Hermite Methods in Time

Abstract

We study time-stepping methods based on Hermite-Birkhoff interpolation. For linear problems, the methods are designed so that their stability characteristics are identical to those of standard implicit Runge–Kutta methods based on Gauss-Legendre quadrature (Hairer and Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Springer, New York, 1996). However, in contrast, only a single nonlinear system whose dimension matches that of the original problem must be solved independent of the method order; that is, they are singly implicit. Besides outlining the construction of the methods and establishing some of their basic properties, we carry out illustrative computations both on standard test problems and a spectral discretization of an initial-boundary value problem for the Schrödinger equation.