Some Numerical Methods For Solving Navier-Stokes Equations

The Navier-Stokes equations are the foundation of fluid mechanics.Since they can help us to know the order of natural,so it is important to study the Navier-Stokes equations.However,it's very difficult to solve the Navier-Stokes equations,up to now, there are only a few simple problems getting the exact solutions.Most of them get the approximation solution by discretization.Thus,we pay more attention to discrete methods for the Navier-Stokes equations.In this thesis we discuss some numerical methods for the Navier-Stokes equations, such as iterative penalty method,two-grid method,operator-splitting method and so on.The unknowns(u,p) of Navier-Stokes equations are coupled together by the incompressibility constraint▽·u=0.Additionally,the Navier-Stokes equations have nonlinearity and sometimes dependent on time,all of these make us troubled when solving the Navier-Stokes equations.Many numerical schemes have been developed for dealing with one or more difficulties above.This thesis has been separated into four chapters.In chapter 1 we briefly introduce some models about the incompressible flows.In fact,the Navier-Stokes equations are the behaviors of conservation of mass,conservation of momentum and the Newton's famous Second Law of Motion in fluid mechanics.Every variable and every relation have realistic meanings,for example,u represents velocity,p is the pressure,▽·u=0 means that divergence free.And we also can know that the Stokes equations are the approximation of the Navier-Stokes equations under some assumptions.In the rest of this chapter we also say something about the finite element method,which is our discrete method in this thesis.Compared with finite difference method,the finite element method is stable and can get error estimates easily.At last,we point out some problems that we meet in coding for finite element method.The Stokes equations should be solved by mixed finite element method,so at the beginning of chapter 2 we get some knowledge of mixed finite element method.In order to guarantee a unique solution to the problem,the spaces we choose must satisfy the LBB condition,so does discrete case.That means we must choose the stable element,such as RT,BDM,Mini and Taylor-Hood pairs.The Stokes equations are equivalent to a constrained minimization problem.The divergence-free is the constrained condition,thus we apply the penalty method related to reduced integration to the problem.Compared the Lagrange multiplier method,penalty method make the variable least.It is well know that the error estimate of the penalty method to Stokes equations is O(ε+R_h),where R_h is obtained by space discretization.Thus,εmust be sufficiently small to yield an accurate approximation of the solution.On the other hand,the condition number of the numerical discretizations of Stokes equations is O(ε^-1h^-2),so the scheme becomes unstable due to round-off error,ifεis too small. Thus,an iterative penalty method is proposed for the Stokes equations,which allows us to use a "not very small" penalty parameter to avoid the unstable computation. However,the finite elements which have been chosen are required to satisfy the LBB condition.The simplest finite elements,i.e.Q₁-P₀ element do not satisfy the LBB condition.We discuss the Q₁-P₀ element for the Stokes equations and present the error estimates for the iterative penalty method using this element.In chapter 3 we introduce two-grid methods and their application to the Navier-Stokes equations.The idea of the two-grid finite element was originally proposed by Jinchao Xu for discretizing nonsymmetric and indefinite partial differential equations, and then has been further investigated for linearization for nonlinear problems,for localization and parallelization for solving a large class of PDEs,for decoupling the systems of PDEs.First we introduce the two-grid method by two examples,and then we apply this idea to solve the Navier-Stokes equations.We improve the scaling for errors in energy norm and L² norm,e.g.,for energy norm it gets to h～H^4-s and for L² norm it gets to h～H²,if k=1,where s=1/2 of d=3 and s=ε＞0 if d=2.The two-grid method we proposed involves solving one small,nonlinear coarse mesh system,and two linear problems on the fine mesh which have the same stiffness matrix with only different right hand side.Numerical example is given to show the convergence of the method.In chapter 4 we introduce fractional-step methods for the approximation of the unsteady incompressible Navier-Stokes equations.At first,we state some popular operator-splitting methods and their error estimates,respectively.We also list their advantage and disadvantage,and then we propose our schemes.We use the Crank- Nicolson time stepping and we not only use the velocity and pressure at t_n but also the pressure at t_n-1.Under mild regularity assumptions on the continuous solution,we obtain second order error estimate in the time step size,both for velocity and pressure. Numerical results in agreement with the error analysis are also presented.