MHD method rigorously PP for both cell averages and interface point values while maintaining DDF throughout.