本文選題：流體及流體耦合問題　切入點：特征線　出處：《南京師范大學(xué)》2016年博士論文

【摘要】：在本文中,我們研究幾類流體及流體耦合問題的有限元方法。流體及流體耦合問題在海洋學(xué)、地球物理學(xué)以及流體力學(xué)中經(jīng)常遇到。例如,低速運動的氣流,水流,地下水污染問題以及大氣-海洋耦合問題等。當(dāng)應(yīng)用通常的有限元方法來數(shù)值求解這些問題,由于對流占優(yōu)特性、高雷諾數(shù)問題及線性和非線性耦合條件,通常的有限元方法會使數(shù)值方法的有效性變差。本文的目的是綜合運用特征方法、變分多尺度方法及穩(wěn)定化有限元方法,設(shè)計有效求解此類問題的數(shù)值方法,給出相應(yīng)的穩(wěn)定性分析和誤差估計。第一章,我們提出了求解對流占優(yōu)對流擴散反應(yīng)問題的特征變分多尺度方法。該格式的構(gòu)造綜合了特征線方法和變分多尺度方法,給出了相應(yīng)格式的穩(wěn)定性分析和誤差估計。該格式不僅降低了時間截斷誤差、可以應(yīng)用較大的時間步長而且還保持了良好的穩(wěn)定性和高精度。二維和三維數(shù)值試驗表明了該格式的有效性。第二章,基于最低階的等階協(xié)調(diào)有限元子空間,我們給出了一類新的特征穩(wěn)定化有限元方法數(shù)值求解不可壓的Navier-Stokes方程。我們運用動量方程的殘量和散度自由方程定義穩(wěn)定化項。特征線方法和穩(wěn)定化有限元方法的自然組合保持了兩類方法的最優(yōu)特點。嚴(yán)格推出了相應(yīng)格式的穩(wěn)定性和誤差估計。最后,數(shù)值試驗驗證了該方法數(shù)值求解非穩(wěn)態(tài)Navier-Stokes方程的有效性。第三章,我們研究了逼近流體耦合問題的數(shù)值方法,考慮一個簡化模型,兩個對流占優(yōu)對流擴散反應(yīng)方程通過界面條件耦合,提出了隱顯時間步進流線擴散法求解該類問題,得到了相應(yīng)的穩(wěn)定性分析和誤差估計,數(shù)值試驗證明了該方法的有效性。第四章,我們分析了局部投影穩(wěn)定化特征解耦格式求解流體耦合問題。我們應(yīng)用特征線方法克服非線性項導(dǎo)致的困難,應(yīng)用局部投影穩(wěn)定化方法來控制偽振蕩,應(yīng)用幾何平均的思想解耦耦合問題。給出相應(yīng)格式的穩(wěn)定性分析,數(shù)值試驗證明了該方法的有效性。
[Abstract]:In this paper, we study the finite element method for several kinds of fluid and fluid coupling problems.Fluid and fluid coupling problems are often encountered in oceanography, geophysics, and fluid dynamics.For example, low-speed moving air flow, water flow, groundwater pollution and atmospheric-ocean coupling problem and so on.When the conventional finite element method is used to solve these problems, due to the convection dominance, the high Reynolds number problem and the linear and nonlinear coupling conditions, the general finite element method will make the effectiveness of the numerical method worse.The purpose of this paper is to design an effective numerical method for solving this kind of problems by means of characteristic method, variational multi-scale method and stabilized finite element method, and give the corresponding stability analysis and error estimation.In chapter 1, we propose a characteristic variational multiscale method for convection-dominated convection-diffusion reaction problems.The construction of the scheme combines the eigenline method and the variational multi-scale method, and gives the stability analysis and error estimation of the corresponding scheme.This scheme not only reduces the time truncation error, but also keeps good stability and high precision.Two-dimensional and three-dimensional numerical experiments show the effectiveness of the scheme.In chapter 2, based on the lowest order equal-order conforming finite element subspace, we give a new type of eigen-stabilized finite element method to solve the incompressible Navier-Stokes equation numerically.We define the stabilization term by using the residual and divergence free equations of momentum equation.The natural combination of the eigenline method and the stabilized finite element method preserves the optimal characteristics of the two methods.The stability and error estimation of the corresponding scheme are strictly derived.Finally, numerical experiments show that the method is effective in solving unsteady Navier-Stokes equations.In chapter 3, we study the numerical method of approximate fluid coupling problem, and consider a simplified model. Two convection-dominated convection-diffusion reaction equations are coupled by interfacial conditions, and a implicit time step streamline diffusion method is proposed to solve this kind of problems.The corresponding stability analysis and error estimation are obtained, and the effectiveness of the method is proved by numerical experiments.In chapter 4, we analyze the characteristic decoupling scheme of local projection stabilization to solve the fluid coupling problem.We apply the eigenline method to overcome the difficulties caused by nonlinear terms, apply the method of local projection stabilization to control the pseudo oscillation, and apply the idea of geometric mean to decouple the coupling problem.The stability analysis of the corresponding scheme is given, and the effectiveness of the method is proved by numerical experiments.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O241.82

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相關(guān)期刊論文 前1條

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本文編號：1709429