A Projection Method With Modular Grad-Div Stabilization for Inductionless Magnetohydrodynamic Equations Based on Charge Conservation

In this paper, we proposed a fully discrete projection method with modular grad-div stabilization for solving the time-dependent inductionless magnetohydrodynamic equations. The method incorporates a minimally intrusive module into the classical projection method, serving as a postprocessing step, thereby enhancing solution accuracy and improving mass conservation. Additionally, a decoupled strategy is employed to separate the magnetic and fluid field functions from the original system. Since we only need to solve several linear subsystems at each time step, the numerical solutions can be obtained efficiently. In the spatial discretization, $$$ {\mathbf{H}}_0\left(\operatorname{div},\Omega \right)\times {L}_0^2\left(\Omega \right) $$$-conforming finite element pair is used to approximate the current density and electric potential, ensuring that the discrete current density is exactly divergence-free. Therefore, the designed numerical scheme maintains the features of linearization, decoupling, unconditional energy stability, charge conservation, and improved mass conservation. The unconditional energy stability and convergence of the algorithm are rigorously analyzed and proven. Numerical results are presented to verify the robustness of the algorithm with respect to the stabilization parameters and to demonstrate the performance of the scheme, particularly in terms of its stability and accuracy.