Provably Positivity-Preserving Constrained Transport (PPCT) Second-Order Scheme for Ideal Magnetohydrodynamics

TL;DR

This paper introduces a novel second-order scheme for ideal MHD that preserves positivity and divergence-free magnetic fields, combining finite volume and finite difference methods with a splitting technique for enhanced stability and accuracy.

Contribution

The paper presents a new positivity-preserving constrained transport scheme for MHD that guarantees divergence-free magnetic fields and positive density and pressure, with rigorous proofs and efficient algorithms.

Findings

Successfully maintains positivity of density and pressure.

Ensures a divergence-free magnetic field in simulations.

Demonstrates robustness in high Mach number MHD jet simulations.

Abstract

This paper proposes and analyzes a robust and efficient second-order positivity-preserving constrained transport (PPCT) scheme for ideal magnetohydrodynamics (MHD) on non-staggered Cartesian meshes. The PPCT scheme ensures two critical physical constraints: a globally discrete divergence-free (DDF) condition on the magnetic field and the positivity of density and pressure. The method is inspired by a novel splitting technique from [T.A. Dao, M. Nazarov and I. Tomas, J. Comput. Phys., 508:113009, 2024], which divides the MHD system into an Euler subsystem with steady magnetic fields and a magnetic subsystem with steady density and internal energy. To achieve these structure-preserving properties, the PPCT scheme combines a positivity-preserving (PP) finite volume method for the Euler subsystem with a finite difference constrained transport (CT) method for the magnetic subsystem via Strang splitting. The finite volume method employs a new PP limiter that retains second-order accuracy and enforces the positivity of density and pressure, with rigorous proof provided using the geometric quasilinearization (GQL) approach [K. Wu and C.-W. Shu, SIAM Review, 65:1031-1073, 2023]. For the magnetic subsystem, we develop an implicit finite difference CT method that conserves energy and maintains a globally DDF constraint. This nonlinear system is efficiently solved to machine precision using an iterative algorithm. Since the CT method is unconditionally energy-stable and conserves steady density and internal energy, the PPCT scheme requires only a mild CFL condition for the finite volume method to ensure stability and the PP property. While the focus is on 2D cases for clarity, the extension to 3D is discussed. Several challenging numerical experiments, including highly magnetized MHD jets with high Mach numbers, validate the PPCT scheme's accuracy, robustness, and high resolution.