A class of low order DIRK methods for a class of DAEs Original Research Article

We study the numerical solution of a DAE described by an implicit differential equation where the state derivative is multiplied by a singular matrix that depends on the state. We consider a class of s-stage DIRK methods having s−1 implicit stages, an explicit first stage and the stiff accuracy property. The DIRKs we consider have global order of at most 3. We determine how many stages are required to meet different order and stability specifications, both for solitary (fixed step size) DIRKs as well as embedded pairs of DIRKs. We present some solitary DIRKs and some embedded DIRK pairs that have appeared in the literature and that are suitable for solving the DAE in question. In addition, we derive some new solitary DIRKs and DIRK pairs. Our tests with embedded pairs show that some pairs may suffer from performance deterioration when the dynamics in the DAE are of different orders of magnitude.