【作者】 孙红；

【导师】 王同科；

【作者基本信息】 天津师范大学 ， 基础数学， 2005， 硕士

【摘要】 对流扩散方程的特征差分方法具有计算稳定,计算效率高等优点,在流体力学的数值模拟中有着广泛的应用。样条插值具有光滑性能好,精度高等优点,在工程实践中得到了广泛的应用。本文综合特征差分方法和样条插值的优点,提出了具有良好计算效果的基于样条插值的特征差分方法并运用离散泛函分析方法给出了其差分解对真解的误差估计式。全文共分三部分。 第一部分:一维对流扩散方程基于三次自然样条插值的特征差分方法,我们对对流项沿特征线方向离散,对扩散项使用二阶中心差商离散,在插值部分运用三次自然样条插值,对得到的差分格式,给出了L₂模的误差估计式,数值算例表明格式在很大程度上消除了插值误差对计算格式的影响。 第二部分是二维对流扩散方程的差分格式,我们是把第一部分进行推广,插值部分用双三次样条插值,并且也给出了L₂模的误差估计式。 本文最后一部分是一维对流扩散方程周期性边值问题,对扩散项采用了更高精度的紧差分格式离散,插值部分用三次周期样条插值,数值算例表明该格式精度确实比较高。更多还原

【Abstract】 The characteristics difference scheme of convection-diffusion equation has virtues that it is very stable to compute and the efficiency is very high. So it has a comprehensive application in numerical value simulation of hydrody-namics.The spline interpolation is very smooth and has a high accuracy.So it has a comprehensive application in the engineering practice. The paper combine the virtue of the characteristics difference method and the spline in-terplation.We obtain characteristics difference method based on cubic spline interpolation for convection-diffusion equtions and analyze the error estimate of exact solution and difference solution by functional analysis methods.The paper has three parts.The first part is characteristics difference method based on cubic natural spline interpolation for one-dimentional convection-diffusion eqution. we disrete the convection term along characteristics and the diffusion term is replaced by the second order centered difference quotients.We use cubic natural spline interpolation in the interpolation part and obtains discrete L₂-norm error estimate. The numerical examples show that to a large extent the scheme cancels the effect of the error in the interpolation.The secend part is characteristics difference method based on cubic natural spline interpolation for two-dimentional convection-diffusion eqution. We extend the first part to the two-dimensional convection-diffusion equation.The last part presents a higher order compact difference scheme and use cubic periodic spline interplation in the interpolation. The numerical example shows that the scheme is very good.更多还原