Numerical Analysis Of Efficient Numerical Method For The Time-dependent Incompressible Natural Convection Equations

The natural convection equations is an important equations in atmospheric dynamics.It includes unsteady or steady,compressible or incompressible situations.In real life and engineering applications,it plays an extremely important role.In this paper,we consider the time-dependent incompressible natural convection problem: it is not only an incompressible and strongly nonlinear problem,but also a problem in which the energy equation and the fluid motion equation are coupled together.Compared with the unsteady incompressible Navier-Stokes equations,it not only has an additional unknown temperature function,but also has a non-linear relationship between velocity and temperature.Finding numerical solution for this problem becomes a more difficult task.The direct difficulty generally encountered in numerical simulation of natural convection problems is the coupling of various unknown variables,which eventually leads to a huge scale of the linear equations formed after the problem is fully discretized.Directly solving the linear equations requires huge computer memory and a lot of CPU time.Thus,it is very necessary to construct a high-precision and high-efficiency numerical algorithm.In this paper,based on the finite element method,we mainly give two second-order time-accuracy numerical algorithms for solving timedependent incompressible natural convection equations,and give theoretical analysis and numerical experiments.The specific research arrangements are as follows:(1)We first consider a simple problem after the degeneration of the natural convection problem: When the temperature ? is equal to zero or constant,the evolution the natural convection problem reduce to the incompressible Navier-Stokes equations.For the time-dependent incompressible Navier-Stokes equations with Dirichlet boundary conditions,based on the finite element method,we combine the Crank-Nicolson Leap Frog(CNLF)method——(implicit-explicit combination)is a second-order time accuracy method,which is widely used in the coupling of atmosphere and ocean currents and climate modeling and prediction——and Grad-Div(GD)stabilization method to give a second-order time precision numerical algorithm(standard CNLF-GD).In addition,on the basis of standard CNLFGD,combined with the operator-splitting method,the fractional-step second-order time precision numerical algorithm(CNLF-FGD)is given,and the corresponding full-discrete algorithm and its corresponding stability and error analysis are given.And numerical experiments are provided to verify theoretical analysis.The advantage of two new algorithms: when the viscosity coefficient ? changes,the accuracy of the numerical solution can be well maintained.(2)For the time-dependent incompressible natural convection equations with Dirichlet boundary conditions,the CNLF method is directly applied to the coupled system.The coupled second-order time precision numerical algorithm(coupled CNLF method)can be obtained.For coupled CNLF method,three unknown variables(velocity,pressure and temperature)must be calculated at the same time.In order to solve this problem more efficiently,we combine the decoupling method with the CNLF method and propose a decoupled CNLF numerical algorithm.The decoupled CNLF algorithm decomposes the original problem into two sub-problems,and solving each sub-problem is much easier than the original problem.We give the stability and error analysis of the numerical format and verify it through numerical experiments.(3)The time filter method is an effective method to improve the accuracy of the numerical solution.When the original value satisfies certain feasible assumptions,compared with the Backward Euler method,The backward Euler method of time filtering can not only improve the accuracy of the numerical solution from the first order to the second order,but also reduce the spurious oscillation of the numerical solution and maintain the stability of the numerical format.It is easier to implement than the second-order backward Euler method.Due to this reason,we give a second-order time precision decoupled time filtered Backward Euler method to solve the time-dependent incompressible natural convection equations,and give corresponding theoretical analysis and numerical experiments.