A Family of ESDIRK Integration Methods

In this paper we derive and analyze the properties of explicit singly diagonal implicit Runge-Kutta (ESDIRK) integration methods. We discuss the principles for construction of Runge-Kutta methods with embedded methods of different order for error estimation and continuous extensions for discrete event location. These principles are used to derive a family of ESDIRK integration methods with error estimators and continuous-extensions. The orders of the advancing method (and error estimator) are 1(2), 2(3) and 3(4), respectively. These methods are suitable for obtaining low to medium accuracy solutions of systems of ordinary differential equations as well as index-1 differential algebraic equations. The continuous extensions facilitates solution of hybrid systems with discrete-events. Other ESDIRK methods due to Kv{\ae}rn{\o} are equipped with continuous-extensions as well to make them applicable to hybrid systems with discrete events.