On the stability function of functionally-fitted Runge--Kutta methods

Abstract

Classical collocation Runge--Kutta methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted Runge--Kutta methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric functionally fitted Runge--Kutta method targeted at such a problem. However, functionally fitted Runge--Kutta methods lead to variable coefficients that depend on the parameters of the problem, the time, the step size, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show that it is possible to characterize the stability region of some special methods in this particular class. Explicit stability functions are given for some representative examples.