A class of orthogonal integrators for stochastic differential equations

The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constraints to Runge–Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.     

High-Order Symplectic and Symmetric Composition Methods for Multi-Frequency and Multi-Dimensional Oscillatory Hamiltonian Systems

The multi-frequency and multi-dimensional adapted Runge-Kutta-Nyström (ARKN) integrators, and multi-frequency and multi-dimensional extended Runge-Kutta-Nyström(ERKN) integrators have been developed to efficiently solve multi-frequency oscillatory Hamiltonian systems. The aim of this paper is to analyze and derive high-order symplectic and symmetric composition methods based on the ARKN integrators and ERKN integrators. We first consider the symplecticity conditions for the multi-frequency and multi-dimensional ARKN integrators. We then analyze the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN integrators and ERKN integrators, respectively. On the basis of the theoretical analysis and by using the idea of composition methods, we derive and propose four new high-order symplectic and symmetric methods for the multi-frequency oscillatory Hamiltonian systems. The numerical results accompanied in this paper quantitatively show the advantage and efficiency of the proposed high-order symplectic and symmetric methods.     

Analysis of trigonometric implicit Runge–Kutta methods

Using generalized collocation techniques based on fitting functions that are trigonometric (rather than algebraic as in classical integrators), we develop a new class of multistage, one-step, variable stepsize, and variable coefficients implicit Runge–Kutta methods to solve oscillatory ODE problems. The coefficients of the methods are functions of the frequency and the stepsize. We refer to this class as trigonometric implicit Runge–Kutta (TIRK) methods. They integrate an equation exactly if its solution is a trigonometric polynomial with a known frequency. We characterize the order and A-stability of the methods and establish results similar to that of classical algebraic collocation RK methods.     

Selectable Set Randomized Kaczmarz

The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz is a variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive “selectable set” and thus yields an improved convergence guarantee. In this article, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.     

Computational inelasticity for loading conditions on multiple time scales by adaptive step size control

This contribution proposes an algorithm based on adaptive step size control for the simulation of inelastic solids and structures undergoing loading conditions at multiple time scales. Adaptivity in time integration of viscoelastic constitutive laws is directed by an refinement indicator which is constructed from integrators of different order, here a fourth-order Runge-Kutta (RK) method and linear Backward-Euler. The key novel aspect is that by virtue of an recently established consistency condition the higher order methods, p ≥ 2, can achieve their full nominal order without fulfilling the weak form of balance of linear momentum in the RK stages, but only at the end of the time interval. A representative numerical example illustrates the performance of the present adaptive method and underpins the computational savings compared with uniform time step sizes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CHARACTERIZATIONS OF SYMMETRIC MULTISTEP RUNGE-KUTTA METHODS

Some characterizations for symmetric multistep Runge-Kutta(RK) methods are obtained. Symmetric two-step RK methods with one and two-stages are presented. Numerical examples show that symmetry of multistep RK methods alone is not sufficient for long time integration for reversible Hamiltonian systems. This is an important difference between one-step and multistep symmetric RK methods.     

Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge-Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can‘t reach order more than s 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s 1 when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.     

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ω h (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.     

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.

     

Numerical treatment for nonlinear Brusselator chemical model

Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant species which interacts with other species is simulated by the Runge-Kutta of order four (RK4) and by Non-Standard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter h. The results are compared with the well-known numerical scheme, i.e. RK4. The developed scheme NSFD gives better results than RK4.     

具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式

构造了一种解具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式．基本思想是离散Landau-Lifshitz方程成常微分方程组,应用李群方法和显式Runge-Kutta方法解常微分方程组．数值试验比较了两方法的保平方守恒特性和精度,得出李群方法(RK-Cayley方法)比相应的Runge-Kutta(RK)方法有更好的精度和保平方守恒特性．     

An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values

We employ B-series to analyze the order of Runge-Kutta (RK) methods that use simple iteration, modified Newton iteration or full Newton iteration to compute their internal stage values. Our assumptions about the initial guess for the internal stage values cover a wide range of practical schemes. Moreover, the analytical techniques developed in this paper can be applied in many other similar contexts.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, the Information Technology Research Centre of Ontario, and the Norwegian Research Council for Science and Humanities.     

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.     

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)     

STABILITIES OF （A,B,C）AND NPDIRK METHODS FOR SYSTEMS OF NEUTRAL DELAY—DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

Consider the following neutral delay-differential equations with multiple delays (NMDDE)$$y'(t)=Ly(t)+\sum_{j=1}^{m}[M_jy(t-\tau_j)+N_jy'(t-\tau_j)],\ \ t\geq 0, (0.1)$$ where $\tau>0$, $L, M_j$ and $N_j$ are constant complex- value $d×d$ matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE is studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods (NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methos. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).     

An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems

Recently, the symplectic exponentially-fitted methods for Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest. As an alternative to them, in this paper, we propose a class of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage Runge–Kutta (RK) methods, and extend the theory of exponentially-fitted RK methods in the context of continuous stage RK methods. Then by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.     

A class of symplectic partitioned Runge–Kutta methods

This paper deals with some relevant properties of Runge–Kutta (RK) methods and symplectic partitioned Runge–Kutta (PRK) methods. First, it is shown that the arithmetic mean of a RK method and its adjoint counterpart is symmetric. Second, the symplectic adjoint method is introduced and a simple way to construct symplectic PRK methods via the symplectic adjoint method is provided. Some relevant properties of the adjoint method and the symplectic adjoint method are discussed. Third, a class of symplectic PRK methods are proposed based on Radau IA, Radau IIA and their adjoint methods. The structure of the PRK methods is similar to that of Lobatto IIIA–IIIB pairs and is of block forms. Finally, some examples of symplectic partitioned Runge–Kutta methods are presented.     

Exponential Rosenbrock integrators for option pricing

In this paper, we are concerned with the time integration of differential equations modeling option pricing. In particular, we consider the Black-Scholes equation for American options. As an alternative to existing methods, we present exponential Rosenbrock integrators. These integrators require the evaluation of the exponential and related functions of the Jacobian matrix. The resulting methods have good stability properties. They are fully explicit and do not require the numerical solution of linear systems, in contrast to standard integrators. We have implemented some numerical experiments in Matlab showing the reliability of the new method.     

Rational Krylov methods in exponential integrators for European option pricing

The aim of this paper is to analyze efficient numerical methods for time integration of European option pricing models. When spatial discretization is adopted, the resulting problem consists of an ordinary differential equation that can be approximated by means of exponential Runge–Kutta integrators, where the matrix‐valued functions are computed by the so‐called shift‐and‐invert Krylov method. To our knowledge, the use of this numerical approach is innovative in the framework of option pricing, and it reveals to be very attractive and efficient to solve the problem at hand. In this respect, we propose some a posteriori estimates for the error in the shift‐and‐invert approximation of the core‐functions arising in exponential integrators. The effectiveness of these error bounds is tested on several examples of interest. They can be adopted as a convenient stopping criterion for implementing the exponential Runge–Kutta algorithm in order to perform time integration. Copyright © 2013 John Wiley & Sons, Ltd.