Recursive regularised lattice Boltzmann method for magnetohydrodynamics

TL;DR

This work presents a recursive regularised lattice Boltzmann method for two-dimensional incompressible MHD using a hybrid double-distribution framework, where the magnetic field evolves with a standard BGK LBM and the fluid solver benefits from Hermite-based recursive regularisation. By employing a fourth-order Hermite expansion on the D2Q9 lattice and reconstructing non-equilibrium moments up to order four, the method filters spurious lattice-artefacts while preserving the incompressible MHD limit, improving stability at low viscosities and across current-sheet forming regimes. Validation on the Orszag–Tang vortex shows that RR-LBM matches reference solutions on well-resolved grids and provides robust stability in turbulent regimes, at the cost of modest extra computational overhead compared to moment-based schemes. The approach offers a systematic path to regularised LBM for multiphysics systems and suggests extensions to 3D and fully regularised magnetic populations for further artefact reduction.

Abstract

We present and test a recursive regularised lattice Boltzmann method for incompressible magnetohydrodynamic (MHD) flows. The approach is based on a double-distribution formulation, in which the magnetic field is evolved using a standard BGK lattice Boltzmann scheme, while the fluid solver is enhanced through a Hermite-based recursive regularisation of the non-equilibrium moments. The method exploits a fourth-order Hermite expansion of the equilibrium distribution on the D2Q9 lattice, allowing higher-order isotropy to be retained while selectively filtering spurious non-hydrodynamic contributions. The regularisation procedure reconstructs the non-equilibrium distribution from physically consistent Hermite coefficients, avoiding explicit evaluation of velocity gradients. The resulting scheme preserves the correct incompressible MHD limit, improves numerical stability at low viscosities, and reduces lattice-dependent artefacts. The proposed formulation provides a robust and versatile framework for MHD simulations and offers a systematic route for extending regularised lattice Boltzmann methods to coupled multiphysics systems.

Figures (5)

• Figure 1: Low-Reynolds-number flow: evolution of the magnetic field at selected time instants obtained by the RR LBM. The sequence illustrates the progressive distortion of the initial large-scale structures, the formation of thin current sheets, and the emergence of a highly intermittent magnetic field as the flow transitions toward fully non-linear MHD dynamics.
• Figure 2: $\mathrm{Re}=5000$: evolution of the magnetic field at selected time instants. Snapshots illustrate the transition from large-scale coherent structures to fully developed turbulent magnetic filaments.
• Figure 3: Temporal evolution of the peak electric current density $j_{\max}=\max_{\Omega}|j_z|$ for the Orszag–Tang vortex at $\mathrm{Re}=200\pi$ (magenta solid line), $\mathrm{Re}=2500$ (brown dashed line), and $\mathrm{Re}=5000$ (olive line with smaller dashes). The initial growth reflects large-scale magnetic field-line stretching, followed by a rapid intensification associated with current-sheet thinning and the onset of magnetic reconnection. Increasing Reynolds number leads to stronger current concentration and sustained intermittency, characteristic of turbulent MHD dynamics. Findings are obtained by the present RR scheme.
• Figure 4: Time evolution of kinetic (solid lines) and magnetic (dashed lines) energies for the Orszag–Tang vortex at $\mathrm{Re}=200\pi$ (orange), $\mathrm{Re}=2500$ (navy), and $\mathrm{Re}=5000$ (dark green). At early times, magnetic energy dominates as the initial field is stretched by the flow. As non-linear interactions intensify, magnetic energy is rapidly converted into kinetic energy through current-sheet formation and magnetic reconnection. Increasing Reynolds number leads to stronger energy transfer, delayed saturation, and enhanced turbulent mixing, with higher-Re cases exhibiting sustained kinetic activity and slower magnetic dissipation characteristic of fully developed MHD turbulence. Findings are obtained by the present RR scheme.
• Figure 5: Computational cost: normalised runtime as a function of $N^2$ for the BGK (solid black line with squares), RMs (dashed red line with circles), CMs (green dotted line with triangles), and RR (blue dash-dotted line with inverted triangles) schemes in log-log scale. Moments-based LBMs consistently provide the most efficient performance.