Magnetic induction and electric potential smoothed particle magnetohydrodynamics for incompressible flows

In order to solve incompressible, nonideal magnetohydrodynamic (MHD) free-surface flows, two weakly compressible smoothed particle hydrodynamics models, with and without the consideration of magnetic induction, are developed. The SPH formulation for magnetic induction magnetohydrodynamics (SPMHD), which is popular in astrophysical studies, is applied for the first time to incompressible free-surface MHD flows, such as liquid metal flows, with the consideration of nonideal MHD effects and boundaries with arbitrary electric conductivity. An SPMHD implementation using the inductionless approximation is also proposed for both electrically conductive and insulating boundaries, in which a Poisson equation is solved to compute the Lorentz force instead of evolving the magnetic induction equation. Both proposed methods are validated against MHD benchmarks, including free-surface MHD cases. The proposed inductionless SPMHD implementation has the advantages of stability and relaxed time-step restrictions, but is only accurate at a low range of Hartmann numbers. For high Hartmann number problems, magnetic induction SPMHD model is more accurate. The computational efficiency and conservation error of the two models are compared and discussed.