Three-stage two-parameter symplectic, symmetric exponentially-fitted Runge-Kutta methods of Gauss type

We construct an exponentially-fitted variant of the well-known three stage Runge-Kutta method of Gauss-type. The new method is symmetric and symplectic by construction and it contains two parameters, which can be tuned to the problem at hand. Some numerical experiments are given

Multiparameter exponentially-fitted methods applied to second-order boundary value problems

Second-order boundary value problems are solved by means of a new type of exponentially-fitted methods that are modifications of the Numerov method. These methods depend upon a set of parameters which can be tuned to solve the problem at hand more accurately. Their values can be fixed over the entire integration interval, but they can also be determined locally from the local truncation error. A numerical example is given to illustrate the ideas

On the Leading Error Term of Exponentially Fitted Numerov Methods

Second-order boundary value problems are solved with exponentially-fitted Numerov methods. In order to attribute a value to the free parameter in such a method, we look at the leading term of the local truncation error. By solving the problem in two phases, a value for this parameter can be found such that the tuned method behaves like a sixth order method. Furthermore, guidelines to choose between multi le possible values for this parameter are given

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

Multi-parameter exponentially fitted, P-stable Obrechkoff methods

We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for second order differential equations. An earlier result for single-parameter exponential fitting is re-examined and extended to multi-parameter, multi-order exponential fitting

Properties and implementation of r-Adams methods based on mixed-type interpolation

We investigate the properties of the coefficients of modified r-Adams methods for the integration of ODEs. The derivation of these methods is, in contrast with the classical Adams methods, not based on a polynomial interpolation theory, but rather starts from a mixed interpolation theory in which a parameter kappa is involved. It will be shown that the coefficients of the modified methods possess properties which make these methods very attractive. Further, we will discuss the role of so-called over-implicit modified r-Adams schemes in the construction of more general linear multistep methods. Our second goal is to show that the modified Adams-Bashforth/Adams-Moulton methods are very well suited to be implemented as a predictor-corrector pair. In particular, are will discuss the choice of the interpolation parameter when such a method is applied to general systems of equations. Numerical tests are performed to support the theory

On the generation of mono-implicit Runge-Kutta-Nyström 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