T. Van Hecke

Exponentially-fitted explicit Runge–Kutta methods

An exponentially-fitted explicit Runge–Kutta method is constructed, which exactly integrates differential initial-value problems whose solutions are linear combinations of functions of the form exp(ωx) and exp(−ωx) (ω∈R or iR); this method is compared to a previously constructed method of Simos. Numerical experiments show the efficiency of the new method.

Exponentially fitted Runge-Kutta methods

Abstract Exponentially fitted Runge–Kutta methods with s stages are constructed, which exactly integrate differential initial-value problems whose solutions are linear combinations of functions of the form {x j exp (ωx),x j exp (−ωx)} , ( ω∈ R or i R , j=0,1,…,j max ), where 0⩽j max ⩽⌊s/2−1⌋ , the lower bound being related to explicit methods, the upper bound applicable for collocation methods. Explicit methods with s∈{2,3,4} belonging to that class are constructed. For these methods, a study of the local truncation error is made, out of which follows a simple heuristic to estimate the ω-value. Error and step length control is introduced based on Richardson extrapolation ideas. Some numerical experiments show the efficiency of the introduced methods. It is shown that the same techniques can be applied to construct implicit exponentially fitted Runge–Kutta methods.

On a class of P-stable mono-implicit Runge-Kutta-Nystro¨m methods

By a detailed investigation of the conditions to be satisfied by a mono-implicit Runge-Kutta-Nystrom (MIRKN) method in order to be P-stable, a subset of candidate P-stable MIRKN methods is firstly isolated. Then it is shown that this subset effectively contains P-stable MIRKN methods with an even number of stages. Finally, P-stable MIRKN methods with 2, 4 and 6 stages are constructed explicitly.

Deferred correction with mono-implicit Runge-Kutta methods for first-order IVPs

Abstract To reach a high order of accuracy for numerical solutions of IVPs with mono-implicit Runge–Kutta (MIRK) methods, the technique of deferred correction is used. Special attention is paid to the possible increase of the order and the stability of such schemes. Several schemes are given.

On the generation of mono-implicit Runge-Kutta-Nystro¨m methods by mono-implicit Runge-Kutta methods

Mono-implicit Runge-Kutta methods can be used to generate implicit Runge-Kutta-Nystrom (IRKN) methods for the numerical solution of systems of second-order differential equations. The paper is concerned with the investigation of the conditions to be fulfilled by the mono-implicit Runge-Kutta (MIRK) method in order to generate a mono-implicit Runge-Kutta-Nystrom method (MIRKN) that is P-stable. One of the main theoretical results is the property that MIRK methods (in standard form) cannot generate MIRKN methods (in standard form) of order greater than 4. Many examples of MIRKN methods generated by MIRK methods are presented.

A variable-step variable-order algorithm for systems of stiff odes

A variable-step variable-order algorithm for stiff ODEs based on previously derived stabilized extended one-step methods is established. The developed code is tested on certain initial-value problems for systems of ODEs contained in the test set proposed by CWI.

A mono-implicit Runge-Kutta-Nyström modification of the Numerov method

We present two two-parameter families of fourth-order mono-implicit Runge-Kutta-Nystrom methods. Each member of these families can be considered as a modification of the Numerov method. We analyze the stability and periodicity properties of these methods. It is shown that (i) within one of these families there exist A-stable (even L-stable) and P-stable methods, and (ii) in both families there exist methods with a phase lag of order six.

P-stable Mono-Implicit Runge-Kutta-Nyström Modifications of the Numerov Method

We present families of fourth-order mono-implicit Runge-Kutta-Nystrom methods. Each member of these families can be considered as a modification of the Numerov method. Some parameters of these new methods are used to optimize the linear stability properties, i.e. to obtain P-stable methods with a minimal phase-lag. Also we show that in some cases there exist P-stable methods with stage-order 3. Since the methods considered are mono-implicit, the computational work needed in each time-step to solve the implicit equations is reduced seriously.

Pareto plot threshold for multiple management factors in design of experiments

Abstract Economical, technical and strategic reasons make unreplicated experimental designs necessary and popular in industrial and management settings. This paper investigates the legitimate use of well known methods that appoint effects as significant when they are larger than a defined threshold. We found that established estimation methods, such as Lenth’s method and Dong’s method, are too tolerant and a larger threshold should be used before an effect can be determined as influencing. This comparing analysis was done by using the degree of freedom that is available when the resolution of the design is one less than the maximum value. Using the Pareto Principle we suggest a more accurate threshold estimator for significantly influencing factors and/or interactions based on analysis of variance.AbstractEconomical, technical and strategic reasons make unreplicated experimental designs necessary and popular in industrial and management settings. This paper investigates the legitimate use of well known methods that appoint effects as significant when they are larger than a defined threshold. We found that established estimation methods, such as Lenth’s method and Dong’s method, are too tolerant and a larger threshold should be used before an effect can be determined as influencing. This comparing analysis was done by using the degree of freedom that is available when the resolution of the design is one less than the maximum value. Using the Pareto Principle we suggest a more accurate threshold estimator for significantly influencing factors and/or interactions based on analysis of variance.

Multistride L-Stable Fourth-Order Methods for the Numerical Solution of ODEs

Recently, Chawla et al. [1] have obtained a one-parameter family of double-stride L-stable methods of fourth order by the coupling of three Linear Multistep Methods (LMMs). Here we present an alternative derivation technique based on mono-implicit Runge-Kutta (MIRK) methods. The MIRK method provides a framework within which all families of double- and triple-stride L-stable methods of fourth order can be expressed. The well-established theory for Runge-Kutta methods can then be used to derive these families in a straightforward manner.