Active flux for ideal magnetohydrodynamics: A positivity-preserving scheme with the Godunov-Powell source term

TL;DR

This paper develops a high-order, positivity-preserving Active Flux scheme for ideal magnetohydrodynamics, effectively maintaining divergence-free conditions and positivity of density and pressure through specialized limiters and source term discretization.

Contribution

It introduces a novel positivity-preserving Active Flux scheme incorporating the Godunov-Powell source term for ideal MHD, ensuring divergence control and positivity with high-order accuracy.

Findings

The scheme achieves third-order accuracy.

It effectively preserves positivity of density and pressure.

The method successfully captures shocks and maintains divergence-free conditions.

Abstract

The Active Flux (AF) is a compact, high-order finite volume scheme that allows more flexibility by introducing additional point value degrees of freedom at cell interfaces. This paper proposes a positivity-preserving (PP) AF scheme for solving the ideal magnetohydrodynamics, where the Godunov-Powell source term is employed to deal with the divergence-free constraint. For the evolution of the cell average, apart from the standard conservative finite volume method for the flux derivative, the nonconservative source term is built on the quadratic reconstruction in each cell, which maintains the compact stencil in the AF scheme. For the point value update, the local Lax-Friedrichs (LLF) flux vector splitting is adopted for the flux derivative, originally proposed in [Duan, Barsukow, and Klingenberg, SIAM Journal on Scientific Computing, 47(2), A811--A837, 2025], and a central difference is used to discretize the divergence in the source term. A parametrized flux limiter and a scaling limiter are presented to preserve the density and pressure positivity by blending the AF scheme with the first-order PP LLF scheme with the source term. To suppress oscillations, a new shock sensor considering the divergence error is proposed, which is used to compute the blending coefficients for the cell average. Several numerical tests are conducted to verify the third-order accuracy, PP property, and shock-capturing ability of the scheme. The key role of the Godunov-Powell source term and its suitable discretization in controlling divergence error is also validated.