A projection method with modular grad-div stabilization for
inductionless magnetohydrodynamic equations based on charge-conservation
Abstract
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 post-processing step, thereby enhancing solution accuracy and
improving mass conservation. Concurrently, a decoupled strategy is
employed to separate the magnetic and fluid field functions from the
original system. Therefore, at each time step, we only need to solve
several linear sub-systems for which the numerical solutions can be
obtained efficiently. For spatial discretization, the current density
and electric potential are discretized by H ( div , Ω ) × L 2 ( Ω )
-conforming finite element pair, which ensures 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 analyzed and derived. Numerical results are presented to
verify that the algorithm exhibit robustness with respect to the
stabilization parameters and demonstrate the performance of the scheme,
particularly with respect to its stability and accuracy.