High-Order Temporally Discretization Schemes With Finite Element Methods For Several Magnetohydrodynamic Equations

The magnetohydrodynamic(MHD)equation is an important class of partial differen-tial equations,which have been widely applied into the industrial production.It consists of the Navier–Stokes equation and Maxwell’s equation,and its theoretical analysis and numerical computation are also the advanced fields in mathematics.In this dissertation,for the MHD equation and its related models,the numerical schemes will be designed together with the theoretical analysis.Finally,for all the schemes,accuracy and stability will be verified through the numerical examples.For the standard incompressible MHD equation,a fully discrete scheme will be pro-posed based on the Crank–Nicolson and finite element methods.Additionally,utiliz-ing the pressure projection method,an artificial variable is introduced to remove the in-compressible restriction from the velocity field equation.In detail,the original MHD equation has been divided into two sub-systems,where the second one,consisting of the velocity field and pressure field,is a Poission-type equation.For the fully discrete scheme,the unique solvability,energy stability and optimal convergence rates are proven.In particular,it is the first time that the error estimate of velocity fields measured in∞(0,T;L~2(Ω))has been proven when adopting the pressure projection methods,even including the Navier–Stokes equations.Taking the further interaction between the magnetic and electric fields into consid-eration,a quad-curl term of the magnetic field is added into the standard MHD equation,which becomes the so called resistive MHD equation.A fully discrete scheme combin-ing the second-order backward differential formulation(BDF2)and finite element meth-ods has been designed,which possesses three following features.Firstly,by defining an intermediate variable equal to the second-order curl of the magnetic field,the original model with a fourth-order curl term will be reduced into a second-order partial differential equation.Secondly,making use of the“zero-energy-contribution”properties,an artificial function satisfying a trivial ordinary differential equation will be introduced to separate the magnetic and velocity fields.Thirdly,based on the pressure projection methods,the pressure could be removed from the velocity equation.Consequently,in the computa-tion only several sub-systems need to be solved,among which except the magnetic field sub-system,the remaining ones could be shown that the computational expense is less by numerical examples.Hence the computational efficiency will be improved.In addition,the numerical scheme is proven to be uniquely solvable,energy-stable and convergent with optimal order of accuracy theoretically.The two phase MHD model,combined with magnetohydrodynamics and phase mod-els,is highly nonlinear and coupled,which leads to lots of difficulties in theoretical anal-ysis and numerical computation.The cube of phase field in the equation results in that,if one would like to remain the unconditional energy stability,the numerical scheme has to be nonlinear.Then the iteration method,such as the newton’s iteration method,will be introduced additionally in the computation,which costs relatively much.In the end,for this fully discrete modified Crank–Nicolson FEM scheme,the unique solvability,energy stability and optimal error estimates could be obtained.