On construction of high order exponentially fitted methods based on parameterized rational approximations to \(\exp (q)\) (Q1969932)

This paper is concerned with the construction of some exponentially fitted methods for the numerical solution of stiff initial value problems for ordinary differential equations. Starting with \((4,4)\) rational approximations to the exponential function with orders \( \geq 6\) and two free parameters they give the necessary and sufficient conditions on them for A-stability and introduce the exponentially fitted requirement. They propose a 4th derivative and a 3rd derivative one step method with two free parameters to be chosen according to stability and fitting requirements. Several choices of the two available parameters are discussed, in particular those that satisfy the so called median properties are recommended due to the fact that the local truncation error satisfies a uniformly bounded estimation.