Quasi stage order conditions for SDIRK methods

Abstract

The stage order condition is a simplifying assumption that reduces the number of order conditions to be fulfilled when designing a Runge Kutta (RK) method. Because a DIRK (diagonally implicit RK) method cannot have stage order greater than 1, we introduce quasi stage order conditions and derive some of their properties for DIRKs. We use these conditions to derive a low-order DIRK method with embedded error estimator. Numerical tests with stiff ODEs and DAEs of index 1 and 2 indicate that the method is competitive with other RK methods for low accuracy tolerances.