Exponentially fitted singly diagonally implicit Runge-Kutta methods

It is the purpose of this paper to derive diagonally implicit exponentially fitted (EF) Runge-Kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations, whose solutions are supposed to exhibit an exponential behaviour. In addition to the standard approach for the derivation of EF methods, we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages. The derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages, as classically done in the standard derivation of multistage EF-based methods. Standard and revised EF methods are then compared in terms of linear stability and numerical performances.