Runge–Kutta Methods and Differential-Algebraic Systems

For the solution of differential-algebraic systems, only those Runge–Kutta formulae with nonsingular coefficient matrix A have been considered up to now. To retain the algorithmic simplicity of DIRK formulae while overcoming “order reduction,” A is allowed here to be singular. Algebraic conditions on the formula are given that are necessary and sufficient for it to solve unambiguously constant-coefficient linear differential-algebraic systems of arbitrary index. For a special class, the regular formulae, an expression is derived for the local discretization error; this expression reduces to the one known when the Runge–Kutta matrix A is nonsingular. The paper concludes with some computational illustrations.