二维分数阶超扩散方程和非局部方程的数值算法
发布时间：2018-08-28 15:14
【摘要】：扩散过程在自然界随处可见.许多学者指出在自然界中存在着反常扩散,即在微观尺度下,粒子的随机过程不是布朗运动,它相应的均方方差可能比在高斯过程中的增长得快(超扩散)或慢(亚扩散).本文主要研究二维分数阶超扩散方程的高效稳定的数值算法和一类带周期边界条件的时空非局部方程的渐近相容数值算法.第一章简要介绍了反常扩散的物理背景并回顾了分数阶扩散方程的数值算法.第二章研究了二维带阻尼项的时间分数阶超扩散方程的ADI Galerkin有限元离散.我们首先提出了两种有效的ADI Galerkin有限元格式求解带阻尼项的时间分数阶超扩散方程,其中时间分数阶导数为阶α的Caputo导数,α∈(1,2).这两种离散格式在时间方向上分别基于修正L1方法和L2-1σ方法,在空间方向上基于Galerkin有限元方法.离散格式的无条件稳定和收敛性都得到严格的证明.数值实验表明这两种全离散格式在时间方向分别具有(3-α)阶和2阶精度.第三章研究了二维Riesz空间分数阶扩散波方程的ADI Galerkin有限元离散.这个分数阶模型可理解为经典的二维波方程的推广.我们通过结合在时间方向上的Crank-Nicolson方法和在空间方向上的有限元方法,发展了一种有效的ADI Galerkin有限元格式求解这个分数阶模型.随后我们在新构造的范数下给出了数值格式的稳定性和收敛性分析.数值结果表明这种全离散格式具有2阶时间精度.第四章研究了二维时空分数阶扩散波方程的ADI Galerkin有限元离散.同样,这个分数阶模型也可理解为经典的二维波方程的推广.类似第二章对时间分数阶导数的离散,我们在时间上利用修正的L1方法,在空间上利用有限元方法,并通过添加适当的小量项得到有效的ADI Galerkin有限元格式.结合第三章关于分数阶Sobolev空间的讨论,我们对提出的数值格式给出了严格的稳定性和收敛性分析.数值结果表明这种全离散格式在时间方向具有(3-β)阶精度,其中β表示时间Caputo导数的阶数且β∈(1,2).第五章研究了一类带周期边界条件的时空非局部方程的渐近相容谱格式.我们首先通过构造合适的函数空间得到了时空非局部方程的适定性,并研究了范围参数δ和σ都趋于0时,非局部方程的局部极限.随后我们提出一种Fourier谱方法求解时空非局部方程,并证明提出的数值格式是稳定的和渐近相容的.最后数值结果验证了理论分析.第六章简要给出了对本论文研究内容的总结和展望.
[Abstract]:Diffusion can be seen everywhere in nature. Many scholars have pointed out that there is anomalous diffusion in nature, that is, at the microscopic scale, the stochastic process of particles is not a Brownian motion, and its mean square variance may grow faster (superdiffusion) or slower (subdiffusion) than that in Gao Si process. In this paper, the efficient and stable numerical algorithms for two-dimensional fractional order superdiffusion equations and the asymptotically consistent numerical algorithms for a class of spatio-temporal nonlocal equations with periodic boundary conditions are studied. In chapter 1, the physical background of anomalous diffusion is briefly introduced and the numerical algorithms for fractional diffusion equations are reviewed. In chapter 2, the ADI Galerkin finite element discretization of time fractional superdiffusion equations with damping term is studied. We first propose two effective ADI Galerkin finite element schemes for solving time fractional superdiffusion equations with damping terms, where the time fractional derivative is the Caputo derivative of order 伪, 伪 鈭，
本文编号：2209769
[Abstract]:Diffusion can be seen everywhere in nature. Many scholars have pointed out that there is anomalous diffusion in nature, that is, at the microscopic scale, the stochastic process of particles is not a Brownian motion, and its mean square variance may grow faster (superdiffusion) or slower (subdiffusion) than that in Gao Si process. In this paper, the efficient and stable numerical algorithms for two-dimensional fractional order superdiffusion equations and the asymptotically consistent numerical algorithms for a class of spatio-temporal nonlocal equations with periodic boundary conditions are studied. In chapter 1, the physical background of anomalous diffusion is briefly introduced and the numerical algorithms for fractional diffusion equations are reviewed. In chapter 2, the ADI Galerkin finite element discretization of time fractional superdiffusion equations with damping term is studied. We first propose two effective ADI Galerkin finite element schemes for solving time fractional superdiffusion equations with damping terms, where the time fractional derivative is the Caputo derivative of order 伪, 伪 鈭，
本文编号：2209769
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