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

Stéphane Balac (1) , Fabrice Mahé (2)

1 FOTON - Fonctions Optiques pour les Technologies de l'informatiON
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Interaction picture method; Embedded Runge-Kutta method; Gross-Pitaevskii equation; Generalised non-linear Schrödinger equation; Split-step method

Domaines: Analyse numérique [math.NA]

Stéphane Balac, Fabrice Mahé. Embedded Runge-Kutta scheme for step-size control in the interaction picture method. Computer Physics Communications, 2013, 184 (4), pp.1211-1219. ⟨10.1016/j.cpc.2012.12.020⟩. ⟨hal-00797190⟩