高振荡积分方程及其数值解法

发布时间：2018-08-06 21:02

【摘要】：高振荡问题已广泛出现于许多科学工程应用领域,例如与生活息息相关的电磁、声波散射问题,而高振荡积分方程是高振荡问题中的一个重要研究方向。然而由于高振荡积分方程中的高振荡性质,传统积分方程数值解法面临极其困难的数值计算挑战,使得高振荡积分方程数值解至今被认为是一项极具挑战性的数值难题。因此,高振荡积分方程及其数值解研究对高振荡理论和解决实际应用问题均有重大意义与价值。本文围绕高振荡积分方程的振荡性质及其有效数值解法展开了广泛和深入的研究。主要工作及创新点体现在以下几个方面:(1)提出了振荡的新概念并定义了新的振荡函数空间。本文从振荡对数值分析的影响的角度出发提出了关于振荡的新概念,它可以刻画影响数值精度的振荡强弱。基于新振荡概念,本文定义了新的振荡函数空间,包括不同振荡阶的振荡函数空间和具有振荡结构的结构化振荡空间。这些空间是分析高振荡积分方程解的振荡性质的重要工具。(2)进行了高振荡积分方程解的振荡性质研究。基于振荡新概念和新振荡函数空间,本文分别研究了两类高振荡Fredholm积分方程和高振荡Volterra积分方程。结果表明,这两类振荡积分方程的解具有振荡结构,在新振荡概念下可以表示为非振荡函数与已知振荡函数的乘积的和,同时在新定义的结构化振荡函数空间中是非振荡的。(3)提出了高振荡积分方程的保振荡Galerkin法和保振荡配置法。保振荡法在标准逼近空间中引进一些简单又能刻画方程解振荡性质的振荡函数,在逼近方程解时保持解的振荡结构,从而使得数值求解精度不受解的高振荡影响。数值实验结果表明这些保振荡法相对于振荡频率具有一致的最优收敛阶并且在振荡频率足够大时在数值计算上是稳定的。(4)提出了多频振荡插值。它是保振荡配置法的基础,其插值函数除了包含经典的多项式函数还包括一组具有不同振荡频率的振荡函数,可以使得在逼近振荡函数时其逼近误差不随振荡频率的增大而增大。(5)利用基于非均匀网格划分的保振荡配置法求解了一类有实际应用背景的带高振荡Bessel积分核的Volterra积分方程。数值实验结果表明无论强制项函数是否振荡,基于非均匀网格划分的保振荡配置法均能有效求解这类高振荡方程而不受高振荡的影响。本文结尾总结了可进一步发展的三个研究方向:复杂高振荡积分方程解的振荡性分析、保振荡数值法的快速算法及并行计算研究以及保振荡法在实际问题中的应用。
[Abstract]:High oscillation problems have been widely used in many scientific engineering applications, such as electromagnetic and acoustic scattering problems, which are closely related to life, and high oscillation integral equations are an important research direction in high oscillation problems. However, due to the high oscillation in the high oscillation integral equation, the traditional numerical solution of the integral equation is faced with a very difficult numerical calculation challenge, so the numerical solution of the high oscillation integral equation is considered to be a challenging numerical problem. Therefore, the study of the high oscillation integral equation and its numerical solution is of great significance and value to the theory of high oscillation and the solution of practical problems. In this paper, the oscillatory properties of high oscillatory integral equations and their effective numerical solutions are studied extensively and deeply. The main work and innovations are as follows: (1) A new concept of oscillation is proposed and a new oscillation function space is defined. From the point of view of the effect of oscillation on numerical analysis, a new concept of oscillation is proposed in this paper, which can describe the intensity of oscillation which affects the numerical accuracy. Based on the new concept of oscillation, a new oscillation function space is defined in this paper, including oscillatory function space with different oscillation order and structured oscillation space with oscillatory structure. These spaces are important tools for analyzing the oscillatory properties of solutions of high oscillatory integral equations. (2) Oscillation properties of solutions of high oscillatory integral equations are studied. Based on the new concept of oscillation and the new oscillatory function space, two kinds of highly oscillatory Fredholm integral equations and highly oscillatory Volterra integral equations are studied in this paper. The results show that the solutions of these two kinds of oscillatory integral equations have oscillatory structures and can be expressed as the sum of the products of nonoscillatory functions and known oscillatory functions under the new concept of oscillation. At the same time, in the newly defined structured oscillatory function space, the nonoscillatory method is proposed. (3) the oscillation-preserving Galerkin method and the oscillation-preserving collocation method for the high oscillation integral equation are proposed. The oscillation-preserving method introduces some simple oscillatory functions in the standard approximation space which can describe the oscillatory properties of the solution of the equation. The oscillatory structure of the solution is maintained when the solution is approximated, so that the accuracy of the numerical solution is not affected by the high oscillation of the solution. Numerical results show that these oscillation-preserving methods have a consistent optimal convergence order relative to the oscillation frequency and are stable in numerical calculation when the oscillation frequency is large enough. (4) Multi-frequency oscillation interpolation is proposed. It is the foundation of the oscillation-preserving collocation method. Its interpolation function includes not only classical polynomial functions, but also a set of oscillatory functions with different oscillation frequencies. It is possible that the approximation error does not increase with the increase of oscillation frequency when the oscillation function is approximated. (5) A class of Volterra integral equations with high oscillatory Bessel integral kernels are solved by using the oscillation-preserving collocation method based on nonuniform grid partitioning. The numerical results show that the Oscillation-preserving collocation method based on nonuniform mesh division can effectively solve this kind of high oscillation equation regardless of whether the mandatory term function is oscillating or not without the influence of high oscillation. At the end of this paper, three research directions that can be further developed are summarized: the oscillation analysis of the solution of complex high oscillation integral equation, the fast algorithm and parallel computation of oscillation-preserving numerical method, and the application of oscillation-preserving method in practical problems.
【学位授予单位】：国防科学技术大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.83

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