Overview
A Finite Volume (FV) scheme has been developed for the purpose of solving the extended magnetohydrodynamic (XMHD) equations. This scheme is designed to produce accurate results across various magnetohydrodynamic regimes, specifically the ideal, resistive, and Hall MHD limits.
Approach
The development of the FV scheme involved re-writing the XMHD equations. This re-formulation enables the algorithm to maintain the use of ideal MHD Riemann solvers. Concurrently, the constrained transport method is employed to preserve divergence-free magnetic fields within the computational framework.
The XMHD model intrinsically introduces numerical stiffness due to the incorporation of electron inertia and displacement current. To address this computational challenge, a semi-implicit FV scheme was adopted. This scheme re-formulates the XMHD model as a relaxation system. Time advancement within this framework is achieved using an explicit 2nd-order Runge-Kutta scheme. Operator splitting is applied to manage the implicit source term updates at each sub-stage of the time integration.
To enhance numerical stability, particularly in regions where non-ideal effects become pronounced, a density-dependent slope limiter has been implemented. This limiter serves to increase flux diffusivity specifically in low-density regions. The entire algorithm has been implemented within a scalable adaptive mesh refinement (AMR) framework.
Findings
The newly developed algorithm retains several fundamental aspects of ideal MHD formulations. This design choice contributes to its natural asymptotic behavior towards the ideal MHD limit.
Moreover, the scheme demonstrated promising results when applied to the resistive and Hall MHD limits. Verification of its performance across these limits was conducted by comparing its outputs against established reference test problems that are specific to ideal, resistive, and Hall MHD scenarios.
Why This Matters
The scheme's ability to accurately resolve XMHD equations across ideal, resistive, and Hall MHD limits is significant for simulations in plasma physics. Its asymptotic-preserving properties and numerical stability features suggest potential for reliable modeling of complex magnetohydrodynamic phenomena.