Fractional magneto-hydrodynamics: Algorithms and applications

Abstract

We present two discretization methods for solving the fractional magneto-hydrodynamic (FMHD) equations with the fractional Laplacian defined in bounded domains. In the first method, we add a pseudo-pressure in the magnetic field equation to enforce that the magnetic field is divergence-free. In the second method, the magnetic field satisfies the divergence-free condition automatically with the no-slip boundary condition for the velocity and the perfectly conducting boundary condition for the magnetic field equation. In addition, we propose a discretization method for solving the modified fractional magneto-hydrodynamic (M-FMHD) equation. The divergence-free condition is preserved by employing the stream function in the M-FMHD system. Numerical experiments with fabricated solutions show that all three methods exhibit second-order accuracy in time for the velocity and magnetic fields. The pseudo-pressure and pressure also exhibit second-order convergence in time for small values of time step, however the pressure exhibits 1.5-order for convergence for large values of time step size when the fractional order α=1. We use the spectral decomposition method for spatial discretization, and we demonstrate that exponential convergence is achieved for all fields and for any fractional order. We also applied these methods to solve the lid-driven cavity flow with and without magnetic field, and we observe new flow patterns emerging as the fractional order varies.