A Vector Potential implementation for Smoothed Particle Magnetohydrodynamics

The development of smooth particle magnetohydrodynamic (SPMHD) has significantly improved the simulation of complex astrophysical processes. However, the preservation the solenoidality of the magnetic field is still a severe problem for the MHD. A formulation of the induction equation with a vector potential would solve the problem. Unfortunately all previous attempts suffered from instabilities. In the present work, we evolve the vector potential in the Coulomb gauge and smooth the derived magnetic field for usage in the momentum equation. With this implementation we could reproduce classical test cases in a stable way. A simple test case demonstrates the possible failure of widely used direct integration of the magnetic field, even with the usage of a divergence cleaning method.