Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates

2019 ◽  
Vol 26 (3) ◽  
pp. 631-653
Author(s):  
Yong Liu
Keyword(s):  
Magnetohydrodynamic Equations ◽  
Cylindrical Coordinates ◽  
Divergence Free ◽  
Ideal Magnetohydrodynamic ◽  
Dg Methods

Solving Magnetohydrodynamic Equations Without Special Treatment for Divergence-Free Magnetic Field

AIAA Journal ◽  
2004 ◽  
Vol 42 (12) ◽  
pp. 2605-2608 ◽  
Author(s):  
Moujin Zhang ◽  
S.-T. John Yu ◽  
Shang-Chuen Lin ◽  
Sin-Chung Chang ◽  
Isaiah Blankson
Keyword(s):  
Magnetic Field ◽  
Special Treatment ◽  
Magnetohydrodynamic Equations ◽  
Divergence Free

A fully divergence-free finite element method for magnetohydrodynamic equations

2018 ◽  
Vol 28 (04) ◽  
pp. 659-695 ◽  
Author(s):  
Ralf Hiptmair ◽  
Lingxiao Li ◽  
Shipeng Mao ◽  
Weiying Zheng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Divergence Free ◽  
Incompressible Magnetohydrodynamic Equations ◽  
Preconditioned Gmres ◽  
Energy Law ◽  
Element Method

We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

Nonlinear fast sausage waves in homogeneous magnetic flux tubes

2015 ◽  
Vol 81 (6) ◽  
Author(s):  
Badma B. Mikhalyaev ◽  
Michael S. Ruderman
Keyword(s):  
Bessel Functions ◽  
Nonlinear Evolution ◽  
Solar Physics ◽  
Modified Bessel Functions ◽  
Magnetohydrodynamic Equations ◽  
Carrier Wave ◽  
Flux Tubes ◽  
Magnetic Flux Tubes ◽  
Ideal Magnetohydrodynamic ◽  
The Ideal

We consider fast sausage waves in straight homogeneous magnetic tubes. The plasma motion is described by the ideal magnetohydrodynamic equations in the cold plasma approximation. We derive the nonlinear Schrödinger equation describing the nonlinear evolution of an envelope of a carrier wave. The coefficients of this equation are expressed in terms Bessel and modified Bessel functions. They are calculated numerically for various values of parameters. In particular, we show that the criterion for the onset of the modulational or Benjamin–Fair instability is satisfied. The implication of the obtained results for solar physics is discussed.

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