Ch. Tsitouras

On modified Runge–Kutta trees and methods

Modifled Runge{Kutta (mRK) methods can have interesting properties as their coe‐cients may depend on the step-length. By a simple perturbation of very few coe‐cients we may produce various function-fltted methods and avoid the overload of evaluating all the coe‐cients in every step. It is known that, for Runge{Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, e‐cient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fltted properties are analyzed for this case and speciflc phase-fltted pairs of orders 8(6) and 6(5) are presented and tested.

Optimized Runge-Kutta pairs for problems with oscillating solutions

Three types of methods for integrating periodic initial value problems are presented. These methods are (i) phase-fitted, (ii) zero dissipation (iii) both zero dissipative and phase fitted. Some particular modifications of well-known explicit Runge-Kutta pairs of orders five and four are constructed. Numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems.

Phase-fitted RungeKutta pairs of orders 8(7)

A new phase fitted RungeKutta pair of orders 8(7) which is a modification of a well known explicit RungeKutta pair for the integration of periodic initial value problems is presented. Numerical experiments show the efficiency of the new pair in a wide range of oscillatory problems.

Explicit Numerov Type Methods with Reduced Number of Stages

We present in this paper a new approach for the derivation of hybrid explicit Numerov type methods. The new methodology does not require the intermediate use of high accuracy inter- polatory nodes, since we only need the Taylor expansion of the internal points. As a consequence, a sixth-order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now. Numerical results over some well-known problems in physics and mechanics indicate the superiority of the new method. (~) 2003 Elsevier Science Ltd. All rights reserved.

Explicit two-step methods for second-order linear IVPs

Abstract We present a new type of method for the integration of systems of linear inhomogeneous initial value problems with constant coefficients. Our methods are of hybrid explicit Numerov type. The methods are constructed without the intermediate use of high accuracy interpolatory nodes, since only the Taylor expansion at the internal points is needed. Then we derive the order conditions taking advantage of the special structure of the problem considered. We present a method with algebraic order seven at a cost of only four stages per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation, indicate the superiority of the new method.

EXPLICIT EIGHTH ORDER TWO-STEP METHODS WITH NINE STAGES FOR INTEGRATING OSCILLATORY PROBLEMS

We present a new explicit hybrid two step method for the solution of second order initial value problem. It costs only nine function evaluations per step and attains eighth algebraic order so it is the cheapest in the literature. Its coefficients are chosen to reduce amplification and phase errors. Thus the method is well suited for facing problems with oscillatory solutions. After implementing a MATLAB program, we proceed with numerical tests that justify our effort.

EXPLICIT NUMEROV TYPE METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTIONS

New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrodinger equation and related problems.1–3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

A general family of explicit Runge-Kutta pairs of orders 6(5)

Explicit Runge–Kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations. So far, several Runge–Kutta pairs of order 6(5) have appeared in the literature. These pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients, depending on the use of the FSAL (first stage as last) device. These equations form a set of necessary and sufficient conditions that a 6(5) Runge–Kutta pair must satisfy. The solution of the latter is achieved by employing various types of simplifying assumptions. In this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions. Using only these simplifying assumptions we define a new family of 6(5) Runge–Kutta pairs. Its main characteristic, which is also a property that no other known family shares, is that all of its nodes (except the last one, which eq...

A Tenth Order Symplectic Runge–Kutta–Nyström Method

A tenth order explicit symmetric and in consequence symplectic Runge–Kutta–Nyström method is presented here. We derive the order conditions needed and solve them for the parameters of the method. Numerical results indicate the superiority of the new method compared to the other high order symplectic methods appeared in the literature until now.

Symbolic derivation of Runge-Kutta order conditions

Tree theory, partitions of integer numbers, combinatorial mathematics and computer algebra are the basis for the construction of a powerful and efficient symbolic package for the derivation of Runge-Kutta order conditions and principal truncation error terms.