On the stability of functionally fitted Runge–Kutta methods

Classical collocation RK 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 RK (FRK) 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 FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the parameters of the problem, the time, the stepsize, 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 how to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example and we provide further insight for separable methods with symmetric collocation points. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60