Exponentially fitted Runge-Kutta methods

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(omegax),x(j) exp(-omegax)}, to (omega is an element of R or iR, j = 0, 1,...,j max), where 0 less than or equal to j max less than or equal to [s/2 - 1], the lower bound being related to explicit methods, the upper bound applicable for collocation methods. Explicit methods with s is an element of {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 omega -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. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 65L05; 65L06; 65L20.