分数阶偏微分方程保高精度谱Galerkin方法以及近场动力学模型快速配置方法研究
发布时间：2018-09-16 19:47
【摘要】：分数阶微积分概念历史悠久,最早源于1695年9月L' Hospital写给莱布尼茨的信件中。在分数阶微积分被提出至今300多年时间中,由于在物理和力学等学科并未获得广泛的关注与应用,而仅仅作为数学领域中的纯理论问题被诸多数学学者研究,这其中包括Euler, Lacroix, Abel, Liouville, Riemann等。随着对复杂物理现象认识程度的加深和计算机模拟能力的提高,力学与工程问题的分数阶导数建模越来越引起人们的重视。尤其对于扩散现象,研究工作者发现越来越多的扩散现象不满足Fick定律,这样的扩散过程称为反常扩散过程。在描述这些复杂系统时,由于反常扩散所具有的历史依赖与全域相关的特性恰好可以由分数阶导数来表示,因此较于整数阶动力学方程,分数阶动力学方程更能有效的描述([18,50,57,5])。无论在理论分析还是数值计算方面,新的分数阶动力学方程都为数学工作者提出了新的挑战。在数值计算方面,现在已经有很多数值求解方法,例如有限差分法([14,42]),有限体积法([91]),有限元法([60,25]),谱方法([35,94])等。这些计算方法已经被广泛的应用于反常扩散的数值模拟中。但是这些方法的误差分析均有很强的正则性假设。通过本文第二章中的反例,我们可以看到即使分数阶方程的扩散系数和右端均充分光滑,我们依然不能保证解的正则性,这是分数阶动力学方程区分于整数阶动力学方程的一个很重要的特性(在整数阶动力学方程情况下,根据方程正则性理论,方程系数和右端的光滑性可以保证方程解的正则性)。通过这一点,我们知道上述数值方法误差分析的假设条件缺少理论支撑,另外在方程解的正则性不强的情况下,即使方程的系数和右端光滑,我们采用高阶差分,高次有限元以及标准谱方法均不能达到很好的误差收敛情况。在计算效率方面,分数阶动力学方程离散得出的系数矩阵通常为满阵,如果我们假设矩阵的阶数或问题的规模为N,则系数矩阵的存储量为O(N2),如果用常用的直接方法求解分数阶动力学方程离散得出的线性系统,则计算复杂度为O(N3)。因此,一直以来,尤其是求解大规模或者多维问题时,分数阶动力学模拟是很费时间的。为解决这一问题,王宏等著名学者通过分析系数矩阵的代数结构,运用快速傅立叶变换,成功的将系数矩阵存储量降为O(N),将求解线性方程组的计算量降为每一步Krylov子空间迭代O(N log N)的计算量([79,80])。固体材料和结构的破坏问题一直是力学研究的经典问题,也是机械、航空航天、土木、水利和化工等领域关注的重点。在近场动力学理论提出之前,随着断裂力学、损伤力学等学科的发展和计算机软件硬件水平的提高,研究者提出了各种不同的力学模型和数值方法来模拟固体材料和渐进破坏的全过程。这些模型均是建立在连续介质假设之上的,他们假设介质所有的内部力均为接触力,最终控制方程绝大多数由偏微分方程所描述。传统的有限元法和有限差分法同样建立在连续介质假设的思想上,在模拟时必须明确知道断裂的位置与尺寸,这在很多现实应用中很难实现,另外随着断裂的发展,传统的有限元法或者有限差分法必须重新划分网格,有很强的网格依赖性([33])。随着不连续有限元方法的发展,在模拟固体材料断裂及发展问题上取得了一些进步,但是在模拟高维复杂断裂系统时仍有很强的局限性。为了克服连续介质力学假定与固体材料不连续这一基本矛盾,2000年,Silling基于非局部作用建模,提出了近场动力学模型,它是用积分思想表述的积分方程([68])。这一模型不在基于连续介质假设和求解微分方程来模拟破坏问题,而是将固体看成由由一些包含所有物质信息的带质量的物质点组成,点与点之间存在着相互作用,随着点与点之间距离的增加,这个作用力在减弱,因此通常人们选取一个点的δ邻域为其作用力的影响域。在该理论框架下,不连续现象自然产生,同时这一理论突破了分子动力学在计算尺度上的局限,在宏、细、微观尺度均可表现出较高的求解精度。在近场动力学提出之后,很多数值方法例如无网格方法([64，63,70])、有限元方法([15]),基于积分的有限差分方法([77])等被提出求解近场动力学模型。在有限元情况下,已经被证明数值解满足最优误差估计。然而这些方法都有一个共同的特点,特别在求解多维问题时,由于离散所得到的系数矩阵为稠密矩阵或者满阵(这取决于影响域δ的大小)。因此,类似分数阶动力学方程,系数矩阵的存储量为O(N2),求解最后线性系统的计算复杂度为O(N3)。另外如果用有限元法求解近场动力学模型,每一个系数矩阵的元素均需要计算2d次重积分,其中d是维数,但由于积分核含有奇性,则计算这一积分是很耗时的。王宏等学者同样根据矩阵的代数结构与快速傅立叶变换,成功的将矩阵存储量降为O(N)，将求解最后线性系统的计算量降为每Krylov子空间迭代O(N log N)的计算量([81，，82])。基于以上考虑,我们分别研究了分数阶扩散方程的保高精度谱Galerkin方法和近场动力学模型的快速配置算法.本文的安排如下：在第一章中,我们给出了在本文剩余部分用到的一些基本概念,包括分数阶导数的Riemann-Liouville导数定义和Caputo定义及其一些基本性质,另外我们给出了一些特殊矩阵的定义及其一些性质。在第二章中我们给出了一种分数阶扩散方程的保高精度谱Galerkin方法,这种方法可以保证在方程系数和右端都充分光滑的条件下,即使真解没有足够的光滑性,我们也可以保证解的高精度。并且数值解比标准谱Galerkin方法得到的数值解精度要好,因为在真解没有足够光滑性的条件下.标准谱Galerkin方法并不能达到高精度。我们同时证明了该方法的误差估计。这一章中给出的算例说明了这一保高精度谱方法的有效性。在第三章中我们提出了求解二维近场动力学模型的快速配置方法。在这一章中,我们仔细分析了由配置法离散得出的系数矩阵,经过分析我们得出系数矩阵与任何向量的乘积可以由三个block-Toeplitz-Toeplitz-block (BT-TB)矩阵与向量的乘积得到,所以系数矩阵与向量的乘法的计算复杂度为O(N log N),如果用Krylov子空间迭代法求解该线性系统,则每一步迭代的计算量可以由O(N2)降为O(NlogN).同时从本文也可以得到该矩阵的计算机存储量可以由O(N2)降为O(N).这一章中给出的算例说明了快速配置方法的有效性。在第四章中针对在近场动力学模型中积分核函数奇性大,运用Krylov子空间迭代求解由于配置法离散得出的线性系统迭代次数比较多的情况,我们提出了两种预条件矩阵。第一种预条件是block-Cireulant-Toeplitz-block (BCTB)型的,第二种预条件是block-Circulant-Circulant-block (BCCB)型的。这两种预条件对降低求解线性系统迭代次数是有效的,并且通过这一章给出的例子我们可以看到第二种预条件因为计算预条件矩阵的逆比较快速,所以计算时间会更快。在第五章中我们运用加罚的思想提出了求解一般凸区域非局部扩散模型的快速配置方法。这种方法是通过加罚将原来凸区域上的问题扩展为包含该凸区域的矩形的问题,经过配置法离散,我们可以看到系数矩阵是一个BTTB矩阵与一个对角矩阵的和,通过这种矩阵结构,我们将系数矩阵的存储由O(N2)降为O(N),将每一步Krylov子空间的计算量由O(N2)降为O(N log N).数值算例说明了这种方法的有效性。
[Abstract]:The concept of fractional calculus has a long history and originated from L'Hospital's letter to Leibniz in September 1695. Since it was put forward more than 300 years ago, it has been studied by many mathematicians only as a pure theoretical problem in the field of mathematics because it has not been widely concerned and applied in physics and mechanics. These include Euler, Lacroix, Abel, Liouville, Riemann and so on. With the deepening of understanding of complex physical phenomena and the improvement of computer simulation ability, the fractional derivative modeling of mechanical and engineering problems has attracted more and more attention. In describing these complex systems, the fractional-order dynamic equation is more effective than the integer-order dynamic equation ([18,50,57,5]) because the historical dependence and global dependence of the anomalous diffusion can be expressed by fractional derivatives. New fractional dynamic equations pose new challenges to mathematicians both in theoretical analysis and numerical calculation. In numerical calculation, there are many numerical methods, such as finite difference method ([14,42]), finite volume method ([91]), finite element method ([60,25]), spectral method ([35,94]) and so on. It is widely used in numerical simulation of anomalous diffusion. But the error analysis of these methods has strong regularity assumptions. Through the counter examples in Chapter 2 of this paper, we can see that even though the diffusion coefficients and the right end of the fractional order equation are smooth enough, we still can not guarantee the regularity of the solution, which is the fractional order dynamics square. A very important property of the equation that distinguishes it from the integer-order dynamic equation (in the case of the integer-order dynamic equation, according to the regularity theory of the equation, the coefficients of the equation and the smoothness of the right-hand end can guarantee the regularity of the solution of the equation). From this point, we know that the assumptions for error analysis of the above numerical methods lack theoretical support, and in addition, in the case of the integer-order dynamic equation, When the regularity of the solution of the equation is not strong, even if the coefficients of the equation and the right end of the equation are smooth, we can not achieve good error convergence by using high-order difference, high-order finite element and standard spectral method. The storage capacity of the coefficient matrix is O(N2) if the order or the scale of the problem is N, and the computational complexity is O(N3) if the linear system is discretized by solving the fractional-order dynamic equation with the usual direct method. Wang Hong and other famous scholars analyzed the algebraic structure of coefficient matrix and used fast Fourier transform to reduce the storage of coefficient matrix to O (N) and the computation of solving linear equations to O (N log N) iteration in every Krylov subspace ([79,80]). Classical problems are also the focus of attention in mechanical, aerospace, civil, hydraulic and chemical fields. Before the theory of near-field dynamics was put forward, with the development of fracture mechanics, damage mechanics and the improvement of computer software and hardware, researchers proposed various mechanical models and numerical methods to simulate solids. These models are based on the continuum assumption that all the internal forces in the medium are contact forces, and the ultimate governing equations are mostly described by partial differential equations. With the development of fracture, the traditional finite element method or finite difference method must be re-meshed, which has strong grid dependence ([33]). With the development of discontinuous finite element method, it is difficult to simulate the fracture and development of solid materials. Some progress has been made, but there are still strong limitations in simulating high-dimensional complex fracture systems. In order to overcome the basic contradiction between continuum mechanics assumption and solid material discontinuity, in 2000, Silling proposed a near field dynamic model based on nonlocal action modeling, which is an integral equation ([68]) expressed by integral theory. It is not based on the continuum hypothesis and the solution of differential equations to simulate the failure problem. Instead, the solid is considered to consist of some mass-bearing points containing all the material information. There is an interaction between points. As the distance between points increases, the interaction force is weakening, so people usually choose a point. In the framework of this theory, discontinuities occur naturally, and this theory breaks through the limitations of molecular dynamics on computational scales. It can show high accuracy in macro, fine and micro scales. Finite element method ([15]), integral-based finite difference method ([77]) and so on are proposed to solve the near-field dynamic model. In the case of finite element method, it has been proved that the numerical solution satisfies the optimal error estimate. Matrix or full matrix (depending on the size of the influence domain delta). Therefore, similar to the fractional-order dynamic equation, the storage of the coefficient matrix is O (N2), and the computational complexity of solving the final linear system is O (N3). In addition, if the finite element method is used to solve the near-field dynamic model, the elements of each coefficient matrix need to compute the second-order multiple integral, where D is According to the algebraic structure of the matrix and the fast Fourier transform, Wang Hong and other scholars succeeded in reducing the storage of the matrix to O (N), and the computation of solving the final linear system to O (N log N) iteration per Krylov subspace ([81,82]). In this paper, we study the high-precision spectral Galerkin method for fractional-order diffusion equations and the fast collocation algorithm for near-field dynamic models. In the first chapter, we give some basic concepts used in the rest of this paper, including the Riemann-Liouville derivative definition and the Caputo definition of fractional-order derivatives. In the second chapter, we give a high-precision spectral Galerkin method for fractional diffusion equations, which guarantees sufficient smoothness of the coefficients and the right end of the equation, even if the true solution does not have enough smoothness. They can also guarantee the high accuracy of the solution, and the numerical solution is more accurate than the standard spectral Galerkin method, because the true solution is not smooth enough. The standard spectral Galerkin method can not achieve high accuracy. We also prove the error estimate of the method. In Chapter 3, we propose a fast collocation method for solving two-dimensional near-field dynamic models. In this chapter, we analyze the coefficient matrices discretized by collocation method carefully. After analysis, we conclude that the product of the coefficient matrix and any vector can be obtained by three block-Toeplitz-Toeplitz-block (BT-BT-BT). The product of TB matrix and vector is obtained, so the computational complexity of the multiplication of coefficient matrix and vector is O (N log N). If the linear system is solved by Krylov subspace iteration method, the computational complexity of each iteration can be reduced from O (N 2) to O (N log N). At the same time, the computer storage of the matrix can be reduced from O (N 2) to O (N). In chapter 4, two preconditioned matrices are proposed to solve the linear system with large number of iterations due to the discretization of the collocation method. The first preconditioned matrix is given. The condition is block-Cireulant-Toeplitz-block (BCTB) type, and the second precondition is block-Circulant-Circulant-block (BCCB) type. These two preconditions are effective in reducing the number of iterations for solving linear systems, and we can see from the example given in this chapter that the second precondition is because the inverse ratio of the preconditioned matrix is calculated. In Chapter 5, we propose a fast collocation method for solving nonlocal diffusion models in convex domains. This method extends the problem on convex domains to a rectangular problem containing the convex domains by adding penalties. After the collocation method is discretized, we can see the system. Number matrix is the sum of a BTTB matrix and a diagonal matrix. By using this matrix structure, we reduce the storage of coefficient matrix from O (N2) to O (N), and the computational complexity of each Krylov subspace from O (N2) to O (N log N). Numerical examples show the effectiveness of this method.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
本文编号：2244599
[Abstract]:The concept of fractional calculus has a long history and originated from L'Hospital's letter to Leibniz in September 1695. Since it was put forward more than 300 years ago, it has been studied by many mathematicians only as a pure theoretical problem in the field of mathematics because it has not been widely concerned and applied in physics and mechanics. These include Euler, Lacroix, Abel, Liouville, Riemann and so on. With the deepening of understanding of complex physical phenomena and the improvement of computer simulation ability, the fractional derivative modeling of mechanical and engineering problems has attracted more and more attention. In describing these complex systems, the fractional-order dynamic equation is more effective than the integer-order dynamic equation ([18,50,57,5]) because the historical dependence and global dependence of the anomalous diffusion can be expressed by fractional derivatives. New fractional dynamic equations pose new challenges to mathematicians both in theoretical analysis and numerical calculation. In numerical calculation, there are many numerical methods, such as finite difference method ([14,42]), finite volume method ([91]), finite element method ([60,25]), spectral method ([35,94]) and so on. It is widely used in numerical simulation of anomalous diffusion. But the error analysis of these methods has strong regularity assumptions. Through the counter examples in Chapter 2 of this paper, we can see that even though the diffusion coefficients and the right end of the fractional order equation are smooth enough, we still can not guarantee the regularity of the solution, which is the fractional order dynamics square. A very important property of the equation that distinguishes it from the integer-order dynamic equation (in the case of the integer-order dynamic equation, according to the regularity theory of the equation, the coefficients of the equation and the smoothness of the right-hand end can guarantee the regularity of the solution of the equation). From this point, we know that the assumptions for error analysis of the above numerical methods lack theoretical support, and in addition, in the case of the integer-order dynamic equation, When the regularity of the solution of the equation is not strong, even if the coefficients of the equation and the right end of the equation are smooth, we can not achieve good error convergence by using high-order difference, high-order finite element and standard spectral method. The storage capacity of the coefficient matrix is O(N2) if the order or the scale of the problem is N, and the computational complexity is O(N3) if the linear system is discretized by solving the fractional-order dynamic equation with the usual direct method. Wang Hong and other famous scholars analyzed the algebraic structure of coefficient matrix and used fast Fourier transform to reduce the storage of coefficient matrix to O (N) and the computation of solving linear equations to O (N log N) iteration in every Krylov subspace ([79,80]). Classical problems are also the focus of attention in mechanical, aerospace, civil, hydraulic and chemical fields. Before the theory of near-field dynamics was put forward, with the development of fracture mechanics, damage mechanics and the improvement of computer software and hardware, researchers proposed various mechanical models and numerical methods to simulate solids. These models are based on the continuum assumption that all the internal forces in the medium are contact forces, and the ultimate governing equations are mostly described by partial differential equations. With the development of fracture, the traditional finite element method or finite difference method must be re-meshed, which has strong grid dependence ([33]). With the development of discontinuous finite element method, it is difficult to simulate the fracture and development of solid materials. Some progress has been made, but there are still strong limitations in simulating high-dimensional complex fracture systems. In order to overcome the basic contradiction between continuum mechanics assumption and solid material discontinuity, in 2000, Silling proposed a near field dynamic model based on nonlocal action modeling, which is an integral equation ([68]) expressed by integral theory. It is not based on the continuum hypothesis and the solution of differential equations to simulate the failure problem. Instead, the solid is considered to consist of some mass-bearing points containing all the material information. There is an interaction between points. As the distance between points increases, the interaction force is weakening, so people usually choose a point. In the framework of this theory, discontinuities occur naturally, and this theory breaks through the limitations of molecular dynamics on computational scales. It can show high accuracy in macro, fine and micro scales. Finite element method ([15]), integral-based finite difference method ([77]) and so on are proposed to solve the near-field dynamic model. In the case of finite element method, it has been proved that the numerical solution satisfies the optimal error estimate. Matrix or full matrix (depending on the size of the influence domain delta). Therefore, similar to the fractional-order dynamic equation, the storage of the coefficient matrix is O (N2), and the computational complexity of solving the final linear system is O (N3). In addition, if the finite element method is used to solve the near-field dynamic model, the elements of each coefficient matrix need to compute the second-order multiple integral, where D is According to the algebraic structure of the matrix and the fast Fourier transform, Wang Hong and other scholars succeeded in reducing the storage of the matrix to O (N), and the computation of solving the final linear system to O (N log N) iteration per Krylov subspace ([81,82]). In this paper, we study the high-precision spectral Galerkin method for fractional-order diffusion equations and the fast collocation algorithm for near-field dynamic models. In the first chapter, we give some basic concepts used in the rest of this paper, including the Riemann-Liouville derivative definition and the Caputo definition of fractional-order derivatives. In the second chapter, we give a high-precision spectral Galerkin method for fractional diffusion equations, which guarantees sufficient smoothness of the coefficients and the right end of the equation, even if the true solution does not have enough smoothness. They can also guarantee the high accuracy of the solution, and the numerical solution is more accurate than the standard spectral Galerkin method, because the true solution is not smooth enough. The standard spectral Galerkin method can not achieve high accuracy. We also prove the error estimate of the method. In Chapter 3, we propose a fast collocation method for solving two-dimensional near-field dynamic models. In this chapter, we analyze the coefficient matrices discretized by collocation method carefully. After analysis, we conclude that the product of the coefficient matrix and any vector can be obtained by three block-Toeplitz-Toeplitz-block (BT-BT-BT). The product of TB matrix and vector is obtained, so the computational complexity of the multiplication of coefficient matrix and vector is O (N log N). If the linear system is solved by Krylov subspace iteration method, the computational complexity of each iteration can be reduced from O (N 2) to O (N log N). At the same time, the computer storage of the matrix can be reduced from O (N 2) to O (N). In chapter 4, two preconditioned matrices are proposed to solve the linear system with large number of iterations due to the discretization of the collocation method. The first preconditioned matrix is given. The condition is block-Cireulant-Toeplitz-block (BCTB) type, and the second precondition is block-Circulant-Circulant-block (BCCB) type. These two preconditions are effective in reducing the number of iterations for solving linear systems, and we can see from the example given in this chapter that the second precondition is because the inverse ratio of the preconditioned matrix is calculated. In Chapter 5, we propose a fast collocation method for solving nonlocal diffusion models in convex domains. This method extends the problem on convex domains to a rectangular problem containing the convex domains by adding penalties. After the collocation method is discretized, we can see the system. Number matrix is the sum of a BTTB matrix and a diagonal matrix. By using this matrix structure, we reduce the storage of coefficient matrix from O (N2) to O (N), and the computational complexity of each Krylov subspace from O (N2) to O (N log N). Numerical examples show the effectiveness of this method.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
【参考文献】
相关期刊论文 前1条
1 HUANG JianFei;NIE NingMing;TANG YiFa;;A second order finite difference-spectral method for space fractional diffusion equations[J];Science China(Mathematics);2014年06期
本文编号：2244599
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