Multisymplectic Variational Integrators And Nonstandard Finite Difference Methods

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.