FLASH MHD Code Overview

• FLASH MHD Code is a modular adaptive mesh refinement framework for solving magnetohydrodynamics equations with applications ranging from astrophysics to fusion design.
• It integrates advanced divergence-control methods and high-performance Riemann solvers to ensure accuracy, stability, and preservation of physical invariants.
• The code’s modular structure, scalable parallelism, and extensible microphysics support efficient simulation of complex MHD phenomena in diverse regimes.

The FLASH MHD Code is a modular, block-structured, adaptive mesh refinement (AMR) framework designed to solve the equations of magnetohydrodynamics (MHD) and related multi-physics problems, with broad application across astrophysics, laboratory plasma, and high-energy density physics. Developed for extensibility and performance on high-performance computing architectures, FLASH implements a suite of MHD solvers—each optimized for particular stability, accuracy, and conservation constraints—supporting both ideal and resistive MHD, radiation hydrodynamics, and a variety of microphysical modules. The code employs diverse divergence-control strategies, entropy-stable schemes, and advanced Riemann solvers, allowing it to target regimes from supersonic turbulence and dynamos to inertial confinement fusion and planetary magnetospheres.

1. Core Mathematical Formulation and Divergence Control

At the heart of FLASH are the conservative MHD equations, advanced in various forms to ensure mass, momentum, energy, and magnetic induction conservation. The code supports both ideal and resistive MHD via equations such as:

$$\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf{v}) = 0\,,\quad \frac{\partial (\rho\mathbf{v})}{\partial t} + \nabla \cdot \left[\rho\mathbf{v}\mathbf{v} + (p + \frac{B^2}{2})\mathbf{I} - \mathbf{B}\mathbf{B} \right]= 0\,,$$

$$\frac{\partial E}{\partial t} + \nabla\cdot\left[(E+p+\frac{B^2}{2})\mathbf{v} - (\mathbf{B}\cdot\mathbf{v})\mathbf{B}\right] = 0\,, \quad \frac{\partial \mathbf{B}}{\partial t} + \nabla \times (\mathbf{v} \times \mathbf{B}) = 0\,, \quad \nabla\cdot\mathbf{B} = 0\,.$$

To maintain $\nabla\cdot\mathbf{B}=0$ on the discrete grid, FLASH implements several divergence-control methods:

• Eight-wave (Powell) and Dedner hyperbolic/parabolic cleaning: Introduces extra flux terms and an auxiliary scalar $\psi$ to propagate and damp divergence errors via an additional cleaning wave, preserving hyperbolicity and empirical stability but breaking strict conservation of total energy at $O((\nabla\cdot\mathbf{B})^2)$ (Brandenburg et al., 2019).
• Constrained transport (CT): Employs staggered-mesh updates of face-centered magnetic fields, maintaining the divergence-free condition to machine precision, especially in the Unsplit Staggered Mesh (USM) solver (Evangelista et al., 2019, Ellison et al., 14 Apr 2025).
• Generalized Lagrange Multiplier (GLM) approach: Extends the hyperbolic cleaning with a thermodynamically consistent scalar field, preserving entropy and Galilean invariance, yielding a nine-wave system suitable for high-fidelity numerical studies (Derigs et al., 2017).
• Janhunen source term and projection schemes: Combines transport of the divergence error with a projection correction step, removing residual numerical monopoles without reliance on a staggered mesh (Derigs et al., 2016).

2. Numerical Schemes: Riemann Solvers and Stability Features

FLASH offers several high-performance Riemann solvers, focusing on physical fidelity and nonlinear stability in regimes of extreme Mach number, low plasma $\beta$, or strong discontinuities:

• Entropy-stable relaxation-based solvers (HLL3R, HLL5R): Ensure positivity of density and internal energy while satisfying discrete entropy inequalities, critical for stable integration of highly compressible turbulence, shocks, and rarefactions. HLL3R resolves the two fastest magnetoacoustic waves with a composite contact/tangential mode, while HLL5R additionally resolves Alfvén and shear waves, providing improved accuracy in turbulent or shear-driven MHD flows (Waagan et al., 2011, Low et al., 2011).
• Positivity-preserving modifications: Limiter functions restrict reconstructed states so that negative pressures or densities cannot be generated, enabling the code to stably operate with Courant numbers up to 0.8 in high-Mach settings and preventing failure in strongly shocked astrophysical or laboratory regimes (Derigs et al., 2016, Low et al., 2011).
• High-order, entropy-stable fluxes: Implementation of entropy-conserving baselines with matrix dissipation allows sharp capture of discontinuities, preservation of physical entropy production, and robustness across AMR levels (Derigs et al., 2016).

The code supports flexible switch-over among Roe, HLLE, HLL3R, HLL5R, and custom entropy-stable (ES) fluxes, with AMR structures dynamically selecting among algorithms to balance accuracy and performance.

3. Modular Structure, AMR Integration, and Parallelism

FLASH is architected around a block-structured AMR hierarchy (PARAMESH), enabling efficient, adaptive resolution of MHD features (e.g., shocks, current sheets, flow instabilities) across multiple spatial and physical scales:

• Uniform block mesh: Each block is a small, regular patch (typically $8^3$ or $16^3$ cells), supporting refinement/restriction at block interfaces and ghost-cell communication for parallel computation (Evangelista et al., 2019, Low et al., 2011).
• Solver modularity: Hydrodynamic, MHD, radiative transfer, and microphysics modules are decoupled via a public interface, allowing rapid substitution or extension (e.g., adding explicit resistivity, viscosity, or new divergence-control schemes) (Derigs et al., 2016).
• Parallelization: All block advances, neighbor exchanges, and AMR adaptation are handled transparently using MPI, with special handling for ghost regions (including MHD-specific variables such as $\mathbf{B}$ and $\psi$) to ensure multi-solver compatibility (Low et al., 2011, Ellison et al., 14 Apr 2025).
• Boundary and rigid-body capability: FLASH supports rigid-body boundaries in both hydrodynamics and MHD, enabling simulations such as planetary obstacles in stellar winds with specialized boundary flags and tailored constraint enforcement for the magnetic field at solid surfaces (Evangelista et al., 2019).

4. Advanced Physics Modules and Microphysics Integration

The FLASH MHD framework is extensible to complex multi-physics regimes:

• Radiation-MHD and three-temperature models: Incorporates evolution of electron, ion, and radiation energies, multi-group diffusion, and interaction terms (Ohmic heating, equilibration, laser energy deposition), essential for inertial confinement fusion (ICF) modeling (Ellison et al., 14 Apr 2025).
• Non-equilibrium ionization (NEI): An optional module uses an eigenvalue-based approach for solving stiff ODEs governing ionization states, coupled with up-to-date atomic data (from AtomDB) and radiative cooling, with demonstrated efficiency improvements and clear diagnostics for the NEI state (Zhang et al., 2019).
• Transport and extended MHD effects: Incorporates magnetized electron/ion conduction, Nernst advection (Braginskii coefficients), volumetric preheat, realistic EOS tables, and nuclear burn physics for hydrogenic plasmas (Ellison et al., 14 Apr 2025).
• Viscosity and resistivity: Viscous and resistive MHD may be included either through explicit viscosity (with Laplacian or full deviatoric stress forms) or implicit Ohmic diffusion using operator-splitting and coupled linear solvers (Evangelista et al., 2019, Ellison et al., 14 Apr 2025).

5. Validation, Physical Regimes, and Limitations

FLASH MHD has been validated over a wide set of canonical and application-driven benchmarks:

• Astrophysical dynamos and helicity: Studies of α²-dynamo action and helicity conservation highlight the distinction between ideal eight-wave solvers (which break exact helicity conservation and introduce numerical resistivity) versus resistive DNS codes (e.g., Pencil Code), with consequences for the reliability of large-scale dynamo modeling at high magnetic Reynolds number (Brandenburg et al., 2019).
• Physical validation: FLASH captures linear and nonlinear instability growth rates (MRT, RMI), implosion trajectories, and neutron yields in Z-facility MagLIF experiments within 10–30% of experimental and leading-code results, demonstrating readiness for ignition-scale ICF target design (Ellison et al., 14 Apr 2025).

Quantitative regimes supported include:

• Mach numbers up to ~100 (positivity-preserving solvers),
• Plasma β ranging from $\lesssim 10^{-3}$ to $\gtrsim 10^{2}$,
• Magnetic monotonicity and discrete entropy production preserved even for complex AMR geometries (Waagan et al., 2011, Low et al., 2011, Derigs et al., 2016).

Known limitations:

• Ideal MHD modules lack explicit resistivity or reconnection physics (reconnection is numerical),
• Numerical dissipation in divergence-cleaning eight-wave schemes is scale-dependent and challenging to parameterize, requiring explicit resistive convergence studies for accurate dynamo or helicity transport results (Brandenburg et al., 2019).
• The positivity-preserving entropy limiters introduce extra diffusion in low-$\beta$ regions,
• Global divergence-cleaning approaches can break exact conservation, and careful choice of cleaning speed is required to avoid CFL constraint.

6. Application Examples and Extensibility

Key areas of application include:

• Astrophysical turbulence and star formation: Sustained, stable, high-Mach turbulence, turbulent dynamo growth rates, and transition to magnetically-dominated states on AMR grids, including inertial-range recoveries at $512^3$ resolution and above (Waagan et al., 2011).
• Planet–wind and MHD–obstacle interactions: Accurate simulation of magnetic draping, bow-shock formation, and recirculation flows past non-magnetized planetary bodies, with detailed control of viscosity, boundary conditions, and field extrapolation (Evangelista et al., 2019).
• Inertial confinement fusion design: Multi-physics coupling of MHD, radiation, and nuclear burn, including magnetic field evolution in current-driven targets (MagLIF), and experimental validation against Z-facility data (Ellison et al., 14 Apr 2025).
• Non-equilibrium plasma diagnostics: Accurate post-processing of SNR and ISM conditions using the NEI module with clear, robust metrics ($d_c$) for the degree of ionization/recombination and relaxation timescales (Zhang et al., 2019).

FLASH’s modular interfaces allow ongoing extension:

• New solvers (e.g., entropy-conserving, GLM, higher-order positivity-preserving),
• Customized microphysics and boundary modules,
• Efficient deployment on massively parallel systems.

7. Comparative Assessment and Best Practices

Comparison across FLASH’s solver suite reveals trade-offs:

• Eight-wave/divergence-cleaning: Simpler and widely used but not thermodynamically consistent and not suitable for studies requiring strict magnetic helicity conservation at high resolution (Brandenburg et al., 2019).
• Constrained transport: Preferred for strict $\nabla\cdot\mathbf{B}=0$ maintenance in AMR but more complex to extend to arbitrary geometries or coupling with certain physics.
• Entropy-stable and positivity-preserving solvers: Essential for highly compressible, high-Mach or low-$\beta$ applications, providing robustness and allowing higher CFL operation (Low et al., 2011, Waagan et al., 2011, Derigs et al., 2016).
• GLM extensions: Offer thermodynamic consistency, local communications (critical for extreme AMR and parallel scaling), and improved error bounds in large-scale, low-divergence applications (Derigs et al., 2017, Derigs et al., 2016).

Best practices in FLASH MHD usage include:

• Explicit diagnostic and convergence analysis for numerical resistivity/helicity,
• Testing of cleaning parameter ranges, floors, and refinement strategies for chosen physical scenario,
• Careful matching of solver and divergence scheme to physics of interest.

FLASH continues active development, incorporating novel algorithms for entropy, stability, and physical fidelity, and supports a broad, reproducible research ecosystem for MHD and plasma physics.