On the stability of functionally fitted Runge–Kutta methods

Abstract

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. (2000): 65L05, 65L06, 65L20, 65L60. AMS subject classification

Functionally fitted explicit pseudo two-step Runge–Kutta methods

“…However, we established in [12,10] that the coefficients of an FRK method that uses a separable basis are timeindependent (in the sense that they only depend on the current stepsize).…”

“…This includes the most common functions, namely monomials, exponentials, trigonometric functions and others. With t 0 = 0 for simplicity, we set [10] for further details on the time-independence of the coefficients. We also draw the attention of the reader to the fact that, similarly to [15,Corollary 1], if we expand the coefficients into Taylor series…”

“…[12,10]). Since the coefficients A(h), b(h) converge to those of the conventional EPTRK method when h → 0, the amplification matrix M h (z) converges to the amplification matrix of the conventional EPTRK method.…”

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

“…These methods are developed to integrate an ODE exactly if the solution of the ODE is a linear combination of some certain basis functions (see, e.g., [2], [7], [8], [12], [13], [14], [17]). For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]).…”

“…For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]). Numerical experiments have shown that trigonometrically-fitted methods are superior to classical Runge-Kutta methods for solving ODEs whose solutions are periodic or nearly periodic functions with known frequencies (see, e.g., [8], [11], [12], [13], [14], [16], [17]). …”