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一维抛物型方程的非多项式三次样条方法
Non-polynomial Cubic Spline Methods for Solving One-Dimensional Parabolic Equations
【作者】 杨涛；
【导师】 赵培浩；
【作者基本信息】 兰州大学 ， 基础数学， 2010， 硕士
【摘要】 基于非多项式三次样条函数,本文首先构造了一族求解一维抛物型方程初边值问题的差分格式.通过选择合适的参数,目前已有的一些差分格式都能从本文的差分格式中推导出来,并且还获得了一个高精度的三层隐式差分格式,该格式在时间和空间方向的精度均为四阶.然后,本文又给出了一个新的处理Neumann边界条件的差分格式,在空间方向边界点的精度达到了四阶.最后的数值实验表明本文的方法是非常有效的.
【Abstract】 In this paper, based on the non-polynomial spline function, a family of difference schemes for solving one dimensional parabolic equation initial boundary value problems are constructed firstly. By choosing suitable parameters, many difference schemes may be derived from our methods, and we obtain a high accuracy three-level difference scheme, the accuracy of this scheme is fourth-order in time and space direction. Secondly, a new method is presented to deal with the Neumann boundary value problem, the accuracy of this method in space direction is improved to fourth-order. Finally, the numerical results show that our methods are very efficient.