The Projection Methods For The Non-steady Natural Convection Problem

Nonlinear partial differential equation is the important mathematical model to describe the law of motion and natural phenomena.Due to the limited understanding for the essence of the nonlinear phenomenon,the numerical simulation becomes a very important research means.There is a big difficulty in direct numerical simulation of nonlinear partial differential equation because of the contradictions between the huge computational scale,long-time integration and limited computing resource.Thus it is extremely important to develop and research high-effective and low-loss algorithm which has long-time stability.A major difficulty for the numerical simulation of incompressible flows is that the velocity and the pressure are coupled by the incompressibility constraint.So we need much more computing time and a large amount of computing storage to solve directly the solution of the nonlinear equations of the discrete problem.The projection method is an efficient numerical scheme for the nonlinear problems and multivariables problems.Using the projection method is to decouple the relationship between the variables,and the nonlinear problem is decomposed into several?two or three?linear subproblems,so as to reduce the size of the solution,decrease the amount of calculation and save the calculation time.In this article,we focus on creating several projection methods to solve the unsteady natural convection problem.The main research contents are as following:In Chapter 3,we consider the first order decoupled methods for the unsteady nat-ural convection problem.Firstly,the temperature?ⁿ⁺¹is solved in parabolic equation.Secondly,the pressure and velocity are decoupled by using the projection method.And then the intermediate velocity uⁿ⁺¹?the velocity uⁿ+1+1 and the pressure pⁿ+1+1 are solved by the relative subproblems.Finally,in order to achieve better convergence order,we construct a modified projection scheme.The advantage is to improve the velocity,temperature and pressure convergence order by the pressure correction,which is the it-erative pressure difference pⁿ⁺¹-pⁿ.Moreover,numerical examples verify the stability and convergence of the new methods.In Chapter 4,we mainly study the second order projection method based on the Euler backward iteration.This method is optimized from the modified first projection scheme of Chapter 3,and its proof method is combined with Taylor expansion technique,mathematical induction,Stokes projection and other techniques.In addition to the advantages of the above projection scheme,it avoids the use of velocity projection step,which reduces the storage space and computing time.In Chapter 5,we analysis the viscosity-splitting method for the unsteady natural convection problem.The difficulties?nonlinear term,incompressibility and variable coupling?of solving the unsteady natural convection problem is separated,getting a series of linear problems,so as to simplify the calculation and enhance the efficiency of the calculation.Theoretical analysis and numerical examples verifies effectiveness about the viscosity-splitting method to solve the unsteady natural convection problem.