Provably positivity-preserving constrained transport scheme for 2D and 3D ideal magnetohydrodynamics

Abstract

This paper proposes and analyzes a robust second-order positivity-preserving constrained transport (PPCT) scheme for ideal magnetohydrodynamics (MHD) on non-staggered Cartesian meshes. The PPCT scheme provably preserves two crucial physical constraints: a globally discrete divergence-free (DDF) condition on the magnetic field and the positivity of density and pressure. Achieving both properties in a single framework is essential for stability and physical fidelity in MHD simulations, yet this has proven to be a challenging endeavor (existing works, e.g., [S. Ding & K. Wu, SIAM J. Sci. Comput., 46: A50–A79, 2024], achieve positivity and only a locally DDF property simultaneously). The PPCT method is motivated by a novel splitting technique proposed in [T.A. Dao, M. Nazarov & I. Tomas, J. Comput. Phys., 508: 113009, 2024], which splits the MHD system into an Euler subsystem with a steady magnetic field and a magnetic subsystem with steady density and internal energy. To achieve structure-preserving properties, the PPCT scheme uses a novel finite volume-finite difference (FV-FD) hybrid approach: a PP finite volume method for the Euler subsystem and a CT finite difference method for the magnetic subsystem. The two are coupled using Strang splitting. The finite volume method is based on a new PP limiter, which is proven to maintain the second-order accuracy of the reconstruction. The PP limiter enforces the positivity of the reconstructed values for density and pressure, as well as an a priori condition for the PP property of the updated cell averages. A rigorous theoretical proof of the PP property is provided using the geometric quasilinearization (GQL) approach [K. Wu & C.-W. Shu, SIAM Review, 65:1031–1073, 2023]. For the magnetic subsystem, we construct an implicit finite difference CT method that conserves energy and preserves a globally DDF constraint on non-staggered Cartesian meshes. The resulting nonlinear algebraic system is solved with an iterative algorithm, reducing the residual error to machine precision within a few iterations. Unique solvability and convergence of this algorithm are theoretically proven under a CFL-like condition. Since the finite difference CT method for the magnetic subsystem is unconditionally energy-stable and preserves steady density and internal energy, the time step for the PP property and stability of the PPCT scheme is restricted only by a mild CFL condition for the scheme of the Euler subsystem. While the primary focus is on the 2D case for clarity, the 3D extension of the proposed PPCT framework is also presented. Several challenging 2D and 3D numerical experiments, including highly magnetized MHD jets with extremely high Mach numbers, validate the accuracy, robustness, and high resolution of the PPCT scheme.