Velocity-Correction Methods For Unsteady Incompressible Magnetohydrodynamics Equations

This paper is mainly devoted to studying two kinds of finite element methods for the incompressible magnetohydrodynamic equations.Firstly,we consider two or three dimensional unsteady incompressible MHD equations with electric field term#12 with the following initial boundary conditions u(x=0)=u0(x),B(x,0)=B0(x),(?)x??,u=0,B·n=0,E×n=0,(?)x?(?)?,t>0,where u represents the velocity of the fluid flow,Re is fluid Reynolds number,Rm is magnetic field,S represents the coupling number,B is magnetic Reynolds number,E represents the electric,j=E+u×B represents the Ohm's law,p represents the pressure,f represents the body force.Secondly,we consider the unsteady incompressible MHD equations as follows#12 with the following initial boundary conditions u(x,0)=u0(x),B(x,0)=B0(x),(?)x??,u=0,B·n=0,n×?×B=0,(?)x?(?)?,t>0,where v represents the kinematic viscosity,? represents the magnetic permeability,g represents the external body force terms.Magnetohydrodynamics mainly studies the dynamics or motion law of the inter-action between the flow field and the magnetic field in the conductive fluid.Its basic equations are composed of Navier-Stokes equations in fluid mechanics and Maxwell equations in electrodynamics.MHD equations has been widely used and developed in astrophysical research,controlled thermonuclear reaction,space physics,heat exchange,new industrial technology and flow control.So it is necessary to study the theory of MHD equations.In this paper,the third chapter mainly discusses the unsteady in-compressible MHD equations of velocity-correction projection method.The key to this method is that it naturally preserves gauss's law.In contrast to most previous ap-proaches that eliminate the electrical field variable E and give a direct discretization of the magnetic field.Our new method preserves the electrical field variable E.First,we construct a new numerical algorithm.Then,we give the stability analysis and error analysis.Finally,we give some numerical examples to prove the effectiveness of this method.In the fourth chapter,we present a characteristics projection finite element method.This method combines the modified characteristics finite element method with the pro-jection method.First of all,we study the system of time discrete and get the stability analysis and error analysis.Secondly,we will do full discretization.Finally,the numer-ical proves that our method is stable and has an optimal convergence rate.