Newton's algorithm for magnetohydrodynamic equations with the initial guess from Stokes-like problem

Abstract The magnetohydrodynamic equations are second order nonlinear partial differential equations which are coupled by fluid velocity and magnetic fields and we consider to apply the Newton’s algorithm to solve them. It is well known that the choice of a proper initial guess is critical to assure the convergence of Newton’s iterations in solving nonlinear partial differential equations. In this paper, we provide a good initial guess for Newton’s algorithm when it is applied for solving magnetohydrodynamic equations.

[1]  Sang Dong Kim,et al.  Newton's method for the Navier-Stokes equations with finite-element initial guess of stokes equations , 2006, Comput. Math. Appl..

[2]  Hidetoshi Hashizume,et al.  Numerical and experimental research to solve MHD problem in liquid blanket system , 2006 .

[3]  T. A. Manteuffel,et al.  First-Order System Least-Squares for the Navier-Stokes Equations , 1996 .

[4]  Catalin Trenchea,et al.  Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows , 2014, Appl. Math. Lett..

[5]  L. Driel-Gesztelyi.  An Introduction to Magnetohydrodynamics , 2004 .

[6]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[7]  M. Gunzburger,et al.  On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics , 1991 .

[8]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[9]  Walter M. Elsasser,et al.  The Hydromagnetic Equations , 1950 .

[10]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[11]  William Layton,et al.  Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows , 2014 .

[12]  Sergey Smolentsev,et al.  MHD thermofluid issues of liquid-metal blankets: Phenomena and advances , 2010 .

[13]  Ricardo G. Durán,et al.  Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate , 2011, Math. Comput..

[14]  Amik St-Cyr,et al.  Optimized Schwarz Preconditioning for SEM Based Magnetohydrodynamics , 2008, 0805.0025.

[15]  Sang Dong Kim,et al.  Error estimate and regularity for the compressible Navier‐Stokes equations by Newton's method , 2003 .

[16]  P. Schmidt.  A GALERKIN METHOD FOR TIME-DEPENDENT MHD FLOW WITH NONIDEAL BOUNDARIES , 1999 .

[17]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[18]  J. Peterson.  On the finite element approximation of incompressible flows of an electrically conducting fluid , 1988 .