多辛变分积分子及非标准有限差分方法
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摘要
具有多辛几何结构的系统在自然界是广泛存在的，尤其是在力学、电磁学等物理范畴的系统中。多辛几何结构是多辛系统的内在结构，而能够保持原系统多辛结构的数值方法往往具有良好的数值表现，能够较好的保持定量定性上的数值特性。比如多辛数值方法具有长时间的稳定性及能较好的保持系统中的各种不变量。目前主要有两类构造多辛数值格式的方法，一类是对多辛Hamilton方程进行直接离散，再试图推导它的离散多辛结构，这种方法被称为多辛Hamilton数值方法；另一类就是基于Lagrange的角度使用变分原理，导出离散Euler–Lagrange方程，同时由变分原理得到其对应的离散多辛结构，这种方法被称为多辛离散变分积分子。后者的优势在于它的构造基于变分原理，而根据离散的变分原理一定可以导出它的离散多辛结构。所以离散变分积分子一定是保持多辛结构的。本文将基于Lagrange的角度使用变分原理研究离散变分积分子及其多辛几何结构，并结合非标准有限差分方法的思想和优势构造非标准有限差分变分积分子。本文还将非标准有限差分方法与复数时间步长复合方法结合起来研究两个生物模型及模型本身的守恒率。
本文首先简要概述了辛和多辛系统的研究历程，介绍了Hamilton系统和Lagrange系统的关系及变分原理的基本思想，并回顾了多辛Hamilton数值方法和离散变分积分子的研究成果。
其次，本文基于Lagrange的角度利用变分原理研究变分积分子及其保持的多辛结构。本文在前人研究Lagrange多辛几何的基础上考虑边界值空间，利用边界Lagrange函数和变分原理，建立了新的推导离散多辛结构——离散多辛形式公式的方法。鉴于Lagrange系统和Hamilton系统在非退化条件下的等价关系，本文也探求了Lagrange变分积分子和多辛Hamilton数值方法之间的关系。针对波动方程，在选择适当的离散下，本文建立了离散变分积分子和经典的多辛Hamilton数值方法之间的等价关系，构造了分别与Euler box数值格式和Preissman box数值格式等价的两个离散变分积分子。
再次，本文结合非标准有限差分方法的思想和前面建立的离散变分积分子的理论，分别构造了线性波动方程和Klein–Gordon方程的非标准有限差分变分积分子。讨论了所构造方法的收敛性，并推导了所构造方法保持的多辛离散结构——离散多辛形式公式。在数值试验中，本文验证了方法的收敛阶，展现了所构造的方法可以很好反映原系统的数值特性。数值实验验证了非标准有限差分变分积分子数值求解多辛系统的可行性和有效性。
接着，本文研究了非线性变系数Schro¨dinger方程。本文分别在三角离散和方形离散下构造了此方程的非标准有限差分变分积分子，讨论了变分积分子的收敛精度，并推导了其保持的多辛离散结构——离散多辛形式公式。数值试验验证了方法的收敛阶，考察了方法的数值稳定性。同时展示了所构造方法可以很好的保持非线性Schro¨dinger方程的模方守恒率，并分别将本文所构造的方法与标准有限差分方法和经典的Crank–Nicolson方法进行比较，展示了本文所构造方法的优势所在。
最后，本文将结合非标准有限差分方法和具有复数时间步长的复合方法去研究浮游生物模型和百日咳传染病模型。针对这两个模型构造的非标准有限差分方法可以很好的保持原系统中的正性和总量守恒率。然后使用具有复数时间步长的复合方法改进了数值解的收敛精度并且继续保持了正性和总量守恒率。这也是首次将复数时间步长的复合方法应用于生物模型的数值求解。复合方法因其简单却有效的格式使得它比Richardson外推方法具有更好的计算效率。数值实验证明了本文对生物模型构造的方法是可行有效的。
The symplectic and multisymplectic systems widely exist in natural world, especial-ly in mechanics and electromagnetics systems etc. Multisymplectic geometric structure isa intrinsic structure of the multisymplectic system. The numerical methods which can p-reserve the multisymplectic structure of original system always perform well and they canpreserve quantitative and qualitative numerical characteristics. For example, multisym-plectic methods have long-time stability and preserve the invariants of original systemswell. There are two main ways to construct multisymplectic numerical schemes. One isto discrete the multisymplectic Hamilton’s equation directly, and try to derive its corre-sponding discrete multisymplectic structure. These methods are called multisymplecticHamiltonian method. The other way is based on variational principle of Lagrangian side.The discrete Euler–Lagrange equation is derived from discrete variational principle, andmeanwhile the corresponding multisymplectic structure is also produced from the dis-crete variational principle. These methods are called multisymplectic discrete variationalintegrators. A advantage of the latter one is that it’s based on the intrinsic variational prin-ciple, and it’s naturally multisymplectic. This dissertation will use the discrete variationalprinciple in Lagrangian field to study the discrete variational integrators and their corre-sponding discrete multisymplectic structures. Furthermore, the idea of nonstandard finitediference methods is applied to construct the nonstandard finite diference variational in-tegrators. This dissertation also combine the nonstandard finite diference methods andcomposition methods with complex time steps to study two population models and theirconservation laws.
Firstly, the research history of symplectic and multisymplectic systems is introduced.The Hamilton’s systems, Lagrange’s systems and the basic idea of variational principleare stated. Some research results of the multisymplectic numerical methods are presented.
Secondly, based on Lagrangian side, variational principle is used to study the varia-tional integrators and corresponding multisymplectic structures. Considering the bound-ary value space, this paper gives a new way to derive the discrete multisymplectic formformula which is preserved by discrete variational integrators. In view of the equivalenceof Lagrangian systems and Hamiltonian systems under nondegenerate condition, this dis-sertation also try to establish equivalency between Lagrangian variational integrators and multisymplectic Hamiltonian numerical methods. For suitable choices of discretizationwhen applied to the wave equation, this paper shows two variational integrators which areequivalent to Euler box method and Preissman box methods respectively.
Thirdly, this paper combines the idea of nonstandard finite diference methods andthe discrete variational integrators to construct nonstandard finite diference variationalintegrators for linear wave equation and Klein–Gordon equation. The convergence ofthe proposed methods are discussed and the discrete multisymplectic form formulas arederived. The numerical experiments proves the feasibility and efectiveness.
Fourthly, nonlinear Schro¨dinger equations with variable coefcients are considered.Nonstandard finite diference variational integrators under triangle and square discretiza-tion are constructed. The convergence of these integrators are discussed, and the discretemultisymplectic structures are presented by discrete multisymplectic form formulas. Inthe numerical experiments, the convergence orders are tested and the stability are shown.Furthermore, the proposed methods could preserve the norm conservation law of nonlin-ear Schro¨dinger equation very well. The comparison between standard finite diferencemethods and the proposed methods, the classical Crank–Nicolson methods and the pro-posed methods in this paper are made.
Finally, nonstandard finite diference methods and composition methods with com-plex time steps are combined together to derive the numerical solutions of two populationmodels. Nonstandard finite diference methods could preserve the positive solutions andthe total population conservation laws of these models. Composition methods with com-plex time steps improve the numerical solutions obtained by nonstandard finite diferencemethods, and continue to preserve the positive values and conservation laws. This is thefirst time to apply composition methods with complex time steps to the numerical biolog-ical systems. Because of the simple form, composition methods are better than Richard-son’s extrapolation method on computational efciency. The numerical experiments showthe feasibility and efectiveness of the proposed methods.
本文首先简要概述了辛和多辛系统的研究历程，介绍了Hamilton系统和Lagrange系统的关系及变分原理的基本思想，并回顾了多辛Hamilton数值方法和离散变分积分子的研究成果。
其次，本文基于Lagrange的角度利用变分原理研究变分积分子及其保持的多辛结构。本文在前人研究Lagrange多辛几何的基础上考虑边界值空间，利用边界Lagrange函数和变分原理，建立了新的推导离散多辛结构——离散多辛形式公式的方法。鉴于Lagrange系统和Hamilton系统在非退化条件下的等价关系，本文也探求了Lagrange变分积分子和多辛Hamilton数值方法之间的关系。针对波动方程，在选择适当的离散下，本文建立了离散变分积分子和经典的多辛Hamilton数值方法之间的等价关系，构造了分别与Euler box数值格式和Preissman box数值格式等价的两个离散变分积分子。
再次，本文结合非标准有限差分方法的思想和前面建立的离散变分积分子的理论，分别构造了线性波动方程和Klein–Gordon方程的非标准有限差分变分积分子。讨论了所构造方法的收敛性，并推导了所构造方法保持的多辛离散结构——离散多辛形式公式。在数值试验中，本文验证了方法的收敛阶，展现了所构造的方法可以很好反映原系统的数值特性。数值实验验证了非标准有限差分变分积分子数值求解多辛系统的可行性和有效性。
接着，本文研究了非线性变系数Schro¨dinger方程。本文分别在三角离散和方形离散下构造了此方程的非标准有限差分变分积分子，讨论了变分积分子的收敛精度，并推导了其保持的多辛离散结构——离散多辛形式公式。数值试验验证了方法的收敛阶，考察了方法的数值稳定性。同时展示了所构造方法可以很好的保持非线性Schro¨dinger方程的模方守恒率，并分别将本文所构造的方法与标准有限差分方法和经典的Crank–Nicolson方法进行比较，展示了本文所构造方法的优势所在。
最后，本文将结合非标准有限差分方法和具有复数时间步长的复合方法去研究浮游生物模型和百日咳传染病模型。针对这两个模型构造的非标准有限差分方法可以很好的保持原系统中的正性和总量守恒率。然后使用具有复数时间步长的复合方法改进了数值解的收敛精度并且继续保持了正性和总量守恒率。这也是首次将复数时间步长的复合方法应用于生物模型的数值求解。复合方法因其简单却有效的格式使得它比Richardson外推方法具有更好的计算效率。数值实验证明了本文对生物模型构造的方法是可行有效的。
The symplectic and multisymplectic systems widely exist in natural world, especial-ly in mechanics and electromagnetics systems etc. Multisymplectic geometric structure isa intrinsic structure of the multisymplectic system. The numerical methods which can p-reserve the multisymplectic structure of original system always perform well and they canpreserve quantitative and qualitative numerical characteristics. For example, multisym-plectic methods have long-time stability and preserve the invariants of original systemswell. There are two main ways to construct multisymplectic numerical schemes. One isto discrete the multisymplectic Hamilton’s equation directly, and try to derive its corre-sponding discrete multisymplectic structure. These methods are called multisymplecticHamiltonian method. The other way is based on variational principle of Lagrangian side.The discrete Euler–Lagrange equation is derived from discrete variational principle, andmeanwhile the corresponding multisymplectic structure is also produced from the dis-crete variational principle. These methods are called multisymplectic discrete variationalintegrators. A advantage of the latter one is that it’s based on the intrinsic variational prin-ciple, and it’s naturally multisymplectic. This dissertation will use the discrete variationalprinciple in Lagrangian field to study the discrete variational integrators and their corre-sponding discrete multisymplectic structures. Furthermore, the idea of nonstandard finitediference methods is applied to construct the nonstandard finite diference variational in-tegrators. This dissertation also combine the nonstandard finite diference methods andcomposition methods with complex time steps to study two population models and theirconservation laws.
Firstly, the research history of symplectic and multisymplectic systems is introduced.The Hamilton’s systems, Lagrange’s systems and the basic idea of variational principleare stated. Some research results of the multisymplectic numerical methods are presented.
Secondly, based on Lagrangian side, variational principle is used to study the varia-tional integrators and corresponding multisymplectic structures. Considering the bound-ary value space, this paper gives a new way to derive the discrete multisymplectic formformula which is preserved by discrete variational integrators. In view of the equivalenceof Lagrangian systems and Hamiltonian systems under nondegenerate condition, this dis-sertation also try to establish equivalency between Lagrangian variational integrators and multisymplectic Hamiltonian numerical methods. For suitable choices of discretizationwhen applied to the wave equation, this paper shows two variational integrators which areequivalent to Euler box method and Preissman box methods respectively.
Thirdly, this paper combines the idea of nonstandard finite diference methods andthe discrete variational integrators to construct nonstandard finite diference variationalintegrators for linear wave equation and Klein–Gordon equation. The convergence ofthe proposed methods are discussed and the discrete multisymplectic form formulas arederived. The numerical experiments proves the feasibility and efectiveness.
Fourthly, nonlinear Schro¨dinger equations with variable coefcients are considered.Nonstandard finite diference variational integrators under triangle and square discretiza-tion are constructed. The convergence of these integrators are discussed, and the discretemultisymplectic structures are presented by discrete multisymplectic form formulas. Inthe numerical experiments, the convergence orders are tested and the stability are shown.Furthermore, the proposed methods could preserve the norm conservation law of nonlin-ear Schro¨dinger equation very well. The comparison between standard finite diferencemethods and the proposed methods, the classical Crank–Nicolson methods and the pro-posed methods in this paper are made.
Finally, nonstandard finite diference methods and composition methods with com-plex time steps are combined together to derive the numerical solutions of two populationmodels. Nonstandard finite diference methods could preserve the positive solutions andthe total population conservation laws of these models. Composition methods with com-plex time steps improve the numerical solutions obtained by nonstandard finite diferencemethods, and continue to preserve the positive values and conservation laws. This is thefirst time to apply composition methods with complex time steps to the numerical biolog-ical systems. Because of the simple form, composition methods are better than Richard-son’s extrapolation method on computational efciency. The numerical experiments showthe feasibility and efectiveness of the proposed methods.
引文
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