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.     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.     

Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order

New SDIRKN methods specially adapted to the numerical integration of second-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion and P-stability conditions for SDIRKN methods which are assumed to be zero dissipative. Two four-stage symplectic and P-stable methods with algebraic order 4 and high order of dispersion are obtained. One of the methods is symmetric and sixth-order dispersive whereas the other method is nonsymmetric and eighth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with other methods derived by Sharp et al. [IMA J. Numer. Anal. 10 (1990) 489–504].     

Certain improvements of Newton’s method with fourth-order convergence

In this paper we present two new schemes, one is third-order and the other is fourth-order. These are improvements of second-order methods for solving nonlinear equations and are based on the method of undetermined coefficients. We show that the fourth-order method is more efficient than the fifth-order method due to Kou et al. [J. Kou, Y. Li, X. Wang, Some modifications of Newton’s method with fifth-order covergence, J. Comput. Appl. Math., 209 (2007) 146–152]. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.     

Equivariant constrained symplectic integration

Summary We use recent results on symplectic integration of Hamiltonian systems with constraints to construct symplectic integrators on cotangent bundles of manifolds by embedding the manifold in a linear space. We also prove that these methods are equivariant under cotangent lifts of a symmetry group acting linearly on the ambient space and consequently preserve the corresponding momentum. These results provide an elementary construction of symplectic integrators for Lie-Poisson systems and other Hamiltonian systems with symmetry. The methods are illustrated on the free rigid body, the heavy top, and the double spherical pendulum.     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

Symplectic exponentially-fitted four-stage Runge–Kutta methods of the Gauss type

The construction of symmetric and symplectic exponentially-fitted Runge–Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions deserves a lot of interest. In previous papers fourth-order and sixth-order symplectic exponentially-fitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators of eighth-order are studied and constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (2004). Numerical experiments for some oscillatory problems are presented.     

Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)