Strong Stability for Additive Runge–Kutta Methods

Abstract

Space discretization of some time‐dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties. In these situations, additive Runge–Kutta (ARK) methods are used. The aim of this paper is to study monotonicity properties (also known as strong stability) for ARK methods. A new definition of absolute monotonicity for ARK methods is given and some of its properties are investigated. With this concept, monotonicity for ARK schemes under certain stepsize restrictions can be ensured. Some ARK methods from the literature are analyzed. As expected, monotonicity for each Runge–Kutte (RK) method does not ensure monotonicity for the ARK scheme. Some numerical examples show the applicability of these results.