Embedded Runge–Kutta scheme for step-size control in the interaction picture method

Abstract

When solving certain evolution type PDEs such as the Schrödinger equation, the Interaction Picture method is a valuable alternative to Split-Step methods. The Interaction Picture method has good computational features when used together with the standard 4th order Runge–Kutta scheme (giving rise to the RK4-IP method). In this paper we present an embedded Runge–Kutta scheme with orders 3 and 4 with the aim to deliver an estimation of the local error for adaptive step-size control purposes in the Interaction Picture method. The corresponding ERK4(3)-IP method preserves the features of the RK4-IP method and provides a local error estimate at no significant extra cost.