求解空间分数阶扩散方程和对流扩散方程的有限差分格式研究

发布时间：2018-04-13 09:39

  本文选题：一维空间分数阶扩散方程 + 一维空间分数阶对流扩散方程　； 参考：《宁夏大学》2016年博士论文

【摘要】：近几十年来,由于分数阶导数可以用来刻画反常扩散现象,而许多复杂动力系统都包含着反常扩散,故分数阶动力学方程是描述复杂系统的有效方法,从而使得分数阶偏微分方程在自然科学和社会科学得到广泛应用,由于其解析解不容易求出来,所以分数阶偏微分方程数值解的研究就显得尤为重要。求解分数阶微分方程数值解需要克服一些困难,其一,分数阶算子具有非局部记忆性质,导致分数阶微分方程的数值求解不稳定；其二,分数阶微分方程的数值求解通常需要求解全系数线性方程组,一般需要用O(N3)的计算时间和O(N2)的存储空间,其中N是网格点数；其三,对于给出的数值格式进行理论分析比较困难。能否构造既稳定又节省计算时间和存储空间的数值方法,而且还方便进行数值格式的理论分析?最大值原理做为整数阶微分方程数值方法中的经典理论分析工具为我们构造这样的格式提供了可能性,它不仅保证了格式的稳定性,而且可以用最大值原理对差分格式做一些先验估计,并且依次可以对差分格式的收敛性、稳定性等做进一步的证明；在此基础上,结合整数阶微分方程的一些经典数值方法,可以构造一些既节省计算时间、存储空间又方便理论分析的稳定的数值方法用以求解分数阶偏微分方程。本文主要基于Riemann-Loiuville分数阶导数定义,以最大值原理为基础,结合一阶精度的Grunwald公式和移位Grunwald公式,构造二阶精度的算子近似分数阶导数,再利用隐式欧拉方法和Saul'ev算法,给出了求解一维空间分数阶扩散方程及对流扩散方程的几种有限差分格式,另外介绍了求解空间分数阶扩散方程的满足最大值原理的中心差分格式,且对格式的稳定性、收敛性等进行了详尽的理论分析。本论文的主要内容及创新点如下：一,介绍了分数阶微分方程数值解相关的研究背景、意义、研究问题的提出以及国内外研究现状,并给出了本文需要用到的理论预备知识,即分数阶导数的几种常用定义及等价性关系、性质和最大值原理的基本知识。二,提出了满足最大值原理的二阶有限差分算子离散分数阶导数,结合隐式欧拉公式构造了解一维单边、双边空间分数阶扩散方程及一维单边、双边空间分数阶对流扩散方程的二阶有限差分格式,并且用最大值原理或者能量不等式法进行了稳定性证明和收敛性分析。三,由上一章提出来的满足最大值原理的二阶算子结合Saul'ev算法,构造了一种对称半隐格式求解空间分数阶扩散方程及空间分数阶对流扩散方程。此格式形式上是隐格式,但是计算过程是显式的,大大节省了计算量和存储量。对此数值方法的稳定性、误差分析进行了详尽的分析证明,并通过理论证明和数值算例均验证了此半隐格式在l2范数意义下的误差估计式为C(△t2h-2(1-α)+△t+h2),其中α是分数阶导数的阶且△t,h分别是时间和空间步长。四,给出一个非整数节点上的二阶有限差分算子离散分数阶导数项,并结合最大值原理,构造了解单边空间分数阶扩散方程的一个定义在非整数节点的有限差分格式,并且用最大值原理进行了稳定性、收敛性分析。
[Abstract]:In recent decades, due to the fractional derivative can be used to describe the anomalous diffusion phenomenon, and many complex dynamical systems contain anomalous diffusion, the fractional kinetic equations is an effective method to describe the complex system, which makes the fractional partial differential equations will be widely used in science, natural science and society, because of its analytical solution is not easy seek out, so the fractional partial differential equation of numerical solution is particularly important. The numerical solution of fractional differential equations need to overcome some difficulties, a fractional order operator with non local memory properties, resulting in the numerical solution of fractional differential equations is not stable; second, the numerical solution of fractional differential equations usually require calculate the total coefficient of linear equations, the general need to use O (N3) of the O (N2) computing time and storage space, where N is the grid points; thirdly, the numerical lattice is given in Type analysis is difficult. Whether the structure is stable and save the numerical method of computing time and storage space, but also facilitate the analysis of numerical theory? The maximum principle for the classical theory of numerical methods for integer order differential equation analysis tools provide the possibility for us to construct such a format, it not only guarantees the format the stability, but also the principle of difference schemes with some priori estimates of maximum value, and can turn on the convergence of the difference scheme, the stability of further proof; on this basis, some classical numerical method combining integer order differential equations, which can be constructed to save computing time, storage space and convenient stable numerical method of theoretical analysis for solving fractional partial differential equations. In this paper the definition of fractional derivative based on Riemann-Loiuville, with the largest value Based on the principle, and shift Grunwald formula with Grunwald formula of first order accuracy, operator two order accuracy approximation of fractional derivative, and then using the implicit Euler method and Saul'ev algorithm, gives several finite difference for solving one-dimensional space fractional diffusion equations and convection diffusion equation format, also introduces the solution space fractional the diffusion equation satisfies the maximum principle of the central difference scheme and stability of the format, the convergence is analyzed theoretically in detail. The main contents of this paper and innovation are as follows: first, this paper introduces the research background, the related numerical solution of fractional differential equations, put forward the research problem and research at home and abroad the status quo, and the need to use the theory of knowledge, namely different definitions and equivalence relation of fractional order derivative, the nature and basic knowledge of the principle of maximum two, Put forward to meet the two order finite difference maximum principle of discrete fractional derivative operator, combined with implicit Euler formula about one-dimensional unilateral, bilateral space fractional diffusion equation and one dimensional unilateral, bilateral two order finite difference space fractional convection diffusion equation and using the format, analyzed to prove the stability and convergence the principle or method of the maximum energy inequality. Three, proposed by the previous chapter meet the two order operator maximum principle based on Saul'ev algorithm, constructed a symmetric semi implicit scheme for solving the space fractional diffusion equation and space fractional convection diffusion equation. This format is implicit, but the calculation process is explicit, greatly reduces the amount of calculation and storage stability. This numerical method, error analysis carried out a detailed analysis of the proof, and through theoretical proof and numerical examples have verified this half The error implicit in the sense of L2 norm estimation for C (delta t2h-2 (1- alpha) + t+h2), where alpha is the fractional derivative order and delta T, h are the time and space step. Four, two order finite difference gives a non integer node of discrete fractional operator derivative, and combined with the maximum principle, construct a definition of unilateral understanding of space fractional diffusion equation in the non integer node finite difference scheme, and the maximum principle for the stability and convergence analysis.

【学位授予单位】：宁夏大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
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本文编号：1743940