保幾何結(jié)構(gòu)算法在等離子體物理中的應(yīng)用
發(fā)布時(shí)間：2019-05-22 14:32
【摘要】：哈密頓系統(tǒng)在物理理論中非常常見,其具有的長期保辛結(jié)構(gòu)特性使得其具有很多守恒性、可以長時(shí)間穩(wěn)定地演化并且不發(fā)散。這些守恒特性有助于我們討論和理解物理系統(tǒng)的長期性質(zhì),并且更加有效地再現(xiàn)物理系統(tǒng)的本質(zhì)。等離子體的四種常見的基本模型(單粒子、無碰撞動理論、理想雙流體與理想磁流體)都是哈密頓系統(tǒng)。對于這些基本模型建立有效的算法以研究復(fù)雜的等離子體行為就顯得尤為重要。然而傳統(tǒng)基于直接對微分方程進(jìn)行離散的算法一般會破壞這些哈密頓系統(tǒng)的保守特性,這使得這些算法在模擬長期多時(shí)間尺度的物理問題時(shí)經(jīng)常會發(fā)散而得不到有用的結(jié)果,在20世紀(jì)80年代由我國著名數(shù)學(xué)家馮康及其學(xué)派提出的保辛結(jié)構(gòu)算法正是為了解決這一問題。不過這一方法在等離子體數(shù)值模擬領(lǐng)域尚未得到廣泛應(yīng)用,這主要是因?yàn)榈入x子體模型多為無窮維非正則哈密頓系統(tǒng),其保結(jié)構(gòu)算法的構(gòu)造相對困難。本文從保辛結(jié)構(gòu)算法的理論出發(fā),簡要介紹了辛算法的特點(diǎn)及構(gòu)造方法,歸納并總結(jié)了最新針對單粒子系統(tǒng)的保結(jié)構(gòu)算法、并提出了針對Vlasov-Maxwell系統(tǒng)、理想雙流體系統(tǒng)與磁流體系統(tǒng)的保辛結(jié)構(gòu)算法。我們還選取了一些基本的物理算例來驗(yàn)證這些算法的正確性與長期保守性。對于帶電粒子在已知外電磁場中運(yùn)動的單粒子模型,由于其一般是一個有限維的正則哈密頓系統(tǒng),所以現(xiàn)成針對此類系統(tǒng)辛算法的理論很豐富。我們首先利用成熟的變分辛方法構(gòu)造了帶電相對論性與非相對論性粒子的保辛結(jié)構(gòu)算法,然后又利用最近新發(fā)現(xiàn)的一種哈密頓分裂法構(gòu)造了針對這兩種單粒子系統(tǒng)的非正則辛算法,最后選取了一個典型的Tokamak中帶電粒子運(yùn)動場景作為算例驗(yàn)證了這些算法的長期守恒性。Vlasov-Maxwell系統(tǒng)是用連續(xù)分布函數(shù)去描述的等離子體系統(tǒng),其非常接近原始等離子體的帶電粒子-電磁場系統(tǒng),因此應(yīng)用也非常廣泛。然而由于它是一個無窮維的非正則哈密頓系統(tǒng),一般而言其保辛結(jié)構(gòu)算法難以實(shí)現(xiàn)。不過由于直接模擬離散的Vlasov-Maxwell系統(tǒng)其計(jì)算量太大,故一般人們使用大量粒子采樣點(diǎn)的Particle-in-Cell方法去模擬Vlasov-Maxwell系統(tǒng)。我們先從粒子-電磁場的拉式量出發(fā)并離散與變分,得到了第一種實(shí)用的變分辛Particle-in-Cell方法。隨后為了構(gòu)造電磁規(guī)范不變(即電荷守恒)與高階顯式的Particle-in-Cell方法,我們創(chuàng)造了方網(wǎng)格多格Whitney插值形式,在此基礎(chǔ)上利用離散外微分與哈密頓分裂法等先進(jìn)的數(shù)學(xué)工具,實(shí)現(xiàn)了顯式高階電荷守恒非正則辛Particle-in-Cell格式。最后實(shí)現(xiàn)了相對論情況的變分與電荷守恒辛Particle-in-Cell格式。同樣,我們也取了 X-Bernstein波色散關(guān)系與Landau阻尼這兩個例子來驗(yàn)證這些算法的正確性與長期守恒性。雙流體系統(tǒng)是一種將帶電粒子視作帶電流體的等離子體模型,在無耗散時(shí)是哈密頓系統(tǒng)。然而同Vlasov-Maxwell系統(tǒng)類似,雙流體系統(tǒng)也是一個無窮維的非正則哈密頓系統(tǒng)。我們使用類似Vlasov-Maxwell系統(tǒng)構(gòu)造辛算法的思路,從雙流體系統(tǒng)的變分理論出發(fā),用方網(wǎng)格多格Wlhitney插值形式、離散外微分以及哈密頓分裂法等方法構(gòu)造了顯式高階電荷守恒非正則辛雙流體格式。我們還用此方法計(jì)算與驗(yàn)證了雙流體系統(tǒng)各種模式的色散關(guān)系以及雙流不穩(wěn)定性。理想磁流體系統(tǒng)是一種等離子簡化模型,通過近似將高頻的電子演化忽略,這樣使得磁流體模型更加適用于低頻問題。該模型是一個較雙流體系統(tǒng)更復(fù)雜的非正則哈密頓系統(tǒng)。這是因?yàn)槠溲莼吮Ｐ两Y(jié)構(gòu)以外,還具有保磁場結(jié)構(gòu)的性質(zhì)(即磁凍結(jié)效應(yīng))。我們從歐拉網(wǎng)格具有約束的磁流體變分原理出發(fā),離散得到辛磁流體算法,并用此驗(yàn)證了磁流體波的色散關(guān)系以及算法的長期守恒性質(zhì)。本文中闡述的等離子體保結(jié)構(gòu)算法實(shí)際上是對等離子體哈密頓模型的保辛結(jié)構(gòu)近似。實(shí)際上根據(jù)辛算法的理論可知這些離散化的系統(tǒng)也是哈密頓系統(tǒng),因而理論上也具有哈密頓系統(tǒng)的長期保守等性質(zhì),這是傳統(tǒng)算法所難以企及的。這些具有優(yōu)良性質(zhì)的算法有助于我們更準(zhǔn)確地模擬和預(yù)測等離子體的行為,了解等離子中的復(fù)雜物理圖像。
[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.
【學(xué)位授予單位】：中國科學(xué)技術(shù)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O53
本文編號：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.
【學(xué)位授予單位】：中國科學(xué)技術(shù)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O53
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