【作者】 陈荣华；

【导师】 吴宗敏；

【作者基本信息】 复旦大学 ， 应用数学， 2005， 博士

【摘要】 本文共分五章。其中,第一章简单介绍已知的解偏微分方程的数值方法。第二章为与本文有关的预备知识。介绍了一些有关径向基函数和双曲型偏微分方程的理论,对已有的四种Multiquadric(MQ)拟插值做出了详细的概括。第三章是我们的研究成果之一。我们给出了一类拟插值是多项式再生的的充分必要条件。此外,还构造了一种新的单变量Multiquadric拟插值,该拟插值在插值区间具有线性再生性及保单调性。数值试验的结果表明:该拟插值的逼近精度较高。第四章论述径向基函数在微分方程数值解中的具体应用。其中,对Multiquadric在微分方程数值解中的应用的介绍比较详细。我们主要就MQ拟插值在双曲型方程和抛物型方程中的应用进行了研究,其基本思路是:利用MQ拟插值的导数逼近微分方程的空间导数,而微分方程的时间导数则采用有限差分逼近。当然,构造数值格式时还利用了一些其它技巧,如引入了一个我们称之为“开关函数”的函数以抑制数值格式的色散等等。从我们给出的数值试验可以看出:我们的方法是可行的。第五章是讨论。该章就我们在求解双曲守恒型方程时所使用的格式及今后将要开展的工作提出我们的初步的看法。更多还原

【Abstract】 This dissertation consists of five chapters. In Chapter 1, we summarize the known numerical methods for solving the partial differential equations(PDE). Chapter 2 includes the preliminary knowledge. In this chapter, we introduce the theory of the radial basis function (RBF) and the hyperbolic partial differential equation. In addition to this, we summarize in detail the knowledge of the konwn four kinds of multiquadric(MQ) quasi-interpolation. Chapter 3 roots in our research. In which, we give the sufficient and necessary conditions for a kind of quasi-interpolation as it is polynomial reproducing. Furthermore, we construct a new univariate MQ quasi-interpolation which possesses the properties of the linear reproducing and preserving monotonicity. The numerical results show that it possesses higher accuracy. In Chapter 4, we discuss the application of the radial basis functions for numerical solving the partial differential equations (PDEs). In which, we give a detailed introduction for applying the MQ to solve PDEs. We investigate mainly solving the hyperbolic and parabolic equations by using the MQ quasi-interpolation. The underlying idea of our means is that: employing the derivative of the MQ quasi-interpolation to approximate the spatial derivative of the PDE, while the approach of the temporal derivative of the PDE is used a finite difference. Of course, we impose other techniques, such as, when we construct the numerical schemes, we import a function which is called "switch function" to damp the dispersion of the numerical schemes and so on. From the results of the numerical experiments given by us, we see that our method is valid. Chapter 5, named discussion. In this chapter, we give the elementary perspectives for更多还原