保几何结构算法在等离子体物理中的应用
发布时间：2019-05-22 14:32
【摘要】：哈密顿系统在物理理论中非常常见,其具有的长期保辛结构特性使得其具有很多守恒性、可以长时间稳定地演化并且不发散。这些守恒特性有助于我们讨论和理解物理系统的长期性质,并且更加有效地再现物理系统的本质。等离子体的四种常见的基本模型(单粒子、无碰撞动理论、理想双流体与理想磁流体)都是哈密顿系统。对于这些基本模型建立有效的算法以研究复杂的等离子体行为就显得尤为重要。然而传统基于直接对微分方程进行离散的算法一般会破坏这些哈密顿系统的保守特性,这使得这些算法在模拟长期多时间尺度的物理问题时经常会发散而得不到有用的结果,在20世纪80年代由我国著名数学家冯康及其学派提出的保辛结构算法正是为了解决这一问题。不过这一方法在等离子体数值模拟领域尚未得到广泛应用,这主要是因为等离子体模型多为无穷维非正则哈密顿系统,其保结构算法的构造相对困难。本文从保辛结构算法的理论出发,简要介绍了辛算法的特点及构造方法,归纳并总结了最新针对单粒子系统的保结构算法、并提出了针对Vlasov-Maxwell系统、理想双流体系统与磁流体系统的保辛结构算法。我们还选取了一些基本的物理算例来验证这些算法的正确性与长期保守性。对于带电粒子在已知外电磁场中运动的单粒子模型,由于其一般是一个有限维的正则哈密顿系统,所以现成针对此类系统辛算法的理论很丰富。我们首先利用成熟的变分辛方法构造了带电相对论性与非相对论性粒子的保辛结构算法,然后又利用最近新发现的一种哈密顿分裂法构造了针对这两种单粒子系统的非正则辛算法,最后选取了一个典型的Tokamak中带电粒子运动场景作为算例验证了这些算法的长期守恒性。Vlasov-Maxwell系统是用连续分布函数去描述的等离子体系统,其非常接近原始等离子体的带电粒子-电磁场系统,因此应用也非常广泛。然而由于它是一个无穷维的非正则哈密顿系统,一般而言其保辛结构算法难以实现。不过由于直接模拟离散的Vlasov-Maxwell系统其计算量太大,故一般人们使用大量粒子采样点的Particle-in-Cell方法去模拟Vlasov-Maxwell系统。我们先从粒子-电磁场的拉式量出发并离散与变分,得到了第一种实用的变分辛Particle-in-Cell方法。随后为了构造电磁规范不变(即电荷守恒)与高阶显式的Particle-in-Cell方法,我们创造了方网格多格Whitney插值形式,在此基础上利用离散外微分与哈密顿分裂法等先进的数学工具,实现了显式高阶电荷守恒非正则辛Particle-in-Cell格式。最后实现了相对论情况的变分与电荷守恒辛Particle-in-Cell格式。同样,我们也取了 X-Bernstein波色散关系与Landau阻尼这两个例子来验证这些算法的正确性与长期守恒性。双流体系统是一种将带电粒子视作带电流体的等离子体模型,在无耗散时是哈密顿系统。然而同Vlasov-Maxwell系统类似,双流体系统也是一个无穷维的非正则哈密顿系统。我们使用类似Vlasov-Maxwell系统构造辛算法的思路,从双流体系统的变分理论出发,用方网格多格Wlhitney插值形式、离散外微分以及哈密顿分裂法等方法构造了显式高阶电荷守恒非正则辛双流体格式。我们还用此方法计算与验证了双流体系统各种模式的色散关系以及双流不稳定性。理想磁流体系统是一种等离子简化模型,通过近似将高频的电子演化忽略,这样使得磁流体模型更加适用于低频问题。该模型是一个较双流体系统更复杂的非正则哈密顿系统。这是因为其演化除了保辛结构以外,还具有保磁场结构的性质(即磁冻结效应)。我们从欧拉网格具有约束的磁流体变分原理出发,离散得到辛磁流体算法,并用此验证了磁流体波的色散关系以及算法的长期守恒性质。本文中阐述的等离子体保结构算法实际上是对等离子体哈密顿模型的保辛结构近似。实际上根据辛算法的理论可知这些离散化的系统也是哈密顿系统,因而理论上也具有哈密顿系统的长期保守等性质,这是传统算法所难以企及的。这些具有优良性质的算法有助于我们更准确地模拟和预测等离子体的行为,了解等离子中的复杂物理图像。
[Abstract]:The Hamiltonian system is very common in the physical theory, and the long-term keeping-in structure of the Hamiltonian system makes the Hamiltonian system have a lot of conservation, can be evolved stably for a long time and does not diverge. These conservation features help us to discuss and understand the long-term nature of the physical system and to more effectively reproduce the nature of the physical system. The four common basic models of plasma (single particle, non-collision theory, ideal dual fluid and ideal magnetic fluid) are Hamiltonian systems. It is particularly important to establish an efficient algorithm for these basic models to study complex plasma behavior. However, the traditional discrete algorithm based on the direct-to-differential equation can generally destroy the conservative characteristics of these Hamiltonian systems, which makes these algorithms often diverge without useful results when simulating the physical problems of the long-term multi-time scale, In the 80 's of the 20th century, by the famous mathematician of our country, Feng Kang and his school put forward the structure of the symplectic structure, which is to solve this problem. However, this method is not widely used in the field of plasma numerical simulation, mainly because the plasma model is an infinite-dimensional non-regular Hamiltonian system, and the structure of the structure-preserving algorithm is relatively difficult. In this paper, the characteristics and construction methods of the symplectic algorithm are briefly introduced from the theory of the structure of the symplectic structure, and the new algorithm for preserving the structure of the single particle system is summarized and summarized, and the structure of the symplectic structure for the Vlasov-Maxwell system, the ideal dual-fluid system and the magnetic fluid system is put forward. We have also selected some basic physical examples to verify the correctness and long-term conservation of these algorithms. As for the single-particle model of the moving of the charged particles in the known external electromagnetic field, since it is generally a regular Hamiltonian system with a finite dimension, the theory of the symplectic algorithm for such systems is very rich. In this paper, we first construct the symplectic structure of the charged relativistic and non-relativistic particles by means of the mature transformation method, and then the non-regular symplectic algorithm for these two single particle systems is constructed with the newly discovered Hamiltonian splitting method. Finally, a typical motion scene of charged particles in Tokamak is selected as an example to verify the long-term conservation of these algorithms. The Vlasov-Maxwell system is a plasma system to be described by a continuous distribution function, which is very close to the charged particle-electromagnetic field system of the original plasma, so it is also very widely used. However, because it is a non-regular Hamiltonian system with infinite dimension, it is generally difficult to realize its structure-keeping structure. However, the Vlasov-Maxwell system is used to simulate the Vlasov-Maxwell system. The first practical variant-in-in-Cell method is obtained from the formula of the particle-electromagnetic field and the dispersion and the variation. in ord to construct that method of the invariant (i. e. the conservation of charge) and the high-order explicit-in-cell method for the construction of the electromagnetic specification, we have created the multi-lattice Whitney interpolation form of the square grid, on the basis of which the advanced mathematical tools such as the discrete external differential and the Hamilton-splitting method are used, The explicit high-order charge conservation non-regular symplectic-in-Cell format is realized. Finally, the variational and charge-conserved symplectic-in-Cell format of the theory of relativity is realized. In the same way, we also take the two examples of the X-Bernstein wave dispersion relation and Landau damping to verify the correctness and long-term conservation of these algorithms. A dual-fluid system is a plasma model that treats charged particles as charged fluids, and is a Hamiltonian system in the absence of dissipation. However, similar to the Vlasov-Maxwell system, the dual-fluid system is also an infinite-dimensional non-regular Hamiltonian system. In this paper, we use the similar Vlasov-Maxwell system to construct the symplectic algorithm, and from the variational theory of the two-fluid system, the explicit high-order charge conservation non-regular two-fluid format is constructed by means of the square-grid multi-lattice Wlnterney interpolation, the discrete external differential and the Hamilton-splitting method. We also use this method to calculate and verify the dispersion relationship and the dual-flow instability of the various modes of the dual-fluid system. The ideal magnetic fluid system is a kind of plasma simplified model, which can ignore the electronic evolution of high frequency, so that the magnetic fluid model is more suitable for low-frequency problems. The model is a more complex non-regular Hamiltonian system of a two-fluid system. This is because it has the properties of the magnetic field structure (i.e., the magnetic freezing effect) in addition to the conformal structure. On the basis of the variational principle of the magnetic fluid with the constraint of the Euler grid, the symplectic magnetic fluid algorithm is obtained, and the dispersion relation of the magnetic fluid wave and the long-term conservation property of the algorithm are verified by this method. The plasma-preserving structure algorithm, which is described in this paper, is in fact approximate to the Basim structure of the plasma Hamiltonian model. In fact, according to the theory of symplectic algorithm, it is known that these discretized systems are also the Hamiltonian system, so the theory also has the long-term conservative property of the Hamiltonian system, which is difficult for the traditional algorithm. These algorithms with good properties help us to more accurately simulate and predict the behavior of the plasma to understand the complex physical image in the plasma.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O53
本文编号：2483009
[Abstract]:The Hamiltonian system is very common in the physical theory, and the long-term keeping-in structure of the Hamiltonian system makes the Hamiltonian system have a lot of conservation, can be evolved stably for a long time and does not diverge. These conservation features help us to discuss and understand the long-term nature of the physical system and to more effectively reproduce the nature of the physical system. The four common basic models of plasma (single particle, non-collision theory, ideal dual fluid and ideal magnetic fluid) are Hamiltonian systems. It is particularly important to establish an efficient algorithm for these basic models to study complex plasma behavior. However, the traditional discrete algorithm based on the direct-to-differential equation can generally destroy the conservative characteristics of these Hamiltonian systems, which makes these algorithms often diverge without useful results when simulating the physical problems of the long-term multi-time scale, In the 80 's of the 20th century, by the famous mathematician of our country, Feng Kang and his school put forward the structure of the symplectic structure, which is to solve this problem. However, this method is not widely used in the field of plasma numerical simulation, mainly because the plasma model is an infinite-dimensional non-regular Hamiltonian system, and the structure of the structure-preserving algorithm is relatively difficult. In this paper, the characteristics and construction methods of the symplectic algorithm are briefly introduced from the theory of the structure of the symplectic structure, and the new algorithm for preserving the structure of the single particle system is summarized and summarized, and the structure of the symplectic structure for the Vlasov-Maxwell system, the ideal dual-fluid system and the magnetic fluid system is put forward. We have also selected some basic physical examples to verify the correctness and long-term conservation of these algorithms. As for the single-particle model of the moving of the charged particles in the known external electromagnetic field, since it is generally a regular Hamiltonian system with a finite dimension, the theory of the symplectic algorithm for such systems is very rich. In this paper, we first construct the symplectic structure of the charged relativistic and non-relativistic particles by means of the mature transformation method, and then the non-regular symplectic algorithm for these two single particle systems is constructed with the newly discovered Hamiltonian splitting method. Finally, a typical motion scene of charged particles in Tokamak is selected as an example to verify the long-term conservation of these algorithms. The Vlasov-Maxwell system is a plasma system to be described by a continuous distribution function, which is very close to the charged particle-electromagnetic field system of the original plasma, so it is also very widely used. However, because it is a non-regular Hamiltonian system with infinite dimension, it is generally difficult to realize its structure-keeping structure. However, the Vlasov-Maxwell system is used to simulate the Vlasov-Maxwell system. The first practical variant-in-in-Cell method is obtained from the formula of the particle-electromagnetic field and the dispersion and the variation. in ord to construct that method of the invariant (i. e. the conservation of charge) and the high-order explicit-in-cell method for the construction of the electromagnetic specification, we have created the multi-lattice Whitney interpolation form of the square grid, on the basis of which the advanced mathematical tools such as the discrete external differential and the Hamilton-splitting method are used, The explicit high-order charge conservation non-regular symplectic-in-Cell format is realized. Finally, the variational and charge-conserved symplectic-in-Cell format of the theory of relativity is realized. In the same way, we also take the two examples of the X-Bernstein wave dispersion relation and Landau damping to verify the correctness and long-term conservation of these algorithms. A dual-fluid system is a plasma model that treats charged particles as charged fluids, and is a Hamiltonian system in the absence of dissipation. However, similar to the Vlasov-Maxwell system, the dual-fluid system is also an infinite-dimensional non-regular Hamiltonian system. In this paper, we use the similar Vlasov-Maxwell system to construct the symplectic algorithm, and from the variational theory of the two-fluid system, the explicit high-order charge conservation non-regular two-fluid format is constructed by means of the square-grid multi-lattice Wlnterney interpolation, the discrete external differential and the Hamilton-splitting method. We also use this method to calculate and verify the dispersion relationship and the dual-flow instability of the various modes of the dual-fluid system. The ideal magnetic fluid system is a kind of plasma simplified model, which can ignore the electronic evolution of high frequency, so that the magnetic fluid model is more suitable for low-frequency problems. The model is a more complex non-regular Hamiltonian system of a two-fluid system. This is because it has the properties of the magnetic field structure (i.e., the magnetic freezing effect) in addition to the conformal structure. On the basis of the variational principle of the magnetic fluid with the constraint of the Euler grid, the symplectic magnetic fluid algorithm is obtained, and the dispersion relation of the magnetic fluid wave and the long-term conservation property of the algorithm are verified by this method. The plasma-preserving structure algorithm, which is described in this paper, is in fact approximate to the Basim structure of the plasma Hamiltonian model. In fact, according to the theory of symplectic algorithm, it is known that these discretized systems are also the Hamiltonian system, so the theory also has the long-term conservative property of the Hamiltonian system, which is difficult for the traditional algorithm. These algorithms with good properties help us to more accurately simulate and predict the behavior of the plasma to understand the complex physical image in the plasma.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O53
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