Some variant of Newton’s method with third-order convergence

We present a new modification of the Newton’s method which produces iterative methods with order of convergence three. A general error analysis providing the higher order of convergence is given, and the best efficiency, in term of function evaluations, of two of this new methods is provided.     

Variable-Order ESIRK Methods for Stiff Differential Equations

ESIRK methods (Effective order Singly-Implicit Runge–Kutta methods) have been shown to be efficient for the numerical solution of stiff differential equations. In this paper, we consider a new implementation of these methods with a variable order strategy. We show that the efficiency of the ESIRK method for stiff problems is improved by using the proposed variable order schemes.     

Schechter Methods of Interpolation and Commutators

We study the relationship between Schechter's methods of complex interpolation and the so called commutator estimates. We obtain new commutator theorems and prove characterizations of the Domain and Range spaces associated with the corresponding quasilogarithmic operator. Our methods also provide a new approach to known results, including the higher order commutator theorems for the complex method recently obtained by R. Rochberg .     

Numerical resolution of linear evolution multidimensional problems of second order in time

We present a new class of efficient time integrators for solving linear evolution multidimensional problems of second‐order in time named Fractional Step Runge‐Kutta‐Nyström methods (FSRKN). We show that these methods, combined with suitable spliting of the space differential operator and adequate space discretizations provide important advantages from the computational point of view, mainly parallelization facilities and reduction of computational complexity. In this article, we study in detail the consistency of such methods and we introduce an extension of the concept of R‐stability for Runge‐Kutta‐Nyström methods. We also present some numerical experiments showing the unconditional convergence of a third order method of this class applied to resolve one Initial Boundary Value Problem of second order in time. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 597–620, 2012     

Analysis of a defect correction method for geometric integrators

We discuss a new variant of Iterated Defect Correction (IDeC), which increases the range of applicability of the method. Splitting methods are utilized in conjunction with special integration methods for Hamiltonian systems, or other initial value problems for ordinary differential equations with a particular structure, to solve the neighboring problems occurring in the course of the IDeC iteration. We demonstrate that this acceleration technique serves to rapidly increase the convergence order of the resulting numerical approximations, up to the theoretical limit given by the order of certain superconvergent collocation methods. This project was supported by the Special Research Program SFB F011 ‘AURORA’ of the Austrian Science Fund FWF.     

On the Numerical Solution of Involutive Ordinary Differential Systems: Higher Order Methods

We analyse some Taylor and Runge—Kutta type methods for computing one-dimensional integral manifolds, i.e. solutions to ODEs and DAEs. The distribution defining the solutions is taken to be defined only on the relevant manifold and hence all the intermediate points occuring in the computations are projected orthogonally to the manifold. We analyse the order of such methods, and somewhat surprisingly there does not appear any new order conditions for the Runge—Kutta methods in our context, at least up to order 4. The analysis shows that some terms appearing in the error expansions can be quite naturally expressed in terms of standard notions of Riemannian geometry. The numerical examples show that the methods work reliably and moreover produce qualitatively correct results for Hamiltonian systems although the methods are not symplectic.This revised version was published online in October 2005 with corrections to the Cover Date.     

Efficient three-step iterative methods with sixth order convergence for nonlinear equations

In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order. The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence order, the computational cost and the efficiency order of our methods with those of the original methods.     

Two-derivative Runge-Kutta methods with optimal phase properties

In this work, we consider two-derivative Runge-Kutta methods for the numerical integration of first-order differential equations with oscillatory solution. We construct methods with constant coefficients and special properties as minimum phase-lag and amplification errors with three and four stages. All methods constructed have fifth algebraic order. We also present methods with variable coefficients with zero phase-lag and amplification errors. In order to examine the efficiency of the new methods, we use four well-known oscillatory test problems.     

New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence

In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) [2], that establishes for an iterative method based on n evaluations an optimal order p=2ⁿ⁻¹ is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, which are 1.587 and 1.682.We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods.     

Moment-free numerical integration of highly oscillatory functions

^** Email: s.olver{at}damtp.cam.ac.uk The aim of this paper is to derive new methods for numericallyapproximating the integral of a highly oscillatory function.We begin with a review of the asymptotic and Filon-type methodsdeveloped by Iserles and Nørsett. Using a method developedby Levin as a point of departure, we construct a new methodthat utilizes the same information as a Filon-type method, andobtains the same asymptotic order, while not requiring the computationof moments. We also show that a special case of this methodhas the property that the asymptotic order increases with theaddition of sample points within the interval of integration,unlike all the preceding methods whose orders depend only onthe endpoints.     

Exponential almost Runge-Kutta methods for semilinear problems

We present a new class of one-step, multi-value Exponential Integrator (EI) methods referred to as Exponential Almost Runge-Kutta (EARK) methods which involve the derivatives of a nonlinear function of the solution. In order to approximate such derivatives to a sufficient accuracy, the EARK methods will be implemented within the broader framework of Exponential Almost General Linear Methods (EAGLMs) to accommodate past values of this nonlinear function and becoming multistep in nature as a consequence. Established EI methods, such as Exponential Time Differencing (ETD) methods, Exponential Runge-Kutta (ERK) methods and Exponential General Linear Methods (EGLMs) become special cases of EAGLMs. We present order conditions which facilitate the construction of two- and three-stage EARK methods and, when cast in an EAGLM format, we perform a stability analysis to enable a comparison with existing EI methods. We conclude with some numerical experiments which confirm the convergence order and also demonstrate the computational efficiency of these new methods.     

Stochastic Runge-Kutta methods with deterministic high order for ordinary differential equations

We consider embedding deterministic Runge-Kutta methods with high order into weak order stochastic Runge-Kutta (SRK) methods for non-commutative stochastic differential equations (SDEs). As a result, we have obtained weak second order SRK methods which have good properties with respect to not only practical errors but also mean square stability. In our stability analysis, as well as a scalar test equation with complex-valued parameters, we have used a multi-dimensional non-commutative test SDE. The performance of our new schemes will be shown through comparisons with an efficient and optimal weak second order scheme proposed by Debrabant and Rößler (Appl. Numer. Math. 59:582–594, 2009).     

On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations

We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383–393, 2014). The order of convergence of new family is increased from 4 of the basic family to \(2+\sqrt {5} \approx 4.24\) by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence \(2+\sqrt {6} \approx 4.45\) is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the theoretical results.     

一类隐式的k+1阶线性k步法

本文首先给出了一类比Adams-Moulton方法的绝对稳定区间大的隐式k+1阶线性k步法基本公式.求出了3-9步新公式的分数形式的精确系数,阶数,局部截断误差主项系数和绝对稳定区间,然后构造了由4阶隐式新公式和同阶显式Nyström公式组合而成的预估-校正方法,比著名的Adams-Bashforth-Moulton和Nyström-Adams-Moulton预估校正方法的绝对稳定区间大,最后用对比数值试验对结果进行了验证.     

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.     

Explicit hybrid six–step,sixth order,fully symmetric methods for solving y ″ = f (x,y)

A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.     

A Structured Secant Method Based on a New Quasi-Newton Equation for Nonlinear Least Squares Problems

In this paper, a new quasi-Newton equation is applied to the structured secant methods for nonlinear least squares problems. We show that the new equation is better than the original quasi-Newton equation as it provides a more accurate approximation to the second order information. Furthermore, combining the new quasi-Newton equation with a product structure, a new algorithm is established. It is shown that the resulting algorithm is quadratically convergent for the zero-residual case and superlinearly convergent for the nonzero-residual case. In order to compare the new algorithm with some related methods, our preliminary numerical experiments are also reported.     

Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data

We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A new two-point scheme for multiple roots of nonlinear equations

In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two-point sixth-order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real-life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.     

MDS Self-Dual Codes over Large Prime Fields

Combinatorial designs have been used widely in the construction of self-dual codes. Recently a new method of constructing self-dual codes was established using orthogonal designs. This method has led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we generalize this method by using generalized orthogonal designs, and we give another new method that creates and solves Diophantine equations over GF(p) in order to find suitable generator matrices for self-dual codes. We show that under the necessary conditions these methods can be applied as well to small and large fields. We apply these two methods to study self-dual codes over GF(31) and GF(37). Using these methods we obtain some new maximum distance separable self-dual codes of small orders.