Research On Second-order Crank-nicolson Scheme For A Class Of Magnetohydrodynamic System

This thesis proposes a second-order semi-implicit Crank-Nicolson scheme for a class of magnetohydrodynamic equations.The magnetohydrodynamics equations are coupling the nonstationary Navier-Stokes equation with the stationary Maxwell's equation.The main characteristics of the scheme are two-fold.One is that the scheme is decoupled.The magnetic field is solved first,and then the velocity field is solved on the same time level.Another is that the two sub-problems are linear,saving the CPU time and easy to be implemented.This thesis is organized by three main parts as follows:1.Using the second-order Crank-Nicolson extrapolation scheme in the time direction,we prove that the algorithm has a second-order convergence rate for magnetic fields,velocity fields,and pressures.2.The Galerkin finite element approximation was applied to the magnetic field-velocity field-pressure,we obtain the spatial finite element error estimate.3.By using the finite element software FreeFEM++,we give the numerical results to verify the theoretical error estimates.