变分与无限维系统的高精度辛格式

1.引 言 冯康和他的研究小组提出的生成函数法[1]系统地解决了象二体问题这样地有限维Hamil-ton系统辛算法的构造问题,该方法也可以自然地推广到无限维Hamilton系统[2].首先在空间方向进行离散,例如采用差分或谱离散,得到有限维Hamilton系统,然后再采用生成函数法离散该系统.这样得到的辛格式是整个一层的格式,对于研究格式的局部性质如多辛性质[3],局部能量守恒性质[5]就相当困难.     

THE CALCULUS OF GENERATING FUNCTIONS AND THE FORMAL ENERGY FOR HAMILTONIAN ALGORITHMS

1.DarbouxTransformationConsidercotangentbundleT*R"acRZnwithnaturalsymplecticstructureandtheproductofcotangentbundles(T*R")x(T*R")=R4nwithnaturalproductsymplecticstructureCorrespondingly,weconsidertheproductspaceR"xR"rsRZn.ItscotangentbunT*(R"xR")=T*Rzn=R'nhasnaturalsymplecticstructure')PreparedbyQinMengzhaoChoosesymplecticcoordinatesz~(p,q)onthesymplecticmanifold,thenforsymplectictransformationg:T*R"~T*R",wehaveitisaLagrangiansubmanifoldofT*R"xT*RninR4n~(R'",J4.).NotethatonR4nth…     

Conformal structure-preserving schemes for damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise

Since many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped-driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped-driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal-symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure-preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.     

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L^∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

任意阶精度蛙跳格式稳定性分析

考虑如下波动方程的初这值问题,设其边界条件为周期的,解具有周期性.如[6](1.1)有两种Hamilton形。一种是经典形式     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

三级四阶显式辛R-K-N格式的完全构造

s级p阶辛Runge-Kutta-Nystr\"om(R-K-N)方法的一种充要条件是用关于参数的非线性方程组来表示的,辛R-K-N格式的构造问题因而转化为该方程组的求解问题. 在一些特殊的限定条件下, 已有该方程组在s=3,p=4时的两组解,即得到了两个三级四阶显式辛格式. 对于s=3,p=4情形,基于吴方法,利用计算机代数系统Maple及软件包wsolve给出了对应的非线性方程组的全部解, 这样就构造了所有的三级四阶显式辛R-K-N格式, 并证明了三级四阶显式辛R-K-N方法所满足的条件方程有冗余. 数值实验结果显示出新的辛格式在一定的条件下有着较好的误差精度.     

Multiplicative Extrapolation Method for Constructing Higher Order Schemes for Ordinary Differential Equations

IntroductionWhenweconstructahigherorderschemeforsystemsofordinarydifferentialequations:y,=f(y)(1)(wherey=y(x),andxisavariable),weoftenusethe"Tayorseriesexpanding"method,butsometimesthismethodisverytediouswhenitisaPpliedtogethigherorderschemes.Thereisanothermethod:Lieseries,itisthemethodweuseinthispaPer.J.Dragt,F,Neri,andStaulySteinberghavedonealotofworkindevelopingthismethod.Fordetails,onecanreferto[4,6,8].WejustaPplythismethodtoourproblem,anddonotneedtocomputeouttheexacttermsofthe"Lieser…     

On Wavewise Entropy Inequalities for High-Resolution Schemes. I: The Semidiscrete Case

We develop a new approach, the method of wavewise entropy inequalities for the numerical analysis of hyperbolic conservation laws. The method is based on a new extremum tracking theory and Volpert's theory of BV solutions. The method yields a sharp convergence criterion which is used to prove the convergence of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.

     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

Time discretizations for numerical optimisation of hyperbolic problems

We consider a class of numerical schemes for optimal control problems of hyperbolic conservation laws. We focus on finite-volume schemes using relaxation as a numerical approach to the optimality system. In particular, we study the arising numerical schemes for the adjoint equation and derive necessary conditions on the time integrator. We show that the resulting schemes are in particular asymptotic preserving for both, the adjoint and forward equation. We furthermore prove that higher-order time-integrator yields suitable Runge-Kutta schemes. The discussion includes the numerically interesting zero relaxation case.     

用Hyperbolic函数构造Schrodinger方程的辛格式

本文利用Ｈｙｐｅｒｂｏｌｉｃ函数ｓｉｎｈ（ｘ）和ｔａｎｈ（ｘ）构造了Ｓｃｈｒｏｄｉｎｓｅｒ方程的任意阶的辛格式并讨论了它们的稳定性．     

A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems

In this paper we consider the necessary conditions of conservation laws of symplectic difference schemes for Hamiltonian systems and give an example which shows that there does not exist any centered symplectic difference scheme which preserves all Hamiltonian energy.     

SYMPLECTIC SCHEMES OF VORTEX SYSTEM IN HALF PLAIN

l introductionThe vortex system is a system of POint voids, and a model Of incompressible inviscidnow inspired by the idea of an almOSt POtence flOW. The voracity in the now is concentratedin N-vortices (i. e., POints at which the vortloty field is singUlar) [4]. An ideal incompressible now can be approximated by the motion Of a ~ system which is not only a usefulheuristic tool in the analysis of the general propelles of solutionS Of Euler equations, but also auseful stachg POint for th…     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.