Improving divergence cleaning in cosmological SPMHD simulations

Technical Summary: Improving Divergence Cleaning in Cosmological SPMHD Simulations

Problem Statement
In cosmological Smoothed Particle Magnetohydrodynamics (SPMHD) simulations, maintaining the solenoidal condition ($\nabla \cdot \mathbf{B} = 0$) is a persistent numerical challenge. Standard schemes in codes like Gadget typically rely on the Powell et al. (1999) "8-wave" approach. While this method advects divergence errors with the fluid flow, it does not actively remove them. In SPH-based codes, residual $\nabla \cdot \mathbf{B}$ errors can reach order unity, leading to unphysical magnetic field amplification and the "tensile instability," where spurious source terms in the momentum equation dominate when magnetic pressure exceeds gas pressure. While the hyperbolic/parabolic divergence cleaning scheme proposed by Dedner et al. (2002) offers a solution, its original formulation has proven unstable in SPMHD, particularly in dynamic, individual-timestep environments with strong density jumps, often acting as an unphysical source of magnetic energy.

Methodology
The authors implement a constrained, energy-conserving version of the Dedner-type cleaning algorithm (Tricco & Price 2012; Tricco et al. 2016a) into the cosmological SPMHD code OpenGadget3. Key methodological adaptations include:

• Energy Conservation: The governing equations are reformulated to guarantee total energy conservation. This involves evolving the cleaning field variable as $\psi/c_h$ (where $\psi$ is the auxiliary scalar and $c_h$ is the cleaning speed) rather than $\psi$ alone, and including a specific source term $-\frac{1}{2}\psi(\nabla \cdot \mathbf{v})$ in the evolution equation. This ensures that the parabolic damping of the cleaning field removes magnetic energy without introducing unbounded growth.
• Cosmological Adaptation: The equations are adapted for comoving coordinates and individual particle timesteps. The cleaning speed $c_h$ is set to the local fast magnetosonic speed ($\sqrt{v_A^2 + c_s^2}$), ensuring the cleaning is intrinsically Courant-stable within the standard SPH timestep constraints.
• Simulation Setup: The authors perform a series of simulations on a massive galaxy cluster ($M_{200c} \approx 10^{15} M_\odot$) seeded with a uniform magnetic field ($10^{-14}$ G comoving). Five distinct runs are compared:
  1. NIFTY-noclean: Reference run using only the Powell advection scheme.
  2. NIFTY-clean: Fiducial run with the full constrained cleaning scheme (including entropy back-reaction).
  3. NIFTY-clean2: Same as above but with an overcleaning factor $\zeta=2$ (cleaning at twice the local wave speed).
  4. NIFTY-advect: Cleaning scheme with the velocity-divergence source term switched off.
  5. NIFTY-heat: Cleaning scheme where the dissipated cleaning energy is discarded rather than fed back into the entropy equation.

Key Results

• Divergence Error Reduction: The primary result is a reduction in relative divergence errors ($|\nabla \cdot \mathbf{B}|/B$) by 2–3 orders of magnitude throughout the cluster volume compared to the Powell-only scheme. This reduction is uniform across the cluster core and outskirts.
• Magnetic Field Amplification: In the cluster outskirts ($r \approx 1\text{--}3 \, h^{-1}$ Mpc), where the magnetic field is weaker and resolution is lower, the active cleaning scheme allows for magnetic field amplification by a factor of 5–10 compared to the Powell-only run. The authors attribute this to the removal of unphysical Lorentz forces caused by divergence errors, which previously suppressed the small-scale dynamo in these regions.
• Core Stability: The global density structure and magnetic field geometry in the cluster core remain largely unchanged between the cleaning and non-cleaning runs, indicating that the cleaning does not artificially alter the saturated state of the core dynamo.
• Thermodynamic Consistency: The inclusion of the entropy back-reaction (feeding dissipated cleaning energy back as heat) is shown to be critical for thermodynamic consistency. Runs without this back-reaction (NIFTY-heat) result in lower central entropy and higher saturated central magnetic fields due to adiabatic compression of a colder core.
• Computational Cost: The implementation introduces no measurable runtime overhead; the improved cleaning is achieved essentially "for free" within the existing Courant timestep constraints.

Significance and Claims
The paper claims that accurate control of divergence errors is essential for capturing magnetic field amplification in the low-density, poorly resolved outskirts of galaxy clusters. The authors argue that the uncontrolled divergence errors in standard Powell schemes act as numerical noise that suppresses the dynamo in sub-saturated regions. By implementing the constrained, energy-conserving cleaning scheme, the simulation achieves a more faithful realization of MHD physics without computational penalty.

The authors conclude that the constrained cleaning scheme:

1. Dramatically reduces numerical divergence errors (2–3 orders of magnitude).
2. Enables more accurate representation of field amplification in under-resolved regions.
3. Produces smoother, more coherent magnetic field structures.
4. Stabilizes magnetic field evolution across different resolutions, improving the physical fidelity of results even at moderate resolutions.

The work recommends that cosmological SPMHD applications adopt this cleaning scheme by default, using the natural cleaning speed ($\zeta=1$) and keeping the parabolic energy back-reaction active to close the energy budget. The implementation is now publicly available in OpenGadget3.