The Decoupled Schemes For Stationary Navier-Stokes Equations And Magneto-hydrodynamic Equations

This paper devoted to developing the decoupled schemes for stationary Navier-Stokes equa-tions and MHD equations.It is well known the Oseen iteration[35]for the stationary Navier-Stokes equations is uncon-ditionally stable.However,it is a coupled type scheme where the velocity u and pressure p are coupled together at each iteration.By treating pressure p explicitly would lead to a decoupled iteration,but this crude and simple treatment is unstable.Thus,we construct a decoupled and un-conditionally stable iteration method to solve the nonlinear and coupled stationary Navier-Stokes equations by adopting the pressure projection method to the temporal disturbed Navier-Stokes system whose solution approximates the steady state solution over time(t?+?).We also rigorously prove its unconditional stability.Numerical simulations demonstrate that our itera-tive method is more efficient and stable than the T-S iteration[20]and extensively used Oseen iteration.Incompressible magneto-hydrodynamics(MHD)describes the flow of a viscous incompress-ible and electrically conducting fluid.And it has extensive applications in the development of geophysics,astrophysics,cosmology and thermal nuclear controlling.The MHD equations couple the incompressible Navier-Stokes equations with the Maxwell equations.Therefore,the MHD equations have the characteristics of nonlinearity,coupling and multiphysics.We propose first-order and second-order linear,unconditionally energy stable,splitting schemes for solving the Magneto-hydrodynamics(MHD)system.These schemes are based on the pro-jection method for Navier-Stokes equations and subtle implicit-explicit treatments for nonlinear coupling terms.We transform a double saddle points problem into a set of elliptic type problems to solve the MHD system.Thence,our schemes are efficient,easy to implement,and stable.We further prove that the time semi-discrete scheme and fully discrete scheme are unconditionally en-ergy stable.Finally,several physical experiments,including Hartmann flow and lid-driven cavity problems,are enforced to test the stability and the accuracy of the schemes.On this basis,we propose a first-order linear,unconditionally energy stable,fully decoupled scheme for solving the MHD system.Because the magnetic field and velocity are still coupled together,the two previous decoupling schemes are only partially decoupled and not fully decou-pled.Thus,we introduce a first order accuracy term to stabilize the decoupling calculation of the magnetic field and the velocity field.We prove the scheme is unconditionally energy stable rigor-ously in both time semi-discrete and fully discrete levels.Finally,various numerical experiments are implemented to demonstrate the stability and accuracy of our scheme.