抛物型方程和空间分数阶扩散方程的两个高阶方法研究 Alternative Title On High Order Methods for Solving Parabolic Equation and Space Fractional Diffusion Equations 周晗 Thesis Advisor 伍渝江 2013-05-19 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword Richardson外推算法 紧ADI格式 抛物方程 Riemann-Liouville分数阶导数 拟紧致差分格式 稳定性和收敛性 空间分数阶方程 Abstract 本文主要探索高阶有限差分方法求解二维抛物型方程，一维和二维空间分数阶扩散方程。在第一章，我们构造了抛物方程的紧致的ADI格式，得到了关于空间方向四阶收敛和时间方向二阶收敛。进一步地，我们设计关于时间和空间的Richardson外推算法，将空间和时间方向的收敛阶提高为六阶。并给出了数值格式的最大模范数的误差估计。在第二章，我们基于WSGD算子，建立了CWSGD算子逼近Riemann-Liouville分数阶导数，并且将其运用于构造差分格式数值求解一维和二维空间分数阶扩散方程，得到关于空间方向三阶收敛。我们给出了格式的无条件稳定性和离散的L^2范数下的收敛性证明。 Other Abstract In this paper we mainly consider the high order finite difference method for solving parabolic equation of two-dimensional and space fractional diffusion equations of one-dimensional and two-dimensional. In the first chapter we construct the compact ADI scheme of parabolic equation. And we achieve the fourth order accuracy and second order accuracy with respect to space and time dimensions, respectively. Furthermore, we design the Richardson extrapolation approach to improve the accuracy order of both time and space to six order. In the second chapter, based on the WSGD operators, we build the CWSGD operators to approximate the Riemann-Liouville fractional derivatives. And we apply them to construct the finite difference schemes for solving one-dimensional and two-dimensional space fractional diffusion equations. The third order accuracy of space direction is obtained. we give the unconditional stability and the convergence with respect to the discrete L^2 norm of the schemes. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224828 Collection 数学与统计学院 Recommended CitationGB/T 7714 周晗. 抛物型方程和空间分数阶扩散方程的两个高阶方法研究[D]. 兰州. 兰州大学,2013.
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