An embedded pair of exponentially fitted explicit Runge–Kutta methods

An embedded pair of exponentially fitted explicit Runge–Kutta (RK) methods for the numerical integration of IVPs with oscillatory solutions is derived. This pair is based on the exponentially fitted explicit RK method constructed in Vanden Berghe et al., and we confirm that the methods which constitute the pair have algebraic order 4 and 3. Some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by Vanden Berghe et al. (J. Comput. Appl. Math. 125 (2000) 107).     

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.     

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 highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems

The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge–Kutta–Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873–1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature.     

Structure preservation of exponentially fitted Runge–Kutta methods

The preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge–Kutta (EFRK) methods is considered. A complete characterisation of EFRK methods that preserve linear or quadratic invariants is given and, following the approach of Bochev and Scovel [On quadratic invariants and symplectic structure, BIT 34 (1994) 337–345], the sufficient conditions on symplecticity of EFRK methods derived by Van de Vyver [A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm. 174 (2006) 255–262] are obtained. Further, a family of symplectic EFRK two-stage methods with order four has been derived. It includes the symplectic EFRK method proposed by Van de Vyver as well as a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. Finally, the results of some numerical experiments are presented to compare the relative merits of several fitted and nonfitted fourth-order methods in the integration of oscillatory systems.     

Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation

A competitive nonstandard semi‐explicit finite‐difference method is constructed and used to obtain numerical solutions of the diffusion‐free generalized Nagumo equation. Qualitative stability analysis and numerical simulations show that this scheme is more robust in comparison to some standard explicit methods such as forward Euler and the fourth‐order Runge‐Kutta method (RK4). The nonstandard scheme is extended to construct a semi‐explicit and an implicit scheme to solve the full Nagumo reaction‐diffusion equation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 363–379, 2003.     

Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Nonstandard finite differences (NSFDs) schemes can improve the accuracy and reduce computational costs of traditional finite difference schemes. In addition NSFDs produce numerical solutions which also exhibit essential properties of solution. In this paper, a class of nonstandard 2-stage Runge–Kutta methods of order two (we call it nonstandard RK2) is considered. The preservation of some qualitative properties by this class of methods are discussed. In order to illustrate our results, we provide some numerical examples.     

Stability analysis of phase fitted variational integrators

In the present work, we discuss the benefits, limitations and restrictions for the high order phase fitted variational integrators, derived specifically for the numerical integration of systems with oscillatory solutions. To do that, we study linear stability properties of these methods using the eigenvalues of the amplification matrix. The obtained stability region verifies their good behavior, when used for the numerical solution of highly oscillatory problems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trigonometric fitted modification of RADAU5

Implicit Runge-Kutta (RK) methods are in common use when addressing stiff initial value problems (IVP). They usually share the property of A-stability that is of crucial importance in solving the latter type of IVP. Radau IIA family of implicit RK methods is among the preferred ones. Especially its fifth-order representative named RADAU5 has received a lot of attention for use with lax accuracies. Here, we try the lesser possible perturbation of its coefficients. Then, we derive a trigonometric fitted modification that is intended to be applied in periodic IVPs. Numerical tests over a variety of problems with oscillatory solutions justify our effort.     

Analysis of higher order phase fitted variational integrators

In the present, the linear stability properties of the higher order phase fitted variational integrators are investigated. Towards this purpose, at first we calculate the eigenvalues of the amplification matrix for each method. Then, since the proposed integrators are derived specifically for the numerical integration of systems with oscillatory solutions, the linear stability analysis verifies their good behavior, when used for these problems. Finally, we test the proposed methods on several numerical examples, first with regard to their stability and secondly concerning the behavior in long term integration of highly oscillatory problems.     

Runge-Kutta methods on Lie groups

We construct generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolve on the correct manifold. Our methods must satisfy two different criteria to achieve a given order.

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

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.     

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].     

Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach

This is a case study of solving the Genesio system by using the classical variational iteration method (VIM) and a newly modified version called the multistage VIM (MVIM). VIM is an analytical technique that grants us a continuous representation of the approximate solution, which allows better information of the solution over the time interval. Unlike its counterpart, numerical techniques, such as the Runge–Kutta method, provide solutions only at two ends of the time interval (with condition that the selected time interval is adequately small for convergence). Furthermore, it offers approximate solutions in a discretized form, making it complicated in achieving a continuous representation. The explicit solutions through VIM and MVIM are compared with the numerical analysis of the fourth-order Runge–Kutta method (RK4). VIM had been successfully applied to linear and nonlinear systems of non-chaotic in nature and this had been testified by numerous scientists lately. Our intention is to determine whether VIM is also a feasible method in solving a chaotic system like Genesio. At the same time, MVIM will be applied to gauge its accuracy compared to VIM and RK4. Since, for most situations, the validity domain of the solutions is often an issue, we will consider a reasonably large time frame in our work.     

Order conditions for RKN methods solving general second-order oscillatory systems

This paper proposes and investigates the multidimensional extended Runge-Kutta-Nyström (ERKN) methods for the general second-order oscillatory system y″ + My = f(y, y′) where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The work forms a natural generalization of our previous work on ERKN methods for the special system y″ + My = f(y) (H. Yang et al. Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794 and X. Wu et al., ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Comm. 181 (2010) 1873–1887). The new ERKN methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the equation brought by the term My into both internal stages and updates. In order to derive the order conditions for the ERKN methods, an extended Nyström tree (EN-tree) theory is established. The results of numerical experiments show that the new ERKN methods are more efficient than the general-purpose RK methods and the adapted RKN methods with the same algebraic order in the literature.     

Specialized Runge-Kutta methods for index differential-algebraic equations

We consider the numerical solution of systems of semi-explicit index differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods specialized Runge-Kutta methods for index DAEs (SRK-DAE).

     

Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations

In functional differential equations (FDEs), there is a class of infinite delay-differential equations (IDDEs) with proportional delays, which aries in many scientific fields such as electric mechanics, quantum mechanics, and optics. Ones have found that there exist very different mathematical challenges between FDEs with proportional delays and those with constant delays. Some research on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. However, up to now, the research for nonlinear case still remains to be done. For this, in the present paper, we deal with nonlinear stability of the Runge-Kutta (RK) methods for a class of IDDEs with proportional delays. It is shown under the suitable conditions that a (k, l)-algebraically stable RK method for this kind of nonlinear IDDE is globally and asymptotically stable.     

Asymptotische Formeln für Eigenwerte des TRICOMIschen Differentialoperators erster Art mit Restgliedabschätzung

A class of n-th order (n > 1) differential equations with deviating arguments is considered and sufficient conditions are established in order that all bounded quickly oscillatory solutions (i.e. oscillatory solutions whose consecutive zeros have distance which approaches zero) tend to zero at ∞.     

Fourth Order Conservative Exponential Methods for Linear Evolution Equations

A few numerical methods for linear evolution equations are developed and analyzed in this paper. These fourth order exponential methods reproduce the exact solutions for equations with time-independent evolution operators. For highly oscillatory problems with evolution operators that vary slowly in time, these methods are often more efficient than the traditional methods, since large step sizes can be used. The methods developed in this paper are also conservative for equations such as the Schrödinger equation, where the evolution operator is skew-selfadjoint.