Decoupled Algorithms And Preconditioning Structure Preserving Method For Magnetohydrodynamics Models

The word“Magnetohydrodynamics”was first used by Swedish physicist Hannes Alflv?en in 1942:“At last some remarks are made about the transfer of momentum from the Sun to the planets,which is fundamental to the theory.The importance of the magnetohy-drodynamic waves in this respect are pointed out.”^[1].Nowadays,MHD has developed into an active research branch of the inter-discipline of physics,mechanics and mathe-matics.It studies the dynamic behaviors of conductive fluid under external magnetic field.The research objects of MHD are plasma,liquid metal,salty water and dielectric.There are extensive applications of MHD in geophysics,astrophysics,cosmology and thermal nuclear controlling,please refer literatures^[2–4].The basic mechanism of MHD is that external magnetic field can induce current in the conductive fluids,meanwhile,the induced current can in turn alter the external magnetic field.The MHD phenomenon can be described by coupling the Navier-Stokes equations and Maxwell equations.As we know,the mathematical research of MHD problem originated from the M.Ser-mange,R.Temam's paper^[5].Up to now,there are many literatures investigating the MHD's mathematical theory,please refer^[6–12].At the same time,the numerical study of MHD are very active.In 1991,M.D.Gunzburger,A.J.Meir and J.P.Peterson studied the Galerkin finite element approximations of steady MHD model in^[13].Later,J.F.Gerbeau,C.Le Bris and T.Leli`evre discussed some mathematical theory and numerical methods for stationary MHD models in the monograph^[14].Besides,there are many papers studying the MHD problem from numerical aspect^[15,17–21].There are three main difficulties in MHD numerical studies:nonlinearity,coupling and multi-physics.Nonlinearity refers the convection part and Lorentz force part in fluid e-quations and convection part in magnetic equations are all nonlinear.The usual method to deal with the nonlinearity is linearization,such as Picard linearization and Newton linearization.But,in the big physical parameters cases,nonlinear iterations converge slowly,even,nonlinear iterations diverge.Coupling is that the Navier-Stokes equation-s and Maxwell equations are coupled together through Lorentz force and Ohm's law.Solving the MHD problems need to find the solutions of velocity,pressure,magnetic field and electric field simultaneously.Therefore,there are huge amount of freedoms in the MHD practical computations,which brings a big challenge to the storage of computers and linear solver design.Multi-physics means that MHD model refers several different physical fields,whose corresponding operators have different features.The algebraic system of MHD model after discretization is block matrix.The different physical fields need different preconditioners.Especially,the efficient and robust strategies of solving preconditioning inverse actions of different physical fields are different as well,which is a big difficulty for devising efficient and stable MHD solver.This thesis mainly focus on tackling the nonlinearity,coupling and multi-physics the three difficulties of MHD problem.Our main work are as follows^*:1.We propose coupled correction and decoupled parallel correction algorithms based on two grids method for solving MHD problems.We establish the accurate conditions of the well-posedness for the MHD equations.Based on this,we find the dependence of the two kinds of algorithms on the MHD parameters via theoretical analysis,which builds the theoretical basis for the two classes of algorithms.Numerical experiments confirm our theoretical analysis and show the high efficiency of our algorithms.2.We propose the decoupled schemes for the time dependent MHD problems.By decoupling the fluid equations and magnetic equations,one can solve the fluid subprob-lem and magnetic subproblem respectively in simulating time evolution MHD problems,which facilitates to make use of fluid codes package and magnetic codes package.We provide the stability and convergence analysis for the decoupled scheme.We prove the decoupled scheme is almost unconditional??t?C?stable and convergent with optimal convergence orders.3.Based on the structure preserving MHD model,we propose three unconditional en-ergy stable schemes,and we provide the stability and convergence analysis for the three schemes.Furthermore,we devise a unified solver for Picard linearised scheme.And we give both theoretical analysis and numerical tests for the solver,which demonstrate that the iterations of our devised solver is independent of time step size,spatial grid size and MHD physical parameters.