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 {xj exp(ωx),xj exp(−ωx)}, (ω∈R or iR, 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.