A grad‐div stabilized method using the Jacobi iteration for the thermally coupled incompressible magnetohydrodynamic system

Abstract

This paper presents a grad-div stabilization with the Jacobi iteration to the thermally coupled incompressible magnetohydrodynamic system, which avoids breakdown of solver and increase of computational time with increasing value of grad-div stabilized parameter. The proposed method includes two steps: a fully discrete first-order Euler semi-implicit scheme based on the mixed finite element method is implemented for the considered problem in Step 1. Then, in Step 2, the Jacobi approximation is applied to the grad-div operator whose each component will be decoupled. Moreover, we obtain theoretical analysis results of the presented method, both unconditional stability and convergence analysis. Finally, some numerical experiments are presented to show the accuracy and verify effectiveness of the proposed method.

Numerical simulation of the ferrohydrodynamics flow using an unconditionally stable second‐order scheme

“…The ferrohydrodynamics problem is in sharp contrast to the magnetohydrodynamics problem [14–16] although they own some common points: the former one considers electrically nonconducting magnetizable fluids, where the flow is influenced by fluid magnetization due to the presence of a magnetic field, while the latter one studies the flow behavior of non‐magnetizable but electrically conducting fluids by ignoring the effect of polarization or magnetization. Besides, the most important body force acting on the fluid in magnetohydrodynamics is the Lorentz force, which generates if electric current follows at an angle to the direction of an applied magnetic field, whereas the dominant one in ferrohydrodynamics is the Kelvin force due to the effect of polarization and nonconducting fluids.…”