【作者】 孙旭东；

【导师】 李松涛；

【作者基本信息】 吉林大学 ， 计算数学， 2008， 硕士

【摘要】 本文对金属凝固过程中的数学模型进行了模拟,该模型可简化为二维常系数抛物型微分方程。文中建立了三种差分格式:向前差分格式、紧交替方向隐式差分格式和改进的紧交替差分格式。同时给出了三种差分格式的收敛阶分别为: O (τ+h~2)、O (τ~2 +h~4)、O (τ~4 +h~6)。利用已知的实验数据对三种格式进行了验证,可以看出改进的紧交替差分格式是模拟金属凝固过程一种有效的且收敛阶较高的数值计算格式。文中给出改进的紧交替方向差分格式在H 1范数下的先验估计式并证明了改进的紧交替方向差分格式的可解性、稳定性和收敛性。更多还原

【Abstract】 This paper presents the actual problem of metal casting,and there is a casting cracks of groove in the middle of metal casting, five percent TiC liquid metal is poured to fill the cracks.The temperature of liquid metal is 983 k, casting temperature is 293 k at room temperature.In order to describe the changes in the temperature of the liquid metal, the mathematical model of the issue has been simplified in the text, that is, considering the following forms of differential equations:Firstly , the forward difference format for doing the model is given in the second chaper of this dissertation:According to Taylor expandedness, we know the interceptive error is R (u )= O(τ+h2)Secondly , the difference format of tight turn vector for doing the model is given:It is analyzed and this paper gives a expression of the interceptive error,the rank of which is O (τ2 + h4)。According to the Extrapolation of Rechardson, the difference format of tight turn direction is gived in the third chapter of this dissertation .It is showed as follows:And this paper gives the improved difference format of tight turn direction under the priori estimation of discrete norm H 1.The first theorem: on the assumption that difference format of tight turn direction which is show as follows:The following theorem is proofed according to the priori estimation:The second theorem: the improved difference format of tight turn direction (6)-(8) have the only one answer.The third theorem: the improved difference format of tight turn direction (6)-(8) is stable for the initial value and the non-homogeneous term.The fourth theorem: supposing that the answer of the improved difference format (6)-(8),then the answer of the improved is convergent.According to the known experimental data, the three format are validated in the last part of this paper , we can see that the tight turn improve differential format is an effective format of Numerical Calculation which haves higher-order convergent rank in the process of analog metal solidification.更多还原