Study On Fractional-step Method For Two Kinds Of Fluid Mechanics Equations

Magnetohydrodynamics fluid(MHD)equations and Navier-Stokes equations are important nonlinear partial differential equations in fluid mechanics.A kind of viscous incompressible fluid motion is closely watched by many mathematicians and physicists.In this thesis,we mainly study two aspects :(1)first-order fractional step algorithm of magnetohydrodynamics fluid(MHD)equations;(2)second-order Crank-Nicolson fractional step algorithm for unsteady navier-stokes equations.Then,the background and current research conditions and the introduction of the magnetohydrodynamics fluid(MHD)equations and navier-stokes equations are briefly described.The content of this master thesis is going to study.Then,respectively on two types of fluid mechanics equation of fractional step algorithm is studied.First,to explore the structure to solve the magnetohydrodynamics liquid(MHD)equations based on the first-order discrete format and time direction has a weak coupling form of fractional step algorithm,and analyze the stability and convergence of the algorithm,and theoretically proves the unconditional stability of fractional step algorithm and time order of convergence is given.Then based on the incompressible MHD equations with first order time discrete format of fractional step algorithm of preliminary understanding,will be based on the discrete time second order Crank-Nicolson format,structure to solve the unsteady incompressible Navier-Stokes equations of second order fractional step algorithm,and select the appropriate finite element space finite element discrete format,proved that the finite element discrete format of stability and convergence,and gives a optimal order error estimate of time and space error estimates.Finally,the fractional step algorithm of two kinds of fluid mechanics equations is summarized.